# Extending Discrete Geometric Singular Perturbation Theory to Non-Hyperbolic
Points

S. Jelbart & C. Kuehn

###### Abstract

We extend the recently developed discrete geometric singular perturbation
theory to the non-normally hyperbolic regime. Our primary tool is the Takens
embedding theorem, which provides a means of approximating the dynamics of
particular maps with the time-1 map of a formal vector field. First, we show
that the so-called reduced map, which governs the slow dynamics near slow
manifolds in the normally hyperbolic regime, can be locally approximated by
the time-one map of the reduced vector field which appears in continuous-time
geometric singular perturbation theory. In the non-normally hyperbolic regime,
we show that the dynamics of fast-slow maps with a unipotent linear part can
be locally approximated by the time-1 map induced by a fast-slow vector field
in the same dimension, which has a nilpotent singularity of the corresponding
type. The latter result is used to describe (i) the local dynamics of two-
dimensional fast-slow maps with non-normally singularities of regular fold,
transcritical and pitchfork type, and (ii) dynamics on a (potentially high
dimensional) local center manifold in $n$-dimensional fast-slow maps with
regular contact or fold submanifolds of the critical manifold. In general, our
results show that the dynamics near a large and important class of
singularities in fast-slow maps can be described via the use of formal
embedding theorems which allow for their approximation by the time-1 map of a
fast-slow vector field featuring a loss of normal hyperbolicity.

Keywords: geometric singular perturbation theory, discrete dynamical systems,
difference equations, singularly perturbed maps, Takens embedding theorem

MSC2020: 37C05, 39A05, 37C10, 34D15, 37G10

##  1 Introduction

Geometric singular perturbation theory (GSPT) is a powerful and established
approach to the mathematical analysis of multiple time-scale systems of
ordinary differential equations (ODEs); see [16, 28, 33, 49] for seminal
works, well-known overviews and books on GSPT. The aim of this article is to
continue the development of a corresponding theory for discrete fast-slow
systems induced by repeated iteration of a map. We follow [26] in referring to
this theory as discrete geometric singular perturbation theory (DGSPT), and we
consider this work to be a natural sequel and complement to the mathematical
formalism developed therein.

DGSPT applies to the study of discrete dynamical systems induced by iteration
of fast-slow maps or difference equations in the general form

$z\mapsto H(z,\varepsilon)=z+h(z)+\varepsilon G(z,\varepsilon),$ (1)

where $z\in\mathbb{R}^{n}$ for an integer $n\geq 2$, the functions
$H(z,\varepsilon)$, $h(z)$ and $\varepsilon G(z,\varepsilon)$ are
$C^{r}$-smooth for a positive integer $r\geq 1$, $\varepsilon$ is a
perturbation parameter satisfying $0<\varepsilon\ll 1$, and the zero set
$\\{z\in\mathbb{R}^{n}:h(z)=0\\}$ is assumed to contain a $C^{r}$-smooth,
$(k<n)$-dimensional critical manifold $S$, which forms a manifold of fixed
points for the limiting map $z\mapsto H(z,0)$. The direct ODE counterpart to
the class of fast-slow maps defined by (1) is the class defined by ODE systems
of the form

$\frac{\textup{d}z}{\textup{d}t}=h(z)+\varepsilon G(z,\varepsilon).$

A recent formulation of GSPT for fast-slow ODE systems in this class can be
found in [49], see also [19, 29, 36, 39, 38]. We emphasise that similarly to
the theory for ODEs, the class of fast-slow maps defined by (1) contains, but
is not limited to, the well-known class of fast-slow maps in standard form

$\begin{split}x&\mapsto x+f(x,y,\varepsilon),\\\ y&\mapsto y+\varepsilon
g(x,y,\varepsilon),\end{split}$ (2)

where $(x,y)\in\mathbb{R}^{n-k}\times\mathbb{R}^{k}$, $0<\varepsilon\ll 1$ and
the functions $f$ and $\varepsilon g$ are $C^{r}$-smooth.

The primary contribution of [26] was to provide the following for general
fast-slow maps (1):

  1. (I)

‘Singular theory’ for the identification and analysis of simplified fast and
slow limiting problems;

  2. (II)

A set of perturbation theorems analogous to those provided by Fenichel theory
[16, 28, 50] in the fast-slow ODE setting, assuming a regularity condition
known as normal hyperbolicity.

The singular theory developed in (I) applies to the identification and
analysis of the layer map, which approximates fast dynamics in regions of
phase space bounded away from the critical manifold $S$, and the reduced map,
which approximates the dynamics near $S$. A notion of normal hyperbolicity was
formulated in terms of the distribution of multipliers associated with the
linearisation of the layer map $z\mapsto H(z,0)$ along $S$. Specifically,
$z\in S$ is normally hyperbolic if the Jacobian matrix $\textrm{D}H(z,0)$ has
$k$ trivial multipliers that are identically equal to $1$ (their corresponding
eigenvectors span the tangent space $\textup{T}_{z}S$), and $n-k$ non-trivial
multipliers $\mu_{j}(z)$ satisfying $\mu_{j}(z)\in\mathbb{C}\setminus S^{1}$,
i.e. $|\mu_{j}(z)|\neq 1$. It was shown that under normally hyperbolic
assumptions, classical invariant manifold theorems dating back to [23] could
be specialised in order to characterise the persistence of compact normally
hyperbolic submanifolds of $S$ as locally invariant slow manifolds
$S_{\varepsilon}$ for all sufficiently small $\varepsilon>0$ (the authors
actually used more recent, and more concrete invariant manifold theory due to
Nipp & Stoffer [44]). The center-stable/unstable manifolds associated to $S$,
as well as their asymptotic rate foliations, were also shown to persist in a
manner which is (for the most part) qualitatively similar to Fenichel theory.

Although we shall also derive new results in the normally hyperbolic regime,
the primary aim of this work is to make steps towards the extension of DGSPT
into the non-normally hyperbolic regime, which is not considered in [26]. More
precisely, we consider local dynamics near non-normally hyperbolic
singularities on $S$. On the linear level, codimension-1 singularities at
$z\in S$ can be classified and divided into three distinct types:

  1. (i)

Fold/contact-type singularities with a single non-trivial multiplier of
$DH(z,0)$ at $1$;

  2. (ii)

Flip/period-doubling type singularities with a single non-trivial multiplier
of $DH(z,0)$ at $-1$;

  3. (iii)

Neimark-Sacker/torus-type singularities with a pair of nontrivial multipliers
of $DH(z,0)$ on $S^{1}$ with non-zero imaginary part.

Due to the coarseness of this ‘classification’, (i) also includes common
singularity types such as transcritical and pitchfork singularities. The same
is true of (ii)-(iii). For analytical purposes, many authors choose to group
singularities in (ii)-(iii) together – they are sometimes referred to
collectively as ‘oscillatory singularities’ – because a number of important
analytical methods and approaches apply equally to the study of dynamics near
both singularity types. They also exhibit dynamical similarities. Numerous
authors have shown that many singularities in (ii)-(iii) are associated with
delayed stability loss; see [3, 4, 17, 18, 41, 42] for important works in this
direction. Similarly to the well-known delayed stability loss phenomena
associated with delayed Hopf bifurcations in fast-slow ODE systems [2, 22, 33,
42, 40], the existence of delay is sensitive to smoothness and noise.

We will focus on the dynamics near fold/contact-type singularities in (i),
which appear to have received less attention in the literature. We believe
that one important reason for this is that the analytical methods used to
study singularities in (ii)-(iii) rely on a certain regularity property which
tends to be violated at points where a non-trivial multiplier of the
linearisation is equal to $1$. Thus, alternative methods are needed. A number
of works have shown that an adaptation of the well-known geometric blow-up
method [10, 25, 30, 32] can be applied to study of discretized fast-slow ODEs
with singularities in (i) [1, 14, 46, 44]. The approach adopted in these works
relies in an important way on scaling properties of the discretization
parameter, but there are significant obstacles to the extension of this
approach to the study of general fast-slow maps, i.e. to the study of maps
that are not obtained via discretization. In order to understand the dynamics
near singularities in (i) in a more general setting, we adopt an alternative
approach which relies on the Takens embedding theorem [6, 7, 20], a formal
embedding theorem which, under certain conditions on the linearisation,
guarantees the existence of an $n$-dimensional vector field with a time-$1$
map which has the same local Taylor expansion as the original map (1). Rather
than tackling the problem of local dynamics directly, we show that in many
cases, the Takens embedding theorem can be used to obtain a local
approximation of the map by a time-1 map induced by the flow of a fast-slow
ODE system in the same dimension. We prove and apply a number of formal
embedding theorems of this kind in both the normally hyperbolic and non-
normally hyperbolic regimes.111It is important to distinguish formal
embeddings, which involve agreement between formal power series associated
with a map and the time$-1$ map of a vector field, from exact embeddings,
which involve an equality (not only in series) between the map and the
time-$1$ map. We refer to Theorem 3.5 and Remark 3.6 for a more precise
distinction.

The most important result we obtain in the normally hyperbolic regime pertains
to the reduced map, which approximates the slow dynamics near normally
hyperbolic submanifolds of $S$ to leading order in $\varepsilon$. We show that
the reduced map can be approximated by the time-1 map induced by the reduced
vector field, which governs the slow dynamics on normally hyperbolic
submanifolds of the critical manifold in fast-slow ODE systems [16, 33, 49].
This is particularly useful in practice due to the fact that the reduced
vector field can be determined explicitly in terms of the functions in (1) and
their derivatives. This shows that in many cases, the slow dynamics in fast-
slow maps can be analysed using the reduced vector field which appears in
fast-slow ODE theory.

The most important result we obtain in the non-normally hyperbolic regime
pertains to the dynamics near singularities in a large and important subclass
of singularities of type (i), known as unipotent singularities, for which the
linearisation $\textrm{D}H(z,0)$ at such a point is unipotent, i.e.
$\textrm{D}H(z,0)=\mathbb{I}_{n}+\Lambda$ for a nilpotent matrix $\Lambda$. We
show that here too, the local dynamics of the map can be approximated by the
time-1 map induced by a fast-slow vector field in the same dimension. The
approximating vector field has a critical manifold which is $C^{r}$-close to
the critical manifold of the map, and (assuming $r$ is large enough) is in
many cases expected to have a (continuous-time) nilpotent singularity of the
corresponding type. We show, for example, that the dynamics of planar fast-
slow maps in standard form (2) with singularities of regular fold,
transcritical and pitchfork type, can be locally approximated by the time-1
map induced by planar fast-slow ODEs in standard form with singularities of
regular fold, transcritical and pitchfork type, respectively. We then combine
these results with (already known) results on the ODE counterparts to these
problems, which can be found in [30, 31], in order to characterise the
extension of the attracting slow manifold through a neighbourhood of the
singularity in the map.

Finally, we show that the dynamics near regular contact type singularities in
$\mathbb{R}^{n}$, which may be viewed as the counterpart to regular fold type
singularities for problems in general non-standard form (1) (they reduce to
regular fold singularities after a local transformation to standard form (2)),
can be understood in a similar way. In general, regular contact points are not
unipotent in $n$-dimensional fast-slow maps with $\textup{codim}(S)=n-k\geq
2$. One can, however, apply a center manifold reduction in order to obtain a
fast-slow map on a $(k+1)$-dimensional center manifold with a regular contact
point on a critical manifold which is codimension-$1$ in $\mathbb{R}^{k+1}$.
This is analogous to the center manifold reduction applied in [48, 49] to
study dynamics near regular folded submanifolds in fast-slow ODEs. In contrast
to the original (higher dimensional) map, the linearised problem within the
center manifold is unipotent, allowing for the application of the embedding-
based methods and results that we derived for unipotent singularities. This
allows us to show that the dynamics on the center manifold can be approximated
by the time-1 map of a $(k+1)$-dimensional fast-slow ODE system with a regular
contact point. This is advantageous, given that the dynamics of the latter
have already been described in [48, 49] (see also [38] for more on contact
points in fast-slow ODEs).

The manuscript is structured as follows. In Section 2 we introduce basic
notions and provide and overview of DGSPT in the normally hyperbolic regime.
We also present two new formal embedding theorems, one of which characterises
the close relationship between the reduced map and the reduced vector field
from continuous-time GSPT. The rest of the paper is devoted to the
approximation of local dynamics near non-normally hyperbolic singularities on
$S$. We begin with the linear classification of codimension-1 singularities in
Section 3. Our main approximation result on the dynamics near unipotent
singularities is stated and proven in Section 3.2. This result is used to
generate approximations and results on the local geometry and dynamics of
$2$-dimensional, standard form fast-slow maps with regular fold, transcritical
and pitchfork type singularities in Section 4. The approximation of dynamics
near regular contact points in $n$-dimensional fast-slow maps in general
(nonstandard) form (1) is treated in Section 5. We conclude in Section 6 with
a summary and outlook. Finally, a number of important results are collected in
the Appendix, including an explicit characterisation the Takens embedding
theorem in the form which is relevant for our purposes in Appendix B.

##  2 DGSPT and formal embeddings for the slow dynamics

We consider $\varepsilon$-families of maps in the form

$z\mapsto\bar{z}=z+h(z)+\varepsilon G(z,\varepsilon),$ (3)

where $z\in\mathbb{R}^{n}$ for $n\geq 2$, $\varepsilon\in[0,\varepsilon_{0}]$
is a small perturbation parameter, and the functions
$h:\mathbb{R}^{n}\to\mathbb{R}^{n}$ and $\varepsilon
G:\mathbb{R}^{n}\times[0,\varepsilon_{0}]\to\mathbb{R}^{n}$ are $C^{r}$-smooth
in $z$ and (for the latter) $\varepsilon$. In the following we require only
that $r\geq 1$, however additional smoothness will be required in later
sections in order to define and study the dynamics near certain singularities.

DGSPT applies to the class of maps (3) defined by the following additional
assumptions, which have direct counterparts in the definition of fast-slow ODE
systems (see e.g. [49]):

######  Assumption 1.

The map (3) is fast-slow in the sense of [26, Def. 2.1]. More precisely, the
set of fixed points $C:=\\{z\in\mathbb{R}^{n}:h(z)=0\\}$ contains a
$(k<n)$-dimensional, regularly embedded submanifold $S\subset\mathbb{R}^{n}$.
We assume for simplicity that $S$ is connected.

######  Assumption 2.

The $\varepsilon$-independent terms can be factorised via

$h(z)=N(z)f(z),$ (4)

where $N(z)$ is an $n\times(n-k)$ matrix with full column rank, and $f(z)$ is
an $(n-k)\times 1$ column vector. Singular points $z_{\ast}\notin S$ which
satisfy $N(z_{\ast})f(z_{\ast})=0$, if they exist, are assumed to be isolated.

Using Assumptions 1 and 2 we may rewrite (3) in the general form

$z\mapsto\bar{z}=H(z,\varepsilon)=z+N(z)f(z)+\varepsilon G(z,\varepsilon),$
(5)

for which the fixed point manifold $S$ described in Assumption 1, which shall
be referred to as the critical manifold, can be written as

$S:=\left\\{z\in\mathbb{R}^{n}:f(z)=0\right\\}.$ (6)

It is worthy to note that locally, Assumption 2 follows from Assumption 1.
This can be shown after a direct application of Hadamard’s lemma; see [36,
Rem. 2] and Appendix A. For this reason, Assumption 2 is not strictly
necessary in the statement of local results. We have decided to retain it here
and throughout, however, since it simplifies the presentation and allows us to
provide an overview of DGSPT which may also apply to the study of global
dynamics.

######  Remark 2.1.

The particular form of the factorisation (4) can often be obtained by
inspection in applications, however we refer to [19, App. 3] for an algebraic
procedure for determining it (locally). Globally, such a factorisation is not
always attainable; see [8, 49] for counterexamples in the continuous-time
setting.

######  Remark 2.2.

Consider the map (5) in the special case that
$z=(x,y)\in\mathbb{R}^{n-k}\times\mathbb{R}^{k}$, with

$N(z)=\begin{pmatrix}\mathbb{I}_{n-k}\\\
\mathbb{O}_{k,n-k}\end{pmatrix},\qquad f(z)=\tilde{f}(x,y,0),\qquad
G(z,\varepsilon)=\begin{pmatrix}\varepsilon^{-1}(\tilde{f}(x,y,\varepsilon)-\tilde{f}(x,y,0))\\\
\tilde{g}(x,y,\varepsilon)\end{pmatrix},$

where $\mathbb{O}_{j,k}$ denotes the $j\times k$ zero matrix for
$(j,k)\in\mathbb{N}_{+}^{2}$ (we shall use this notation throughout). In this
case one obtains a fast-slow map in the well-known standard form

$\begin{split}x\mapsto\bar{x}&=x+\tilde{f}(x,y,\varepsilon),\\\
y\mapsto\bar{y}&=y+\varepsilon\tilde{g}(x,y,\varepsilon).\end{split}$ (7)

Thus the class of maps defined by (5) and Assumptions 1 and 2 includes, but is
not limited to, the well-known class of fast-slow maps in standard form.

###  2.1 DGSPT in the normally hyperbolic regime

Here we provide an overview of the theory established in [26], which applies
to general fast-slow maps (5) under Assumptions 1 and 2.

####  2.1.1 The layer map

The limiting problem associated to the fast dynamics is obtained by taking
$\varepsilon\to 0$ in (5), leading to the so-called layer map

$z\mapsto\bar{z}=z+N(z)f(z).$ (8)

The critical manifold $S$, recall (6), is contained in the fixed point set of
(8). Since $S$ is $(k<n)$-dimensional, the linearisation along $S$, i.e.

$\textrm{D}(z+N(z)f(z))=\mathbb{I}_{n}+N(z)\textrm{D}f(z),\qquad z\in S,$ (9)

has at least $k$ multipliers which are identically equal to $1$. These are
referred to as the trivial multipliers. We denote the remaining $n-k>0$ non-
trivial multipliers by $\mu_{j}(z)$, $j=1,\ldots,n-k$. Linear stability along
$S$ is characterised and classified in terms of the distribution of the non-
trivial multipliers $\mu_{j}(z)$ in the complex plane.

######  Definition 2.3.

We say that a point $z\in S$ is normally hyperbolic if there are no non-
trivial multipliers on the unit circle, i.e. if

$|\mu_{j}(z)|\neq 1,\qquad\forall j=1,\ldots,n-k.$

A subset or submanifold of $S$ is normally hyperbolic if it consists entirely
of normally hyperbolic points.

For calculations it is worthy to note that the $n-k$ non-trivial multipliers
can be identified with the multipliers of the $(n-k)\times(n-k)$ square matrix
$\mathbb{I}_{n-k}+\textrm{D}f(z)N(z)$, by [26, Prop. 2.10]. Thus, the problem
of checking for normal hyperbolicity, or equivalently, identifying a loss of
normal hyperbolicity, reduces to the question of whether or not the spectrum
of $\mathbb{I}_{n-k}+\textrm{D}f(z)N(z)$ contains points on the unit circle.

The local dynamics of the map (5) is also effected by superstability, which is
related to non-invertibility of the map (5), and occurs near points on $S$
with at least one non-trivial multiplier equal to zero.

######  Proposition 2.4.

Consider a point $z\in S$. The map (5) is a diffeomorphism on a neighbourhood
$\mathcal{U}$ about $z$ in $\mathbb{R}^{n}$ for all
$\varepsilon\in[0,\varepsilon_{0})$ if and only if $\varepsilon_{0}>0$ is
sufficiently small and

$\mu_{j}(z)\neq 0,\qquad\forall j=1,\ldots,n-k.$ (10)

###### Proof.

Let $z\in S$. By the inverse function theorem, the extended map

$F(z,\varepsilon):=\begin{pmatrix}z+N(z)f(z)+\varepsilon G(z,\varepsilon)\\\
\varepsilon\end{pmatrix}$

is a local diffeomorphism on a neighbourhood
$\mathcal{U}\times[0,\varepsilon_{0})$ about $(z,0)$ in
$\mathbb{R}^{n}\times\mathbb{R}_{\geq 0}$ if and only if the determinant of
the Jacobian

$\textrm{D}F(z,0)=\mathbb{I}_{n+1}+\begin{pmatrix}N\textrm{D}f(z)&G(z,0)\\\
\mathbb{O}_{1,n}&1\end{pmatrix}$

is non-zero. Now let $\\{\mu_{i}(z)\\}_{i=1}^{n+1}$ denote the multipliers of
$\textrm{D}f(z,0)$, where the first $n-k$ denote the non-trivial multipliers.
The remaining multipliers are given by $\mu_{i}(z)=1$, where
$i=n-k+1,\ldots,n+1$. Using this fact together with (10) yields

$\det\textrm{D}f(z,0)=\prod_{i=1}^{n+1}\mu_{i}(z)=\prod_{i=1}^{n-k}\mu_{i}(z)\neq
0,$

which implies the result. ∎

It is worthy to note that superstable points $z\in S$, for which (5) is non-
invertible, may be normally hyperbolic. The notion of normal hyperbolicity
does not depend on invertibility.

####  2.1.2 Slow manifolds and dynamics for $0<\varepsilon\ll 1$

Similarly to Fenichel theory for fast-slow ODEs in the continuous-time setting
(see e.g. [16, 28, 33, 49, 50]), the theory developed in [26] applies under
normally hyperbolic assumptions. For the purposes of this overview, we
restrict ourselves to the statement pertaining to the persistence of compact
normally hyperbolic submanifolds of $S$ as locally invariant slow manifolds
for $0<\varepsilon\ll 1$. For detailed statements on the persistence and
perturbation of the local center-stable/unstable manifolds
$W_{\textrm{loc}}^{\textrm{s}/\textrm{u}}(S)$, as well as their asymptotic
rate foliations, we refer to [26, Sec. 3]. In the following we write

$S_{\textrm{nh}}:=\left\\{z\in S:|\mu_{j}(z)|\neq 1,\ \forall j=1,\ldots
n-k\right\\},$ (11)

which defines the normally hyperbolic subset of $S$. Generically,
$S_{\textrm{nh}}\subseteq S$ is a union of connected $k$-dimensional
submanifolds of $S$ such that $\overline{S_{\textrm{nh}}}=S$.

######  Theorem 2.5.

[26, Thm. 3.1] Consider the map (5) under Assumptions 1 and 2, and let
$\widetilde{S}\subseteq S_{\textup{nh}}$ be a compact, connected submanifold
of $S_{\textup{nh}}$. Then there is an $\varepsilon_{0}>0$ such that for all
$\varepsilon\in(0,\varepsilon_{0})$, there exists a compact and connected slow
manifold $\widetilde{S}_{\varepsilon}$ which is

  * (i)

$O(\varepsilon)$-close and diffeomorphic to $\widetilde{S}$;

  * (ii)

$C^{r}$-smooth in both $z$ and $\varepsilon$;

  * (iii)

locally invariant under (5), i.e. the restricted map
(5)$|_{\widetilde{S}_{\varepsilon}}$ is invertible and satisfies the
following:

    * (a)

If $z\in\widetilde{S}_{\varepsilon}$ and $\bar{z}^{j}\in\mathcal{U}$ for all
$j=1,\ldots,l$, where $\mathcal{U}$ is a neighbourhood of $\widetilde{S}$,
then $\bar{z}^{j}\in\widetilde{S}_{\varepsilon}$ for all $j=1,\ldots,l$;

    * (b)

If $z\in\widetilde{S}_{\varepsilon}$ and $\bar{z}^{-j}\in\mathcal{U}$ for all
$j=1,\ldots,l$, where $\mathcal{U}$ is a neighbourhood of $\widetilde{S}$,
then $\bar{z}^{-j}\in\widetilde{S}_{\varepsilon}$ for all $j=1,\ldots,l$.

The existence of locally invariant slow manifolds allows for a rigorous fast-
slow decomposition of the dynamics. To leading order, the fast dynamics can be
analysed using properties of the layer map (8) described in Section 2.1.1
above. We turn now to the slow dynamics, which can be analysed after
restriction to the lower dimensional slow manifolds described by Theorem 2.5.

####  2.1.3 The reduced map

The reduced map, which governs the dynamics on slow manifolds to leading order
in $\varepsilon$, can be defined using the following result from [26].

######  Proposition 2.6.

[26, Prop. 2.16] Let $\widetilde{S}_{\varepsilon}$ denote a slow manifold
perturbing from a compact submanifold $\widetilde{S}\subseteq
S_{\textup{nh}}$, as described by Theorem 2.5. Then for all
$\varepsilon\in[0,\varepsilon_{0})$ with $\varepsilon_{0}>0$ sufficiently
small we have

$z\mapsto
z+\varepsilon\Pi_{\mathcal{N}}^{S_{\textup{nh}}}G(z,0)+O(\varepsilon^{2}),\qquad
z\in\widetilde{S},$ (12)

where $\Pi_{\mathcal{N}}^{S_{\textup{nh}}}$ denotes the oblique projection
along the linear fast fibre bundle

$\mathcal{N}:=\bigcup_{z\in
S_{\textup{nh}}}\textup{span}\\{N^{i}(z)\\}_{j=1,\ldots,n-k}$

onto the tangent bundle $\textup{T}S_{\textup{nh}}$, where $N^{i}(z)$ denotes
the $i$’th column of the matrix $N(z)$. It has the matrix representation

$\Pi_{\mathcal{N}}^{S_{\textup{nh}}}=\mathbb{I}_{n}-N(\textup{D}fN)^{-1}\textup{D}f\big{|}_{S_{\textup{nh}}}.$
(13)

Proposition 2.6 provides the explicit asymptotic form of the map (5) after
restriction to a locally invariant slow manifold
$\widetilde{S}_{\varepsilon}$, up to $O(\varepsilon^{2})$. This provides a
means for defining the reduced map as the leading order approximation for the
slow dynamics, i.e. by truncating (12) at order $O(\varepsilon)$:

$z\mapsto z+\varepsilon\Pi_{\mathcal{N}}^{S_{\textrm{nh}}}G(z,0),\qquad
z\in\widetilde{S}.$ (14)

Interestingly, the reduced map (14) coincides with the Euler discretization of
the ODE problem

$z^{\prime}=\frac{dz}{dt}=\varepsilon\Pi_{\mathcal{N}}^{S_{\textrm{nh}}}G(z,0),\qquad
z\in\widetilde{S},$ (15)

with step-size $h=\varepsilon$ or, equivalently,

$\dot{z}=\frac{dz}{d\tau}=\Pi_{\mathcal{N}}^{S_{\textrm{nh}}}G(z,0),\qquad
z\in\widetilde{S},$ (16)

with step-size $h=1$ and slow time-scale $\tau=\varepsilon t$. The ODE (16)
defines the so-called reduced problem in continuous-time GSPT, see e.g. [49,
def. 3.8], which may be viewed as the direct counterpart to the reduced map in
DGSPT. Because of this, a number of important dynamical questions about the
reduced map (14) can be reformulated as questions about the reduced vector
field (16). We illustrate the point with the following simple result.

######  Proposition 2.7.

The reduced map (14) has a fixed point at $z=z_{0}$ if and only if the reduced
vector field (16) has an equilibrium at $z=z_{0}$. The multipliers of the
linearised reduced map satisfy the hyperbolicity condition $||\mu_{i}|-1|>c>0$
for an arbitrarily small but fixed constant $c$ if and only if the $z_{0}$ is
a hyperbolic equilibrium of the reduced vector field (16).

###### Proof.

Since

$z=z_{0}\mapsto
z_{0}+\varepsilon\Pi^{S_{\textrm{nh}}}_{\mathcal{N}}G(z_{0},0)=0\quad\iff\quad\Pi^{S_{\textrm{nh}}}_{\mathcal{N}}G(z_{0},0)=0,$

the reduced map (14) has a fixed point at $z=z_{0}$ if and only if the reduced
vector field has an equilibrium at $z=z_{0}$. The multipliers of the reduced
map at $z=z_{0}$ are given by $\mu_{j}=1+\varepsilon\lambda_{j}$, where
$j=1,\ldots,n$ and $\lambda_{j}$ are the eigenvalues of the matrix
$\textrm{D}(\Pi_{\mathcal{N}}^{S_{\textrm{nh}}}G(z,0))(z_{0})$. The
hyperbolicity statement follows from the fact that

$|\mu_{i}|^{2}=1+2\varepsilon\textup{Re}(\lambda_{i})+O(\varepsilon^{2}),$

as $\varepsilon\to 0$. ∎

In the following section we shall derive an even stronger relationship between
the map (12) – and therefore also the reduced map (14) – and the reduced
vector field (16).

######  Remark 2.8.

The maps (12) and (14) are diffeomorphisms even if the original map (5) is not
invertible, i.e. (12) and (14) are also invertible at superstable points on
$S$. This feature follows from the fact that
(5)$|_{\widetilde{S}_{\varepsilon}}$ is invertible due to Theorem 2.5
Assertion (iii). It can also be seen as a direct consequence of the inverse
function theorem, since the Jacobian matrices associated with the restricted
maps in (12) and (14) are $O(\varepsilon)$-close to $\mathbb{I}_{n}$. To
illustrate the point, consider the simple example
$(x,y)\mapsto(x^{2},y+\varepsilon)^{\textnormal{T}}$, which can be written in
general form (5) with $N(x,y)=(1,0)^{\textnormal{T}}$, $f(x,y)=-x+x^{2}$ and
$G(x,y,\varepsilon)=(0,1)^{\textnormal{T}}$. The critical manifold $S$ has a
branch along $x=0$ with a single non-trivial multiplier $\mu\equiv 0$. Thus
$S$ is superstable and the map is non-invertible along $x=0$. However, the
corresponding reduced map is given by
$(0,y)^{\textnormal{T}}=(0,y+\varepsilon)^{\textnormal{T}}$, which is clearly
invertible.

######  Remark 2.9.

The domain of definition of the reduced map (14) can be smoothly extended to
the set $S_{\textup{nh}}\cup S_{\textup{ns}}$, where

$S_{\textup{ns}}:=\left\\{z\in S:\mu_{j}(z)\in S^{1}\setminus\\{1\\},\ \forall
j=1,\ldots n-k\right\\},$ (17)

where the subscript “ns” stands for “Neimark-Sacker”. Like $S_{\textup{nh}}$,
the set $S_{\textup{nh}}\cup S_{\textup{ns}}$ is generically a union of
$k$-dimensional submanifolds of $S$ satisfying $\overline{S_{\textup{nh}}\cup
S_{\textup{ns}}}=S$. Such an extension is possible because the projection
operator $\Pi_{\mathcal{N}}^{S}$ remains well-defined at non-normally
hyperbolic points with $\mu_{j}(z)\neq 1$. Note, however, that
$\Pi_{\mathcal{N}}^{S}$ is not well-defined on $S\setminus(S_{\textup{nh}}\cup
S_{\textup{ns}})$, since the matrix
$\textup{D}fN|_{S\setminus(S_{\textrm{nh}}\cup S_{\textup{ns}})}$ is not
invertible.

###  2.2 Formal embeddings in the normally hyperbolic regime

Our aim in this section is to derive formal embedding theorems which can be
used to approximate the dynamics of the map (5) by the time-1 map induced by a
flow in the same dimension. We state two different results. The first of these
applies to the dynamics in a neighbourhood about in a given point $z\in
S_{\textrm{nh}}$. The second can be used in order to relate the dynamics of
the reduced map (14) to the dynamics of the (continuous-time) reduced vector
field defined in (16).

We shall assume throughout this section that $0\in S_{\textrm{nh}}$ (this can
be achieved after a simple coordinate translation), and consider the expansion
about $(z,\varepsilon)=(0,0)\in\mathbb{R}^{n+1}$ obtained from (5) after
appending the trivial map $\varepsilon\to\varepsilon$. We shall also assume
that (5) is $C^{r}$-smooth with $r\geq 2$. Similarly to the proof of
Proposition 2.4, we write

$F(z,\varepsilon):=\begin{pmatrix}H(z,\varepsilon)\\\
\varepsilon\end{pmatrix}=A\begin{pmatrix}z\\\
\varepsilon\end{pmatrix}+F^{(2)}(z,\varepsilon)+\cdots+F^{(r)}(z,\varepsilon)+o(|(z,\varepsilon)|^{r}),$
(18)

where each $F^{(j)}(z,\varepsilon)$ is a homogeneous polynomial of degree
$j\in\\{2,\ldots,r\\}$, and

$A:=\textrm{D}f(0,0)=\begin{pmatrix}\mathbb{I}_{n}+N\textrm{D}f(0)&G(0,0)\\\
\mathbb{O}_{1,n}&1\end{pmatrix}.$ (19)

Since every linear transformation on $\mathbb{R}^{n+1}$ has a Jordan-Chevalley
decomposition, the matrix $A$ can be written as

$A=B\left(\mathbb{I}_{n+1}+M\right),$ (20)

where $B$ is semi-simple, $M$ is nilpotent, and $BM=MB$. In particular, there
exists a (generally complex) invertible $(n+1)\times(n+1)$ matrix
$\mathcal{S}$ such that

$A=\mathcal{S}^{-1}\mathcal{D}\mathcal{S}^{-1}+\mathcal{S}^{-1}\left(J(A)-\mathcal{D}\right)\mathcal{S},$
(21)

where $\mathcal{D}:=\textup{diag}(\mu_{1},\ldots,\mu_{n-k},1,\ldots,1)$ is the
matrix of (generally complex) multipliers of $A$, which has $k+1$ trivial
multipliers equal to $1$. $J(A)$ denotes the Jordan decomposition of $A$,
which is block diagonal with Jordan blocks associated to the non-trivial
multipliers in the upper left part, and $\mathbb{I}_{k+1}$ in the bottom right
block. As usual, the matrix $\mathcal{S}$ can be constructed in applications
using the eigenvectors associated to $A$. Comparing (20) with (21), we have

$B=\mathcal{S}^{-1}\mathcal{D}\mathcal{S},\qquad
M=\mathcal{S}^{-1}\left(\mathcal{D}^{-1}J(A)-\mathbb{I}_{n+1}\right)\mathcal{S}.$
(22)

The semi-simple matrix $B$ will play an important role in the approximation of
the local dynamics of the map (5) by a flow. In order to state this result
precisely, however, we need two more important notions and an additional
assumption.

######  Definition 2.10.

Consider a $C^{r}$-smooth function $\phi:\mathbb{R}^{n}\to\mathbb{R}^{m}$ with
the following finite order Taylor expansion at $z=z_{0}$:

$\phi(z)=\sum_{j=0}^{r}\phi^{(j)}(z-z_{0})+o(|z-z_{0}|^{r}),$

where $\phi^{(0)}(z-z_{0})=\phi(z_{0})$ and the coefficient functions
$\phi^{(j)}:\mathbb{R}^{n}\to\mathbb{R}^{m}$ are vectors of degree $j$
homogeneous polynomials defined via Taylor expansion. For each $l=1,\ldots,r$
we define the $l$-jet of $\phi(z)$ and the truncation operator $j^{l}$ at
$z=z_{0}$ by

$j^{l}\phi(z)=\sum_{j=0}^{l}\phi^{(j)}(z-z_{0}).$

######  Definition 2.11.

Consider a $C^{r}$-smooth map $\phi:\Omega\to\mathbb{R}^{n}$ such that
$\phi(0)=0$, where $\Omega$ is a neighbourhood of $0$ in $\mathbb{R}^{n}$. The
map $\phi(z)$ is said to be in local Poincaré normal form up to order $l$ if
the corresponding $l$-jet $j^{l}\phi(z)$ can be written as a linear
combination of “resonant monomials” of the form

$z_{1}^{m_{1}}\cdots z_{n}^{m_{n}}e_{j},$

where $z=(z_{1},\ldots,z_{n})\in\mathbb{R}^{n}$, $e_{j}$ is a standard basis
vector in $\mathbb{R}^{n}$, $j\in\\{1,\ldots,n\\}$, and the $m_{i}$ are
positive integers which satisfy $\sum_{i=1}^{n}m_{i}\geq 2$ and the resonance
condition

$\lambda_{1}^{m_{1}}\cdots\lambda_{n}^{m_{n}}=\lambda_{j}.$

Finally, we assume the following local invertibility property.

######  Assumption 3.

The origin $0\in S$ is not superstable, i.e. the non-trivial multipliers
$\mu_{j}(0)$ satisfy the local invertibility condition

$\mu_{j}(0)\neq 0,\qquad\forall j=1,\ldots,n-k.$

If Assumption 3 is satisfied, then Proposition 2.4 ensures that (5) is a
diffeomorphism on a suitable neighbourhood about $\mathcal{U}\ni 0$ in
$\mathbb{R}^{n}$.

We now state and prove two formal embedding theorems, which can be used in
order to approximate the dynamics of fast-slow maps close to a normally
hyperbolic point on $S$. The proofs rely on the Takens embedding theorem; see
Appendix B for a formulation based on [7, Thm. 8.1] which is suitable for our
purposes.

######  Theorem 2.12.

Assume that $0\in S_{\textup{nh}}$ and consider the extended map (18), where
Assumptions 1, 2 and 3 apply to $H(z,\varepsilon)$. There exists and
$\varepsilon_{0}>0$ such that for each $l\in\\{1,\ldots,r\\}$ there exists a
neighbourhood $\mathcal{U}_{l}\times[0,\varepsilon_{0})$ about
$\tilde{z}=(z,\varepsilon)^{\textnormal{T}}=(0,0)^{\textnormal{T}}$ in
$\mathbb{R}^{n}\times\mathbb{R}_{\geq 0}$, a diffeomorphism
$\psi_{l}:\mathcal{U}_{l}\times[0,\varepsilon_{0})\to\mathbb{R}^{n+1}$, and a
vector field
$X:\mathcal{U}_{l}\times[0,\varepsilon_{0})\to\textup{T}\mathbb{R}^{n+1}\cong\mathbb{R}^{n+1}$
such that

  1. 1.

$j^{l}(\psi_{l}\circ F\circ\psi_{l}^{-1})$ is a Poincaré normal form of $F$ up
to order $l$;

  2. 2.

$X(B\tilde{z})=BX(\tilde{z})$ for any
$\tilde{z}=(z,\varepsilon)\in\mathcal{U}_{l}\times[0,\varepsilon_{0})$, where
$B$ is the semi-simple matrix defined in (22);

  3. 3.

$j^{l}(\psi_{l}\circ
F\circ\psi_{l}^{-1})(\tilde{z})=j^{l}(\Phi_{X}^{1}(B\tilde{z}))$, where
$\Phi_{X}^{t}(\tilde{z})$ denotes the flow of $X(\tilde{z})$ by time $t$.

  4. 4.

The $l$-jet of the vector field $j^{l}X(\tilde{z})$ is uniquely determined by
the $l$-jet of the map $j^{l}F(\tilde{z})$.

If the matrix $\textup{D}fN(0)$ has purely real eigenvalues, the preceding
assertions hold for $B=A$, where $A$ is the Jacobian matrix defined in (19).

###### Proof.

Assertions 1-4 follow after a direct application of Takens’ embedding theorem;
we refer again to Appendix B.

If all $n-k$ eigenvalues $\lambda_{j}(0)$ of the matrix $\textrm{D}fN(0)$ are
real, then the $n-k$ non-trivial multipliers $\mu_{j}(0)=1+\lambda_{j}(0)$
corresponding to eigenvalues of the matrix $\mathbb{I}_{n-k}+\textrm{D}fN(0)$
are also real. Since the remaining $k+1$ eigenvalues of
$\mathbb{I}_{n}+N\textrm{D}f(0)$ are identically $1$, it follows that the
matrix $A$ defined in (19) has purely real eigenvalues, implying that the
semisimple part of $A$ is $A$, i.e. that $B=A$. ∎

Theorem 2.12 shows that in a neighbourhood of a normally hyperbolic, non-
superstable point on $S$, the original map (5) is $C^{r}$-conjugate to a map
which can be formally approximated by the time-1 flow of an
$\varepsilon$-dependent, $C^{r}$-smooth formal vector field in the same
dimension. The matrix $B$ can be determined algorithmically, and the
diffeomorphisms $\psi_{l}$ can be determined in a recursive and systematic
fashion using normal form theory, see e.g. [7, 20]. In practice, however, this
is often a non-trivial and computationally expensive task, which is expected
to lead to difficulties when it comes to question of whether Theorem 2.12 can
be used as an approximation tool in applications.

From a applied point of view, it may be preferable to consider approximations
obtained via a formal embedding of the restricted slow map (12), which governs
the dynamics on $k$-dimensional slow manifolds $S_{\varepsilon}$. In order to
state our results, we denote the right-hand side of the map (12) by
$\mathcal{G}(z,\varepsilon)$, i.e. we write

$\mathcal{G}(z,\varepsilon)=z+\varepsilon\Pi_{\mathcal{N}}^{S_{\textrm{nh}}}G(z,0)+O(\varepsilon^{2}),\qquad
z\in\widetilde{S},$ (23)

where $\widetilde{S}$ is a compact submanifold of $S_{\textrm{nh}}$, and let
$\Phi^{t}_{\Pi_{\mathcal{N}}^{S_{\textrm{nh}}}G(z,0)}(z,\varepsilon)$ denote
the flow induced by the reduced problem (16). Our second approximation theorem
shows that $\mathcal{G}(z,\varepsilon)$ can be locally approximated by the
time-$1$ map
$\Phi^{1}_{\Pi_{\mathcal{N}}^{S_{\textrm{nh}}}G(z,0)}(z,\varepsilon)$.

######  Theorem 2.13.

Consider the map (23) and let $z\in S_{\textup{nh}}$. There is an
$\varepsilon_{0}>0$ and $\varepsilon$-independent neighbourhood $\Omega\ni z$
in $S_{\textup{nh}}$ such that for all
$(z,\varepsilon)\in\Omega\times[0,\varepsilon_{0})$ we have

$j^{1}\mathcal{G}(z,\varepsilon)=j^{1}\Phi^{1}_{\Pi_{\mathcal{N}}^{S_{\textup{nh}}}G(z,0)}(z,\varepsilon),\qquad
j^{l}\mathcal{G}(z,\varepsilon)=j^{l}\Phi^{1}_{\Pi_{\mathcal{N}}^{S_{\textup{nh}}}G(z,0)}(z,\varepsilon)+O(\varepsilon^{2}),$
(24)

where the latter holds for all $l=2,\ldots,r$.

The proof of Theorem 2.13 is deferred to Appendix C for brevity. The existence
of a unique formal embedding, i.e. a unique formal vector field
$X(z,\varepsilon)$ whose time-$1$ map has a Taylor expansion coinciding with
the Taylor expansion of the slow map $\mathcal{G}(z,\varepsilon)$, follows
from the Takens embedding theorem. However, additional arguments are needed in
order to gain information about the particular form of $X(z,\varepsilon)$, and
in particular, to show that the unique formal vector field $X(z,\varepsilon)$
is locally approximated by the reduced vector field, i.e. that

$j^{l}X(z,0)=j^{l}\Pi_{\mathcal{N}}^{S_{\textrm{nh}}}G(z,0)$

for each $l=2,\ldots,r$. This is shown by a recursive argument based on the
formal matching of Taylor coefficients of $H(z,\varepsilon)$ and
$\Phi^{1}_{\Pi_{\mathcal{N}}^{S_{\textrm{nh}}}G(z,0)}(z,\varepsilon)$, where
the Taylor coefficients of the latter are obtained via Picard iteration. This
is similar to the procedure for approximating a map by a flow applied in e.g.
[34, 35].

The practical advantage of Theorem 2.13 over Theorem 2.12 is that preliminary
transformations are not necessary (there is no need to identify the
transformations $\psi_{l}$ appearing in Theorem 2.12), and that the
approximating vector field is known; it is precisely the reduced vector field
(16). It is also worthy to note that Theorem 2.13 still applies if the
original map (5) is orientation-reversing, or even non-invertible (Assumption
3 is not necessary). This is a simple consequence of the fact that the
dynamics on the slow manifold $\widetilde{S}_{\varepsilon}$ itself is
orientation-preserving and invertible, even if (5) is not invertible on a
neighbourhood in about $z\in S_{\textrm{nh}}$ in $\mathbb{R}^{n}$ (we refer
back to Remark 2.8).

######  Remark 2.14.

The $O(\varepsilon^{2})$ term in the second equation in (24) is calculated
explicitly in the proof of Theorem 2.13, and given by equation (80). In fact,
the proof is formal and constructive, allowing for the calculation of
corrections at $O(\varepsilon^{l})$ for all integers $l\geq 2$.

######  Remark 2.15.

There are certain situations in which an exact embedding can be obtained, as
opposed to the finite order approximations in Theorems 2.12 and 2.13. For
example, in [9] the author shows that planar fast-slow maps

$\begin{pmatrix}x\\\ y\end{pmatrix}\mapsto\begin{pmatrix}x\\\
y\end{pmatrix}+\begin{pmatrix}N^{x}(x,y)\\\
N^{y}(x,y)\end{pmatrix}f(x,y)+\varepsilon\begin{pmatrix}G^{x}(x,y,\varepsilon)\\\
G^{y}(x,y,\varepsilon)\end{pmatrix},$

which are in the general form (5), can be embedded into a $2$-dimensional flow
in a neighbourhood $\mathcal{U}\subseteq\mathbb{R}^{2}$ about any point
$(x,y)\in S$ for which the conditions

$\mu(x,y)\in(0,1),\qquad N^{x}(x,y)G^{y}(x,y,0)-N^{y}(x,y)G^{x}(x,y,0)\neq 0,$
(25)

are satisfied. The first condition in (25) implies that the map is locally
orientation-preserving and that $S$ is normally hyperbolic and attracting near
$(x,y)$. The second guarantees that there are no fixed points in the reduced
map. The question of whether or not exact embeddings of this kind are possible
under similar assumptions in higher dimensions is not considered in this work.

##  3 Formal embeddings in the non-normally hyperbolic regime

We now turn to the study of dynamics in the non-normally hyperbolic regime. We
begin in Section 3.1 with basic definitions and a linear classification of
local singularities associated to the loss of normal hyperbolicity along $S$.

###  3.1 Non-normally hyperbolic singularities

As in continuous-time GSPT, local non-normally hyperbolic singularities can
often be defined and classified on the ‘singular level’, in our case, in terms
of local conditions on the layer and reduced maps (8) and (14) respectively.
For our present purposes if suffices to consider conditions on the layer map,
which can be derived using classical bifurcation theory, see e.g. [34], since
non-normally hyperbolic singularities correspond to bifurcations in the layer
map if the problem is viewed in suitable (standard form) coordinates.

In the classification of codimension-$1$ non-normally hyperbolic
singularities, there are three important cases; see Figure 1:

  * •

Fold/Contact-type singularities in the submanifold

$\mathcal{C}:=\\{z\in S:\mu_{1}(z)=1,|\mu_{l}(z)|\neq 1,l=2,\ldots,n-k\\},$

corresponding to a single real non-trivial multiplier $\mu_{1}(z)$ equal to
$1$;

  * •

Flip/Period-doubling-type singularities in the submanifold

$\mathcal{O}_{\textup{f}}:=\\{z\in S:\mu_{1}(z)=-1,|\mu_{l}(z)|\neq
1,l=2,\ldots,n-k\\},$

corresponding to a single real non-trivial multiplier $\mu_{1}(z)$ equal to
$-1$;

  * •

Neimark-Sacker/Torus-type singularities in the submanifold

$\mathcal{O}_{\textup{ns}}:=\\{z\in S:\mu_{1}(z)=\overline{\mu_{2}(z)}\in
S^{1}\setminus\\{\pm 1\\},|\mu_{l}(z)|\neq 1,l=3,\ldots,n-k\\},$

corresponding to a pair of complex conjugate non-trivial multipliers
$\mu_{1,2}(z)$ lying on $S^{1}\setminus\\{\pm 1\\}$.

We refer to points in $\mathcal{C}$ as fold or contact-type singularities,
since these are in certain sense the ‘least degenerate’ points in
$\mathcal{C}$. This will be made more precise below. However, we emphasise
that our terminology includes singularities which would not typically be
referred to as fold or contact singularities in $\mathcal{C}$. This includes
pitchfork and transcritical singularities as important examples. Similar
comments apply to distinctions between different singularity types in
$\mathcal{O}_{\textup{f}}$ and $\mathcal{O}_{\textup{ns}}$.

(a) Fold/contact. (b) Flip/period-doubling. (c) Neimark-Sacker/torus.

Figure 1: A linear classification of codimension-$1$ singularities of the map
(5) can be given in terms of the distribution of multipliers of the matrix
$\textrm{D}H(z,0)=\mathbb{I}_{n}+N\textrm{D}f(z)$ in the complex plane. Three
different cases are distinguished according the point at which a real non-
trivial multiplier $\mu_{1}(z)$ or pair of complex conjugate non-trivial
multipliers $\mu_{1,2}(z)$ (shown in green) intersect(s) $S^{1}$. (a):
Fold/contact type point in $\mathcal{C}$ with $\mu_{1}(z)=1$. (b):
Flip/period-doubling type point in $\mathcal{O}_{\textup{f}}$ with
$\mu_{1}(z)=-1$. (c): Neimark-Sacker/torus type point in $\mathcal{C}$ with
complex conjugate multipliers $\mu_{1,2}(z)\in S^{1}\setminus\\{\pm 1\\}$. In
each case, we also sketch the $k$ trivial multipliers equal to $1$ in shaded
green, as well as examples of stable and unstable multipliers which do not lie
on $S^{1}$ (shaded blue and red respectively).

For analytical purposes, it is often useful to group the oscillatory
singularities in $\mathcal{O}_{\textup{f}}$ and $\mathcal{O}_{\textup{ns}}$
together. In fact, points in
$\mathcal{O}_{\textup{f}}\cup\mathcal{O}_{\textup{ns}}$ are sometimes referred
to as oscillatory singularities [4]. Multiple authors have shown that
oscillatory singularities are associated with $O(1)$ delayed stability loss if
suitable non-degeneracy conditions are satisfied and the original map is
analytic (or more precisely, Gevrey-1) [3, 4, 17, 18, 41, 42]. Similarly to
delayed Hopf bifurcation in the fast-slow ODE setting, the delay effect is
lost if the map is only $C^{r}$ (including $r=\infty$), or if noise is
introduced; see e.g. [2, 40, 42]. In either case, existing analyses of these
problems depend crucially on the local invertibility of the matrix
$\textrm{D}fN$. Generically, however, the matrix $\textrm{D}fN$ is not
invertible at points in $\mathcal{C}$, which necessitates the need for a
different approach. The remainder of this work is devoted to the map (5) near
contact-type singularities in $\mathcal{C}$.

######  Remark 3.1.

In the present context, the ‘codimension’ of a singularity is equal to the
codimension of the submanifold of $S$ along which the relevant defining
condition is satisfied. The submanifolds $\mathcal{C}$,
$\mathcal{O}_{\textup{f}}$ and $\mathcal{O}_{\textup{ns}}$ are generically
codimension-$1$ in $S$, i.e. they are $(k-1)$-dimensional.

######  Remark 3.2.

See [1, 12, 14, 15, 44, 45, 46] for detailed geometric analyses of planar
fast-slow maps with singularities in class $\mathcal{C}$. These works
demonstrate the applicability of the geometric blow-up method in order to
study (in these cases Euler) discretizations of fast-slow ODE systems with
non-normally hyperbolic singularities. This approach relies on non-trivial
scaling properties of the step-size parameter associated to the
discretization, and does not extend directly for the study of general maps,
i.e. it does not apply to the study of maps which cannot be obtained via
discretization.

###  3.2 Formal embeddings for unipotent singularities

In this section we derive formal embedding theorems which can be used to
approximate the dynamics of (5) near unipotent points, which form an important
subset of singularities in $\mathcal{C}$. Such problems are characterised by
the fact that the linearisation of the layer map (8) along $S$, as given by
(9), is ‘close’ to the identity map. More precisely, we consider general fast-
slow maps (5) under Assumptions 1, 2 and 3, locally near a point $z\in S$ for
which the linearisation (9) is unipotent, i.e. for which the matrix
$N\textrm{D}f(z)$ is nilpotent with $(N\textrm{D}f(z))^{l}=\mathbb{O}_{n,n}$
for some integer $l\geq 2$.

######  Remark 3.3.

Generically, the Jacobian matrix (9) appearing in the linearisation of the
layer map associated to a fast-slow map (5) will not be unipotent at a point
in $\mathcal{C}$ if $\textup{codim}(S)=n-k\geq 2$. This follows from the
presence of non-zero eigenvalues of the matrix $\textup{D}fN$, which implies
that $N\textup{D}f$ has non-zero eigenvalues, and therefore that
$N\textup{D}f$ is not nilpotent (nilpotent matrices have only zero
eigenvalues). Nevertheless, as we shall see in Section 5.1 below, the study of
dynamics near singularities in $\mathcal{C}$ can often be reduced to the study
of singularities in fast-slow maps with a codimension-$1$ critical manifold
after center manifold reduction. In this case, every singularity in
$\mathcal{C}$ has a corresponding linearisation with a unipotent Jacobian
matrix (9).

Our first result reduces computational difficulty by establishing an
equivalence between nilpotency of the $n\times n$ square matrix
$N\textrm{D}f|S$ and the smaller $(n-k)\times(n-k)$ square matrix
$\textrm{D}fN|_{S}$. It also shows that nilpotency of $N(z)\textrm{D}f(z)$
implies that $z\in\mathcal{C}$, i.e. that nilpotency of $N(z)\textrm{D}f(z)$
is correlated with the existence of at least one non-trivial multiplier with
$\mu_{j}(z)=1$ at a non-normally hyperbolic contact point.

######  Lemma 3.4.

Let $z\in S$. The $n\times n$ matrix $\textup{D}(Nf)(z)=N\textup{D}f(z)$ is
nilpotent of index $l+1$ if and only if the $(n-k)\times(n-k)$ matrix
$\textup{D}fN(z)$ is nilpotent of index $l$, where $l\in\mathbb{N}_{+}$. In
particular, $\mu_{j}(z)=1$ for at least one $j\in\\{1,\ldots,n-k\\}$.

###### Proof.

It follows from the proof of [26, Prop. 2.8] that local coordinates can be
chosen near $S$ such that

$N\textrm{D}f|_{S}=\begin{pmatrix}\mathbb{O}_{k,k}&\tilde{N}^{x}\\\
\mathbb{O}_{n-k,k}&\textrm{D}fN\end{pmatrix}\bigg{|}_{S},$

where $\tilde{N}^{x}(z)$ is a $k\times(n-k)$ matrix with full column rank by
Assumption 2. Direct matrix multiplication yields

$(N\textrm{D}f)^{l}|_{S}=\begin{pmatrix}\mathbb{O}_{k,k}&\tilde{N}^{x}(\textrm{D}fN)^{l-1}\\\
\mathbb{O}_{n-k,k}&(\textrm{D}fN)^{l}\end{pmatrix}\bigg{|}_{S},$

for all $l\in\mathbb{N}_{+}$. It follows that
$(\textrm{D}fN)^{l}|_{S}=\mathbb{O}_{n-k,n-k}\implies(N\textrm{D}f)^{l+1}|_{S}=\mathbb{O}_{n,n}$.
The converse
$(N\textrm{D}f)^{l+1}|_{S}=\mathbb{O}_{n,n}\implies(\textrm{D}fN)^{l}|_{S}=\mathbb{O}_{n-k,n-k}$
follows from $\tilde{N}^{x}(\textrm{D}fN)^{l}|_{S}=\mathbb{O}_{k,n-k}$, since
$\tilde{N}^{x}(z)$ has full column rank.

In order to see that $\mu_{j}(z)=1$ for some $j\in\\{1,\ldots,n-k\\}$ we
recall that the non-trivial multipliers have the form
$\mu_{j}(z)=1+\lambda_{j}(z)$, where $\lambda_{j}(z)$ are eigenvalues of
$\textrm{D}fN(z)$. Since $\textrm{D}fN(z)$ is nilpotent we have
$\lambda_{j}(z)=0$ for at least one $j\in\\{1,\ldots,n-k\\}$, thereby implying
the result. ∎

We are now in a position to state our main result, namely, a formal embedding
theorem which can be used to approximate the dynamics of the map (5) in a
neighbourhood of a unipotent singularity by the time-1 map of a vector field
with known properties. As before, our primary tool for proving the existence
of a formal embedding near such points $z\in\mathcal{C}\subset S$ is the
Takens embedding theorem; we refer again to Appendix B for details.

######  Theorem 3.5.

Consider the map (5) denoted by $z\mapsto H(z,\varepsilon)$ under Assumptions
1 and 2. Assume additionally that $0\in\mathcal{C}\subseteq S$ and that the
$(n-k)\times(n-k)$ square matrix $\textup{D}fN(0)$ is nilpotent with index
$l\in\mathbb{N}_{+}$. Then there exists a neighbourhood $\mathcal{U}\ni 0$ and
an $\varepsilon_{0}>0$ such that for all
$(z,\varepsilon)\in\mathcal{U}\times[0,\varepsilon_{0})$ and
$m\in\\{1,\ldots,r\\}$ we have

$j^{m}H(z,\varepsilon)=j^{m}\Phi^{1}_{V}(z,\varepsilon),$ (26)

where $\Phi^{1}_{V}(z,\varepsilon)$ denotes the time-$1$ map induced by the
flow of an $\varepsilon$-family of vector fields $V(z,\varepsilon)$ on
$\mathbb{R}^{n}$ satisfying

$j^{r}V(z,\varepsilon)=\widehat{N}(z)j^{r}f(z)+\varepsilon\widehat{G}(z,\varepsilon),$
(27)

where $\widehat{N}(z)$ is a $C^{r}$-smooth $n\times(n-k)$ matrix with full
column rank, $\widehat{G}(z,\varepsilon)$ is a $C^{r}$-smooth column vector,
and $j^{r}f$ is the $r$-jet associated to the Taylor expansion of $f(z)$ about
$z=0$. We also have that

$\widehat{N}(0)=N(0),\qquad\widehat{G}(0,0)=G(0,0).$ (28)

The vector field (27) is fast-slow with a $k$-dimensional critical manifold

$j^{r}S:=\left\\{z\in\mathbb{R}^{n}:j^{r}f(z)=0\right\\}$ (29)

which is $C^{r}$-close to $S$ on $\mathcal{U}$, i.e.

$\textup{dist}(S,j^{r}S):=\sup_{z\in\mathcal{U}}|f(z)-j^{r}f(z)|=o(|z|^{r}).$
(30)

###### Proof.

We want to apply the version of Takens’ embedding theorem provided in
Corollary B.2. In order to do so, we first append the map (5) with the trivial
map $\varepsilon\to\varepsilon$ and consider the Taylor expansion about
$(z,\varepsilon)=(0,0)$, i.e.

$\begin{pmatrix}z\\\
\varepsilon\end{pmatrix}\mapsto\begin{pmatrix}H(z,\varepsilon)\\\
\varepsilon\end{pmatrix}=(\mathbb{I}_{n+1}+\Lambda)\begin{pmatrix}z\\\
\varepsilon\end{pmatrix}+H^{(2)}(z,\varepsilon)+H^{(3)}(z,\varepsilon)+\ldots+H^{(r)}(z,\varepsilon)+o(|(z,\varepsilon)|^{r}),$
(31)

where

$\Lambda:=\begin{pmatrix}N\textrm{D}f(0)&G(0,0)\\\
\mathbb{O}_{1,n}&0\end{pmatrix}$

and the functions $H^{(l)}(z,\varepsilon)$ are homogeneous polynomials of
degree $l\geq 2$. Repeated matrix multiplication leads to the expression

$\Lambda^{l+2}=\begin{pmatrix}(N\textrm{D}f)^{l+2}(0)&(N\textrm{D}f)^{l+1}(0)G(0,0)\\\
\mathbb{O}_{1,n}&0\end{pmatrix}=\mathbb{O}_{n+1,n+1},$

where we used the fact that $(\textrm{D}fN)(0)$ is nilpotent with index $l$ so
that by Lemma 3.4, $(N\textrm{D}f)^{l+1}(0)=\mathbb{O}_{n,n}$. Thus $\Lambda$
is also nilpotent, with index $l+2$ if
$(N\textrm{D}f)^{l}(0)G(0,0)\neq\mathbb{O}_{n,1}$ and index $l+1$ if
$(N\textrm{D}f)^{l}(0)G(0,0)\neq\mathbb{O}_{n,1}$. Moreover, the linearisation
$\mathbb{I}_{n+1}+\Lambda$ has $n+1$ eigenvalues equal to $1$. Therefore, the
inverse function theorem applies, and guarantees that the map (31) is a local
diffeomorphism.

The preceding arguments show that Corollary B.2 applies. Applying it yields
the approximation property (26) for a uniquely determined family of formal
vector fields $V(z,\varepsilon)$ satisfying

$j^{r}V(z,\varepsilon)=(N\textrm{D}f)(0)z+\varepsilon
G(0,0)+V^{(2)}(z,\varepsilon)+V^{(3)}(z,\varepsilon)+\ldots
V^{(r)}(z,\varepsilon),$ (32)

where the $V^{(l)}(z,\varepsilon)$ are smooth, homogeneous polynomials of
degree $l=2,\ldots,r$.

We now show that the truncated formal vector field (32) is fast-slow with
general form (27). Notice that (i) $V(0,0)=0$, and (ii)

$j^{r}H(z,0)=j^{r}\Phi^{1}_{V}(z,0)=z,$

locally for all $z\in j^{r}S$. These two facts imply that $j^{r}V(z,0)=0$ for
all $z\in j^{r}S$, i.e. that the zero sets (critical manifolds) of the
truncated map $j^{r}H(z,0)$ and the truncated vector field $j^{r}V(z,0)$
coincide in $\mathcal{U}$; both are given by (29). Therefore, Hadamard’s lemma
(see again Appendix A) implies that

$j^{r}V(z,0)=\widehat{N}(z)j^{r}f(z),$

for an $n\times(n-k)$ matrix $\widehat{N}(z)$ with full column rank, as
required. This shows that $j^{r}V(z,\varepsilon)$ is in the form (27), and the
linear part of (32) implies the two equalities in (28). Equation (30) follows
directly from the definition of $r$-jets and the truncation operator $j^{r}$
in Definition 2.10. ∎

Theorem 3.5 provides a means for approximating the dynamics of the map (5)
near a unipotent singularity in $\mathcal{C}$, via the time-1 map induced by
the flow of an $\varepsilon$-dependent family of fast-slow formal vector
fields $V(z,\varepsilon)$ in the same dimension. The linear part of
$V(z,\varepsilon)$ is determined by (27) and (28), and higher order terms in
the series expansion can be determined in a formal but systematic fashion; we
refer to Appendices C and D where this procedure is carried out in detail for
particular proofs. Similarly to the problem of explicit determination of the
transformations in Theorem 2.12, this can be a computationally demanding task.
Fortunately, an explicit form for the approximating vector field
$V(z,\varepsilon)$ is not needed in many cases, since we obtain a sufficient
amount of information about the qualitative dynamics without an explicit
representation. This will be demonstrated with numerous applications in
Sections 4 and 5 below. At present, it suffices to emphasise and reiterate the
following points:

  1. (i)

The truncated vector field $j^{r}V(z,\varepsilon)$ is fast-slow in the general
form (27);

  2. (ii)

The critical manifold $j^{r}S$ of the map $j^{r}H(z,\varepsilon)$ coincides
with the critical manifold of the vector field $j^{r}V(z,\varepsilon)$, and
$j^{r}S$ is $C^{r}$-close to $S$, the critical manifold of the original map
$H(z,\varepsilon)$;

  3. (iii)

The linear part of $j^{r}V(z,\varepsilon)$ is known explicitly, due to (28).

Properties (i)-(iii) imply that in many cases, the local defining conditions
on the layer map at unipotent, non-normally hyperbolic singularities in
$\mathcal{C}\subset S$ are satisfied if and only if a set of corresponding
conditions are satisfied in the approximating vector field (27). In
particular, if the map (5) has a unipotent, non-normally hyperbolic
singularity, then the vector field (27) has a nilpotent, non-normally
hyperbolic singularity. Moreover, due to the $C^{r}$-closeness of the critical
manifolds $j^{r}S$ and $S$ and the close relationship between the reduced map
and the reduced vector field described in Theorem 2.13, the satisfaction of
defining conditions for local unipotent singularities in $\mathcal{C}\subset
S$ which are formulated in terms of $N$, $f$, $G$ and their partial
derivatives with respect to $z$, are expected to imply the satisfaction of
analogous conditions for vector field (27). Thus, unipotent singularities of
the map (5) will typically imply nilpotent singularities of the corresponding
type in the vector field (27). This can be a powerful tool in practice, since
it allows for the indirect study of the map (5) using methods and techniques
which are only developed or applicable in the context of fast-slow ODEs.

######  Remark 3.6.

Theorem 3.5 also applies in the $C^{\infty}$ case, i.e. for $r=\infty$. In
this case, equation (26) implies that the Taylor expansions of
$H(z,\varepsilon)$ and $\Phi_{V}^{1}(z,\varepsilon)$ about
$(z,\varepsilon)=(0,0)$ coincide. It is important to stress, however, that
this does not imply an exact embedding

$H(z,\varepsilon)=\Phi_{V}^{1}(z,\varepsilon).$ (33)

Rather, we have that

$H(z,\varepsilon)=\Phi_{V}^{1}(z,\varepsilon)+R(z,\varepsilon),$ (34)

for an error term $R(z,\varepsilon)$ which is flat (beyond all orders) in
$(z,\varepsilon)$ (if $r<\infty$ then
$R(z,\varepsilon)=o(|(z,\varepsilon)|^{r})$). An important example for which
the error term cannot be removed arises in the case of saddle-node bifurcation
in $1$-dimensional maps [24]; see also Remark 4.8 below. Terminologically, we
distinguish “exact embeddings” (33) from “formal embeddings” (34), and we say
that the latter allow for an approximation of the map by a flow. Similar
terminology can be found in e.g. [6, 34, 35].

##  4 Fold, transcritical and pitchfork points in 2 dimensions

In this section we consider dynamics near three different types of unipotent
singularities in two-dimensional fast-slow maps, i.e. in the lowest
dimensional setting. This will help to demonstrate the use of Theorem 3.5 in
practice. For simplicity, we consider maps in fast-slow standard form

$\begin{pmatrix}x\\\ y\end{pmatrix}\mapsto
H(x,y,\varepsilon)=\begin{pmatrix}x\\\
y\end{pmatrix}+\begin{pmatrix}\tilde{f}(x,y,\varepsilon)\\\
\varepsilon\tilde{g}(x,y,\varepsilon)\end{pmatrix},$ (35)

in line with the earlier notation in (7). We shall assume that $H$ is
$C^{r}$-smooth with $r\geq 3$, and that $H(0,0,0)=(0,0)^{\textnormal{T}}$,
i.e. $(0,0)\in S$. Since $\textup{codim}(S)=1$, every point in $\mathcal{C}$
is a unipotent singularity; recall Remark 3.3. Thus, Theorem 3.5 can be
applied near any point in $\mathcal{C}$. We return to the problem of
approximation in higher dimensions, in which case $\mathcal{C}$ contains
singularities that are not unipotent, in Section 5.

###  4.1 Definitions and singular geometry

We consider non-normally hyperbolic singularities of regular fold,
transcritical and pitchfork type. We begin with definitions and a description
of the limiting dynamics and geometry for $\varepsilon=0$ in each case,
starting with the fold.

####  4.1.1 Regular fold points

Regular fold points can be defined as follows.

######  Definition 4.1.

The fast-slow map (35) has a regular fold point at $(x,y)=(0,0)\in S$ if

$\tilde{f}(0,0,0)=0,\qquad\frac{\partial\tilde{f}}{\partial x}(0,0,0)=0,$ (36)

and

$\frac{\partial^{2}\tilde{f}}{\partial x^{2}}(0,0,0)\neq
0,\qquad\frac{\partial\tilde{f}}{\partial y}(0,0,0)\neq
0,\qquad\tilde{g}(0,0,0)\neq 0.$ (37)

Definition 4.1 is analogous to the definition of a regular fold point in
planar fast-slow ODEs; see [30, 33]. The local geometry implied by the
defining conditions (36) and (37) is sketched in Figure 2(a). The critical
manifold $S$ is locally quadratic at the fold point $(0,0)$, which separates
two normally hyperbolic branches. We sketch the case with

$\frac{\partial^{2}\tilde{f}}{\partial
x^{2}}(0,0,0)>0,\qquad\frac{\partial\tilde{f}}{\partial
y}(0,0,0)<0,\qquad\tilde{g}(0,0,0)<0.$ (38)

This choice fixes the orientation and stability of the branches, as well as
the orientation of the reduced flow defined by the reduced vector field on
$S\setminus\\{(0,0)\\}$. There is a normally hyperbolic and attracting
(repelling) branch $S^{a}$ ($S^{r}$) in $\\{x<0\\}$ ($\\{x>0\\}$), and the
reduced flow is oriented towards the fold point; see Figure 2(a).

######  Remark 4.2.

Due to the close relationship between the slow map (12) and the reduced vector
field (16) described in Theorem 2.13, we permit ourselves to speak of a
‘reduced flow’ in the context of fast-slow maps. The advantage is that the
reduced vector field (16) retains important information about the dynamics on
$S$ when $\varepsilon=0$, despite the fact that the slow map (12) is trivial
for $\varepsilon=0$ (similarly to the reduced vector field on the fast time-
scale, recall (15)). We shall also use the reduced vector field to provide a
simpler representation of the slow dynamics as $\varepsilon\to 0$ in figures,
for example in Figure 2.

(a) Regular fold. (b) Transcritical.

(c) Pitchfork.

Figure 2: Singular geometry and dynamics in a neighbourhood $\mathcal{U}$
(shaded grey) about a non-hyperbolic point of (a) regular fold type, (b)
transcritical type, and (c) pitchfork type. Shaded blue (purple) curves
indicate attracting (repelling) branches of the critical manifold $S$, and the
regular fold, transcritical and pitchfork points are indicated in red and
denoted by $F$, $T$ and $P$ in Figures (a), (b) and (c) respectively. The
orientation of the reduced vector field on $S$ is indicated with single
arrows, and sample iterates of the layer map along fast fibers are sketched in
green, with black arrows to indicate the direction of forward iteration.

####  4.1.2 Transcritical points

We now define transcritical points.

######  Definition 4.3.

The fast-slow map (35) has a transcritical singularity at $(x,y)=(0,0)\in S$
if

$\tilde{f}(0,0,0)=0,\qquad\frac{\partial\tilde{f}}{\partial
x}(0,0,0)=0,\qquad\frac{\partial\tilde{f}}{\partial y}(0,0,0)=0,$ (39)

and

$\frac{\partial^{2}\tilde{f}}{\partial x^{2}}(0,0,0)\neq
0,\qquad\det\begin{pmatrix}\frac{\partial^{2}\tilde{f}}{\partial
x^{2}}(0,0,0)&\frac{\partial^{2}\tilde{f}}{\partial x\partial y}(0,0,0)\\\
\frac{\partial^{2}\tilde{f}}{\partial y\partial
x}(0,0,0)&\frac{\partial^{2}\tilde{f}}{\partial
y^{2}}(0,0,0)\end{pmatrix}<0,\qquad\tilde{g}(0,0,0)\neq 0.$ (40)

Definition 4.3 is directly analogous to the definition of a transcritical
singularity in planar fast-slow ODEs; see e.g. [31]. The local geometry
implied by the defining conditions (39) and (40) is sketched in Figure 2(b).
The critical manifold $S$ is given by the union of two curves which intersect
transversally at $(0,0)$ (the transcritical point). There are four normally
hyperbolic branches which emanate from (but do not include) $(0,0)$. We sketch
the case with

$\frac{\partial^{2}\tilde{f}}{\partial
x^{2}}(0,0,0)>0,\qquad\tilde{g}(0,0,0)>0.$ (41)

The first condition implies that the two branches on the left (right) are
attracting (repelling). We denote the upper and lower attracting branches,
shown in blue in Figure 2(b), by $S^{a,+}$ and $S^{a,-}$ respectively. Upper
and lower repelling branches are shown in purple and denoted by $S^{r,+}$ and
$S^{r,-}$ respectively.

####  4.1.3 Pitchfork points

Finally, we define pitchfork points.

######  Definition 4.4.

The fast-slow map (35) has a pitchfork singularity at $(x,y)=(0,0)\in S$ if

$\tilde{f}(0,0,0)=0,\qquad\frac{\partial\tilde{f}}{\partial
x}(0,0,0)=0,\qquad\frac{\partial\tilde{f}}{\partial
y}(0,0,0)=0,\qquad\frac{\partial^{2}\tilde{f}}{\partial x^{2}}(0,0,0)=0,$ (42)

and

$\frac{\partial^{3}\tilde{f}}{\partial x^{3}}(0,0,0)\neq
0,\qquad\frac{\partial^{2}\tilde{f}}{\partial x\partial y}(0,0,0)\neq
0,\qquad\tilde{g}(0,0,0)\neq 0.$ (43)

Definition 4.4 is analogous to the definition of a pitchfork singularity in
planar fast-slow ODEs; we refer again to [31]. The local geometry implied by
the defining conditions (42) and (43) is sketched in Figure 2(c). The critical
manifold $S$ is given by the union of two curves which intersect transversally
at $(0,0)$ (the pitchfork point). There are four normally hyperbolic branches
which emanate from (but do not include) $(0,0)$. We sketch the case with

$\frac{\partial^{2}\tilde{f}}{\partial x\partial
y}(0,0,0)>0,\qquad\frac{\partial^{3}\tilde{f}}{\partial x^{3}}(0,0,0)<0.$ (44)

This choice fixes the orientation and stability of the branches, and the
latter in particular restricts us to the supercritical case. The orientation
of the slow dynamics can be fixed by choosing $\tilde{g}(0,0,0)>0$ or
$\tilde{g}(0,0,0)<0$. We do not specify a choice at this point because we
intend to state results for both cases. There is an attracting lower branch
$S^{a}$ in $\\{y<0\\}$, a repelling upper branch $S^{r}$ and two attracting
outer branches $S^{a,\pm}$ in $\\{y>0\\}$; see Figure 2(c).

###  4.2 Formal embedding near fold, transcritical and pitchfork points

In the following we state and prove a formal embedding theorem which allows
for the local approximation of the map (35) near a regular fold, transcritical
or pitchfork point via the time-1 map induced by a planar system of fast-slow
ODEs with a singularity of the corresponding type.

######  Proposition 4.5.

Assume that the map (35) has a unipotent singularity at $(x,y)=(0,0)\in S$.
Then there exists a neighbourhood $\mathcal{U}\ni(0,0)$ and an
$\varepsilon_{0}>0$ such that for all
$(x,y,\varepsilon)\in\mathcal{U}\times[0,\varepsilon_{0})$ and
$l\in\\{1,\ldots,r\\}$ we have

$j^{l}H(x,y,\varepsilon)=j^{l}\Phi^{1}_{V}(x,y,\varepsilon),$ (45)

where $\Phi^{1}_{V}(x,y,\varepsilon)$ denotes the time-$1$ flow of an
$\varepsilon$-family vector fields $V(x,y,\varepsilon)$ on $\mathbb{R}^{2}$.
The truncated vector field $j^{r}V(x,y,\varepsilon)$ is fast-slow in the
standard form

$j^{r}V(x,y,\varepsilon)=\begin{pmatrix}\mathcal{V}_{1}(x,y,\varepsilon)\\\
\varepsilon\mathcal{V}_{2}(x,y,\varepsilon)\end{pmatrix}=\begin{pmatrix}K(x,y)j^{r}\tilde{f}(x,y,0)+O(\varepsilon)\\\
\varepsilon(\tilde{g}(0,0,0)+O(x,y,\varepsilon))\end{pmatrix},$ (46)

where the function $K(x,y)$ satisfies $K(0,0)=1$. The vector field (46) has a
critical manifold

$j^{r}S=\left\\{(x,y)\in\mathcal{U}:j^{r}\tilde{f}(x,y,0)=0\right\\},$ (47)

for which $(0,0)\in j^{r}S$ is a nilpotent singularity. In particular, $(0,0)$
is a

  1. 1.

Regular fold point if $(0,0)$ is a regular fold point of the map (35);

  2. 2.

Transcritical point if $(0,0)$ is a transcritical point of the map (35);

  3. 3.

Pitchfork point if $(0,0)$ is a pitchfork point of the map (35).

###### Proof.

We want to apply Theorem 3.5. In order to do so, we need to check Assumptions
1, 2 and 3, and show that $(0,0)$ is a unipotent singularity of (35).

Following Remark 2.2, we write the (35) in the general form (5) by defining

$N(x,y)=\begin{pmatrix}1\\\ 0\end{pmatrix},\qquad
f(x,y)=\tilde{f}(x,y,0),\qquad
G(x,y,\varepsilon)=\begin{pmatrix}\varepsilon^{-1}\left(\tilde{f}(x,y,\varepsilon)-\tilde{f}(x,y,0)\right)\\\
\tilde{g}(0,0,0)\end{pmatrix}.$

The point $(0,0)$ is a unipotent singularity if $\textrm{D}fN(0,0)$ is
nilpotent, which in this case ($n-k=1$) is equivalent to
$(0,0)\in\mathcal{C}$. Using (36), (39) or (42), we have

$\textrm{D}fN(0,0)=\frac{\partial\tilde{f}}{\partial x}(0,0,0)=0,$

implying nilpotency. Assumptions 2 and 3 can be verified directly (the latter
follows from the inverse function theorem), as can all of the conditions in
Assumption 1 except for the requirement that $S$ can be regularly embedded as
a submanifold in $\mathbb{R}^{2}$, which is not true if
$\textrm{D}f(0,0)=(0,0)$. Such situations arise at transcritical and pitchfork
singularities due to a self-intersection at $(0,0)$ (for example).
Fortunately, Theorem 3.5 applies regardless, since the proof does not rely on
the property $\textrm{D}f(0,0)\neq(0,0)$. After applying Theorem 3.5, we
obtain the existence of an $\varepsilon$-dependent, $2$-dimensional vector
field $V(x,y,\varepsilon)$ whose time-1 map $\Phi^{1}_{V}(z,\varepsilon)$
satisfies the approximation property in (45).

It follows from (45) that the truncated formal vector field
$j^{r}V(x,y,\varepsilon)$ takes the standard form

$j^{r}V(x,y,\varepsilon)=\begin{pmatrix}\mathcal{V}_{1}(x,y,\varepsilon)\\\
\varepsilon\mathcal{V}_{2}(x,y,\varepsilon)\end{pmatrix},$

where the right-hand side is a formal power series about
$(x,y,\varepsilon)=(0,0,0)$, truncated at order $r$. It follows by Theorem 3.5
that the critical manifold is given by (47). In order to derive the right-most
expression in (46), note that Hadamard’s lemma (Appendix A) implies that

$\mathcal{V}_{1}(x,y,0)=K(x,y)\tilde{f}(x,y,0)$

for a smooth function $K:\mathcal{U}\to\mathbb{R}$, assuming that we choose
the neighbourhood $\mathcal{U}\ni(0,0)$ in $\mathbb{R}^{2}$ sufficiently
small. The right-hand side in (46) follows from this, together with the fact
that equation (28) implies

$K(0,0)=1,\qquad\mathcal{V}_{2}(0,0,0)=\tilde{g}(0,0,0).$

The fact that $(0,0)$ is a nilpotent singularity on $j^{r}S$ follows from

$\frac{\partial\mathcal{V}_{1}}{\partial
x}(0,0,0)=K(0,0)\frac{\partial\tilde{f}}{\partial x}(0,0,0)+\frac{\partial
K}{\partial x}(0,0)\tilde{f}(0,0,0)=0.$

The right-most expression in (46) can be used to directly verify that a
regular fold, transcritical or pitchfork point at $(0,0)$ implies a regular
fold, transcritical or pitchfork point of $j^{r}V(x,y,\varepsilon)$ at $(0,0)$
respectively; one simply checks the conditions in Definitions 4.1, 4.3 and 4.4
for each case using $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ in place of
$\tilde{f}$ and $\tilde{g}$, respectively. ∎

Thus, the local dynamics near unipotent, non-normally hyperbolic singularities
of the map (35) on $S$ can be formally approximated by the time-1 map of a
planar fast-slow vector field. Moreover, if the singularity of the map is of
regular fold, transcritical or pitchfork type, then the approximating vector
field also has a regular fold, transcritical or pitchfork singularity
respectively. Since the local dynamics near regular fold, transcritical and
pitchfork singularities in planar fast-slow ODE systems are already well
understood [30, 31], Proposition 4.5 can be used to approximate the dynamics
of the map.

######  Remark 4.6.

Every non-normally hyperbolic point in a planar fast-slow ODE system is
nilpotent, due to the fact that there are no oscillatory singularities in
planar fast-slow systems. Oscillatory singularities can occur in planar fast-
slow maps, however, due to the possibility of flip/period-doubling
singularities in $\mathcal{O}_{\textup{f}}$. Combining this observation with
Proposition 4.5, we find that every nilpotent singularity of a planar fast-
slow ODE system has a corresponding unipotent singularity in a planar fast-
slow map, but the converse is not true, i.e. not every non-normally hyperbolic
singularity in a planar fast-slow map has a corresponding niplotent
singularity in a planar fast-slow ODE system.

######  Remark 4.7.

Results for planar fast-slow maps with regular fold, transcritical and
pitchfork singularities obtained after Euler discretizations of a planar fast-
slow system with a singularities of the corresponding types have been derived
using a variant of the geometric blow-up method in [44, 46], [14] and [1]
respectively. As already noted in Remark 3.2, the step-size parameter
associated to the discretization plays an important role in these analyses,
and it is not presently known if similar methods can be extended to the study
of general fast-slow maps without a step-size parameter, such as those
considered in this work.

######  Remark 4.8.

Consider (45) with a regular fold point at $(0,0)$. Proposition 4.5 applies to
this problem with $r=\infty$ if the original map $H(x,y,\varepsilon)$ is
$C^{\infty}$-smooth, but existing results for classical (parameter-dependent)
fold bifurcations in 1-dimensional maps due to Ilyashenko & Yakovenko [24]
imply that there is no exact embedding for the layer map
$(x,y)^{\textnormal{T}}\mapsto H(x,y,0)$ which holds over an entire
neighbourhood $\mathcal{U}\ni(0,0)$. A discrepancy arises because the
extension of the exact embeddings about the two normally hyperbolic branches
on either side of $(0,0)$ can be shown to disagree by an exponentially small
amount in a particular sector of the $(x,y)$-plane containing $(0,0)$.

###  4.3 Dynamics near regular fold, transcritical and pitchfork points

In the following we combine well-known results from the theory of planar fast-
slow ODE systems with Proposition 4.5 in order to describe the extension of
slow manifolds through a neighbourhood of regular fold, transcritical and
pitchfork points of the map (35). The relevant results in the fast-slow ODE
setting have been derived using the geometric blow-up method in [30, 31].

####  4.3.1 Regular fold dynamics

Theorem 2.5 implies that compact submanifolds of $S^{a}$ perturb to
$O(\varepsilon)$-close slow manifolds, which we denote by
$S^{a}_{\varepsilon}$ (we refer again to [26] for details). We are interested
in the extension of $S_{\varepsilon}^{a}$ through a sufficiently small but
fixed and $\varepsilon$-independent neighbourhood
$\mathcal{U}\subset\mathbb{R}^{2}$ about the fold point. We assume without
loss of generality that $\mathcal{U}=[-\rho,\rho]^{2}$ for a fixed constant
$\rho>0$, as in Figure 2(a).

######  Corollary 4.9.

Consider the $C^{r}$-smooth fast-slow map (35) with a regular fold point
$(0,0)\in S$ satisfying (38). For $\rho>0$ sufficiently small there exists an
$\varepsilon_{0}>0$ such that for all $\varepsilon\in(0,\varepsilon_{0})$ the
extension of the attracting slow manifold $S_{\varepsilon}^{a}$ leaves
$\mathcal{U}$ at a point $(x,y)=(\rho,Y_{out}(\rho,\varepsilon))$, where

$Y_{out}(\rho,\varepsilon)=-c\varepsilon^{2/3}+o\left(\varepsilon\ln\varepsilon,\rho^{r}\right)$

for a constant $c>0$.

Figure 3: Local geometry and dynamics near a regular fold point $F$ (shown in
red) of the map (35), as described by Corollary 4.9. The extension of the
attracting slow manifold of the map, denoted $S_{\varepsilon}^{a}$ and shown
in green, is $C^{r}$-close to the extension of the attracting slow manifold
$j^{r}S_{\varepsilon}^{a}$ of the approximating vector field
$j^{r}V(x,y,\varepsilon)$, shown in blue, which is obtained via Proposition
4.5. We also sketch the singular limit (layer and reduced) dynamics for the
approximating vector field, which has a critical manifold
$j^{r}S=j^{r}S^{a}\cup F\cup j^{r}S^{r}$, where $j^{r}S^{a}$ ($j^{r}S^{r}$) is
normally hyperbolic and attracting (repelling), shown here in shaded blue
(purple). The intersection of the right-most boundary of $\mathcal{U}$ with
the forward extensions of $j^{r}S^{a}_{\varepsilon}$ and $S^{a}_{\varepsilon}$
are indicated by dark grey disks. $S_{\varepsilon}^{a}$ leaves $\mathcal{U}$
at a point which is $C^{r}$-close to $(\rho,O(\varepsilon^{2/3}))$.

###### Proof.

Assuming that $\rho>0$ is fixed sufficiently small, Proposition 4.5 implies
the existence of an approximating vector field $j^{r}V(x,y,\varepsilon)$ on
$\mathcal{U}$ which is given by (46). Moreover, $j^{r}V(x,y,\varepsilon)$ is
fast-slow with a regular fold point at $(x,y)=(0,0)$. The local dynamics for
this problem is described by [30, Thm. 2.1], after applying a positive
rescaling of $x$, $y$, $\varepsilon$ and $t$ in order to obtain a local normal
form. Note that the rescaling in time does not effect the location of the slow
manifold $j^{r}S_{\varepsilon}^{a}$ of this ODE system, or its extension
through $\mathcal{U}$. The above mentioned result from [30] implies that the
extension of $j^{r}S^{a}_{\varepsilon}$ leaves $\mathcal{U}$ at a point
$(\rho,-c\varepsilon^{2/3}+O(\varepsilon\ln\varepsilon))$, where $c>0$. After
undoing the normal form transformation (a simple rescaling) we obtain the
analogous result for the truncated map $j^{r}H(x,y,\varepsilon)$, which has
the same attracting slow manifold $j^{r}S^{a}_{\varepsilon}$ (and extension
thereof) due to local invariance and the approximation property (45).

The preceding arguments show that the forward extension of the slow manifold
of the truncated map $j^{r}H(x,y,\varepsilon)$ leaves $\mathcal{U}$ at a point
$(x,y)=(\rho,-c\varepsilon^{2/3}+O(\varepsilon\ln\varepsilon))$. The extension
of the slow manifold $S_{\varepsilon}^{a}$ of the original map (35) leaves
$\mathcal{U}$ at a point which is $C^{r}$-close to this point, due to the fact
that

$|H(x,y,\varepsilon)-j^{r}H(x,y,\varepsilon)|=o(|(x,y,\varepsilon)|^{r})$ (48)

for all $(x,y,\varepsilon)\in\mathcal{U}\times[0,\varepsilon_{0})$, which
implies that
$\textup{dist}(S^{a}_{\varepsilon},j^{r}S^{a}_{\varepsilon})=o(|(x,y,\varepsilon)|^{r})$.
∎

The situation is sketched in Figure 3. As is shown by the proof, the geometry
is very similar to the geometry of the continuous-time counterpart considered
in [30]. This is a direct consequence of the formal embedding result in
Proposition 4.5.

######  Remark 4.10.

Despite the similarities exhibited by discrete and continuous-time systems
near a regular fold point, as described in Corollary 4.9, additional
complications arise in the discrete setting when it comes to tracking the
location of individual iterates of the map (35). The problem stems from the
fact that for most initial conditions close to or on $S_{\varepsilon}^{a}$,
the ‘last iterate’ in $\mathcal{U}$ appears in an $\varepsilon$-dependent
neighbourhood of $(0,0)$. The location of the next iterate depends on higher
order terms, and therefore on global properties of the map which are not
captured in a local expansion or normal form. Such an issue is expected to
arise in the analysis near singularities which feature a ‘fast escape’ or
‘jump’; see also [27] for an example which arises after considering the
Poincaré map of a continuous-time fast-slow problem with a global singularity.

####  4.3.2 Transcritical dynamics

Assume that the map (35) has a transcritical singularity at $(x,y)=(0,0)\in
S$. By Theorem 2.5, compact submanifolds of $S^{a,\pm}$ perturb to
$O(\varepsilon)$-close slow manifolds, which we denote by
$S^{a,\pm}_{\varepsilon}$. The following result describes the extension of
$S_{\varepsilon}^{a,-}$ through a neighbourhood
$\mathcal{U}\subset\mathbb{R}^{2}$ about the transcritical point.

######  Corollary 4.11.

Consider the two-dimensional fast-slow map (35) with a transcritical point
$(0,0)\in S$ satisfying (41), and let

$\lambda:=\frac{1}{|g_{0}|\sqrt{\beta^{2}-\gamma\alpha}}\left(\delta\alpha+g_{0}\beta\right),$

where

$\alpha:=\frac{1}{2}\frac{\partial^{2}\tilde{f}}{\partial
x^{2}}(0,0,0),\quad\beta:=\frac{1}{2}\frac{\partial^{2}\tilde{f}}{\partial
x\partial
y}(0,0,0),\quad\gamma:=\frac{1}{2}\frac{\partial^{2}\tilde{f}}{\partial
y^{2}}(0,0,0),\quad\delta:=\frac{\partial\tilde{f}}{\partial\varepsilon}(0,0,0),\quad
g_{0}:=\tilde{g}(0,0,0).$

There exists an $\varepsilon_{0}>0$ such that for all
$\varepsilon\in(0,\varepsilon_{0})$ we have the following:

  1. 1.

If $\lambda<1$ then the extension of $S_{\varepsilon}^{a,-}$ is $C^{r}$-close
to $S_{\varepsilon}^{a,+}$ when it leaves $\mathcal{U}$.

  2. 2.

If $\lambda>1$ then the extension of $S_{\varepsilon}^{a,+}$ is $C^{r}$-close
to a point which is $O(\varepsilon^{1/2})$-close to the critical fiber along
$y=0$ when it leaves $\mathcal{U}$.

(a) $\lambda<1$ stability exchange. (b) $\lambda>1$ fast escape.

Figure 4: Local geometry and dynamics near a transcritical point $T$ (shown in
red) of the map (35), as described by Corollary 4.11. We sketch (a) the
exchange of stability case for $\lambda<1$, and (b) the fast escape case for
$\lambda>1$. In both cases, the extension of the attracting slow manifold
$S^{a,-}_{\varepsilon}$ of the map, shown here in green, is $C^{r}$-close to
the extension of the attracting slow manifold $j^{r}S_{\varepsilon}^{a,-}$ of
the approximating vector field $j^{r}V(x,y,\varepsilon)$, shown in blue, which
is obtained via Proposition 4.5. We also sketch the singular limit (layer and
reduced) dynamics for the approximating vector field, which has a critical
manifold $j^{r}S=j^{r}S^{a,-}\cup j^{r}S^{a,+}\cup t\cup j^{r}S^{r,-}\cup
j^{r}S^{r,+}$, where $j^{r}S^{a,\pm}$ ($j^{r}S^{r,\pm}$) are normally
hyperbolic and attracting (repelling), shown here in shaded blue (purple).

###### Proof.

By Proposition 4.5, the truncated map $j^{r}H(x,y,\varepsilon)$ coincides
locally with the time-$1$ map induced by the fast-slow vector field
$j^{r}V(x,y,\varepsilon)$ in (46), which has a transcritical point at $(0,0)$.
The local dynamics are described after an orientation-preserving linear
coordinate transformation and positive rescaling of $\varepsilon$ by [31, Thm.
2.1]. This result implies the following, in the original coordinates (prior to
the linear transformation and rescaling of $\varepsilon$):

  1. 1’

If $\lambda<1$ then the forward extension of $j^{r}S_{\varepsilon}^{a,-}$ is
$O(\textrm{e}^{-c/\varepsilon})$-close to $j^{r}S_{\varepsilon}^{a,+}$ when it
leaves $\mathcal{U}$, for a constant $c>0$.

  2. 2’

If $\lambda>1$ then the forward extension of $j^{r}S_{\varepsilon}^{a,+}$ is
$O(\varepsilon^{1/2})$-close to the critical fiber along $y=0$ when it leaves
$\mathcal{U}$.

The approximation property (45) implies that the slow manifolds
$j^{r}S_{\varepsilon}^{a,\pm}$ and their extensions in $\mathcal{U}$ coincide
with those of the truncated map $j^{r}H(x,y,\varepsilon)$. Thus, both 1’ and
2’ are also true for the map $j^{r}H(x,y,\varepsilon)$. Assertions 1 and 2 in
the Corollary follow for the map $H(x,y,\varepsilon)$ after accounting for the
fact that an $o(|(x,y,\varepsilon)|^{r})$ error is introduced when
approximating $H(x,y,\varepsilon)$ by $j^{r}H(x,y,\varepsilon)$; recall
equation (48) in the proof of Corollary 4.9. ∎

The situation is sketched in Figure 4. Due to Proposition 4.5, the local
geometry is similar to the local geometry of the continuous-time counterpart
considered in [31].

######  Remark 4.12.

If $\lambda>1$, then the dynamics near a transcritical point of the map (35)
is complicated by the presence of a ‘last iterate problem’ similar to that
appearing near regular fold points; c.f. Remark 4.10.

######  Remark 4.13.

The authors in [31] showed that fast-slow ODEs with a transcritical point
feature canard solutions due to an intersection of the attracting and
repelling (extended) slow manifolds for a value of $\lambda$ which lies in an
$\varepsilon$-dependent neighbourhood about the threshold value $\lambda=1$.
Similar features are expected to arise in the map (2). Such considerations are
omitted in this work because (i) they would take us significantly beyond the
scope of the present investigation, and (ii) canard solutions are often
associated with exponentially small phenomena that may not be treatable with
formal approximations of the kind developed herein. We refer to [11, 13, 15,
17] and the references therein for more on canard dynamics in fast-slow maps.

####  4.3.3 Pitchfork dynamics

We can understand the local dynamics near a pitchfork singularity in a similar
way. Consider (35) with a pitchfork point at $(0,0)$ satisfying (44). Compact
submanifolds of the attracting branches of the critical manifold $S^{a}$ and
$S^{a,\pm}$ perturb to $O(\varepsilon)$-close slow manifolds, which we denote
by $S^{a}_{\varepsilon}$ and $S^{a,\pm}_{\varepsilon}$ respectively. We are
interested in the extension of particular slow manifolds through a
neighbourhood $\mathcal{U}\subset\mathbb{R}^{2}$ about the pitchfork point.

######  Corollary 4.14.

Consider the two-dimensional fast-slow map (35) with a pitchfork point
$(0,0)\in S$ satisfying (44), and let

$\lambda:=\frac{(\delta\alpha+\beta
g_{0})\sqrt{-\gamma}}{\alpha|g_{0}||\alpha|},$

where

$\alpha:=\frac{\partial^{2}\tilde{f}}{\partial x\partial
y}(0,0,0),\quad\beta:=\frac{1}{2}\frac{\partial^{2}\tilde{f}}{\partial
y^{2}}(0,0,0),\quad\gamma:=\frac{1}{6}\frac{\partial^{3}\tilde{f}}{\partial
x^{3}}(0,0,0),\quad\delta:=\frac{\partial\tilde{f}}{\partial\varepsilon}(0,0,0),\quad
g_{0}:=g(0,0,0).$

There exists an $\varepsilon_{0}>0$ such that for all
$\varepsilon\in(0,\varepsilon_{0})$ we have the following:

  1. 1.

If $g_{0}>0$ and $\lambda<0$ then the extension of $S_{\varepsilon}^{a}$ is
$C^{r}$-close to $S_{\varepsilon}^{a,-}$ when it leaves $\mathcal{U}$.

  2. 2.

If $g_{0}>0$ and $\lambda>0$ then the extension of $S_{\varepsilon}^{a}$ is
$C^{r}$-close to $S_{\varepsilon}^{a,+}$ when it leaves $\mathcal{U}$.

  3. 3.

If $g_{0}<0$ then the extension of both $S_{\varepsilon}^{a,\pm}$ are
$C^{r}$-close to $S_{\varepsilon}^{a}$ when they leave $\mathcal{U}$.

(a) $g_{0}>0$, $\lambda<0$. (b) $g_{0}>0$, $\lambda>0$. (c) $g_{0}<0$,
$\lambda>0$.

Figure 5: Local geometry and dynamics near a pitchfork point $P$ (shown in
red) of the map (35), as described by Corollary 4.14. We sketch the cases (a)
$g_{0}>0$, $\lambda<0$, (b) $g_{0}>0$, $\lambda>0$, and (c) $g_{0}<0$,
$\lambda>0$ (the case $g_{0}<0$, $\lambda<0$ is similar to (c)). The extension
of the attracting slow manifold, denoted $S^{a}_{\varepsilon}$ in (a)-(b) and
$S^{a,+}_{\varepsilon}$ in (c), is shown in green, and $C^{r}$-close to the
extension of the corresponding attracting slow manifold
($j^{r}S_{\varepsilon}^{a}$ in (a)-(b) and $j^{r}S_{\varepsilon}^{a,+}$ in
(c)) of the approximating vector field $j^{r}V(x,y,\varepsilon)$, which is
shown in blue. We also sketch the singular limit (layer and reduced) dynamics
for the approximating vector field, which has a critical manifold
$j^{r}S=j^{r}S^{a}\cup j^{r}S^{a,-}\cup j^{r}S^{a,+}\cup P\cup j^{r}S^{r}$,
where $j^{r}S^{a}$ and $j^{r}S^{a,\pm}$ ($j^{r}S^{r}$) are normally hyperbolic
and attracting (repelling), shown here in shaded blue (purple).

###### Proof.

The overall approach is again similar to the proofs of Corollaries 4.9 and
4.11, so we include fewer details. In this case, [31, Thm. 4.1] implies that
the truncated approximating ODE system (46) is such that the following
assertions hold for some constant $c>0$:

  1. 1’

If $g_{0}>0$ and $\lambda<0$ then the extension of $j^{r}S_{\varepsilon}^{a}$
is $O(\textrm{e}^{-c/\varepsilon})$-close to $j^{r}S_{\varepsilon}^{a,-}$ when
it leaves $\mathcal{U}$.

  2. 2’

If $g_{0}>0$ and $\lambda>0$ then the extension of $j^{r}S_{\varepsilon}^{a}$
is $O(\textrm{e}^{-c/\varepsilon})$-close to $j^{r}S_{\varepsilon}^{a,+}$ when
it leaves $\mathcal{U}$.

  3. 3’

If $g_{0}<0$ then the extension of both $j^{r}S_{\varepsilon}^{a,\pm}$ is
$O(\textrm{e}^{-c/\varepsilon})$-close to $j^{r}S_{\varepsilon}^{a}$ when it
leaves $\mathcal{U}$.

The result follows from the fact that the (extended) slow manifolds
$j^{r}S_{\varepsilon}^{a}$ and $j^{r}S_{\varepsilon}^{a,\pm}$ are the same for
the truncated vector field $j^{r}V(x,y,\varepsilon)$ and the truncated map
$j^{r}H(x,y,\varepsilon)$, and the fact that
$\textup{dist}(S_{\varepsilon},j^{r}S_{\varepsilon})=o(\|(x,y,\varepsilon)\|^{r})$.
∎

The situation is sketched in Figure 5. The local geometry is similar to the
local geometry of the continuous-time counterpart considered in [31], again
due to Proposition 4.5.

######  Remark 4.15.

Similarly to the transcritical case, the authors in [31] showed that planar
fast-slow ODEs with a pitchfork singularity feature canard solutions for a
value of $\lambda$ which lies in an $\varepsilon$-dependent interval about the
threshold value $\lambda=0$. We omit the consideration of canards near
pitchfork singularities in planar fast-slow maps in this work for the reasons
provided in Remark 4.13.

Corollaries 4.9, 4.11 and 4.14 describe the extension of attracting and
locally invariant slow manifolds through a neighbourhood of non-hyperbolic
points of regular fold, transcritical and pitchfork type in planar fast-slow
maps in standard form (35). The qualitative similarity to the dynamics of the
corresponding ODE problems in the same dimension, which have been studied in
[30] and [31], is a consequence of a formal embedding result in Proposition
4.5 (or more generally Theorem 3.5). It demonstrates that up to an error
determined by the smoothness, many important aspects of the local dynamics
near unipotent singularities of fast-slow maps can be understood via the study
of a corresponding ODE problem. This is advantageous, since there are many
techniques, in this case geometric blow-up, which apply in the ODE setting not
directly to maps.

##  5 Regular contact points in arbitrary dimensions

The results in Section 4 demonstrate the breadth and applicability of Theorem
3.5, but they are restrictive in these sense that they only apply to two-
dimensional fast-slow maps in standard form (35). In what follows we show that
Theorem 3.5 can also be used to approximate the dynamics in higher dimensional
maps, either directly in the case of maps with $\textup{codim}(S)=1$, or
indirectly after center manifold reduction in the case of maps with
$\textup{codim}(S)\geq 2$. We focus on the dynamics near an important class of
contact points which can be viewed as the higher dimensional counterpart of
regular fold points in (generally non-standard form) fast-slow maps on
$\mathbb{R}^{n}$, but our approach can be generalised in order to study other
codimension-1 singularities in $\mathcal{C}$.

Regular contact points can be defined analogously to regular contact points in
continuous-time GSPT; see [38] and [49, Sec. 4.1 and Sec. 4.2].

######  Definition 5.1.

(Regular contact point) Consider a $C^{r}$-smooth map (5) under Assumptions 1
and 2, where $r\geq 3$. We say that $z\in\mathcal{C}\subset S$ is a regular
contact point if

$\textup{rk}\left(\textup{D}fN(z)\right)=n-k-1$ (49)

and the following genericity conditions are satisfied:

  * •

Transversality, i.e. $\textup{rk}(\textup{D}f)(z)=n-k$;

  * •

Nondegeneracy, i.e.
$l\cdot((\textup{D}^{2}f)(Nr,Nr)+\textup{D}f\textup{D}N(Nr,r))(z)\neq 0$;

  * •

Slow regularity, i.e. $Nrl\textup{D}fG(z,0)\neq\mathbb{O}_{n,1}$,

where $l$ and $r$ denote left and right null vectors of the matrix
$\textup{D}fN(z)$ respectively, and $(D^{2}f)(Nr,Nr)$ and
$\textup{D}f\textup{D}N(Nr,r)$ are bilinear forms with the following
componentwise definitions:

$\begin{split}[\textup{D}^{2}f]_{i}(Nr,Nr)&=\sum_{l,m=1}^{n}\sum_{j,s=1}^{n-k}\frac{\partial^{2}f_{i}}{\partial
z_{l}\partial z_{m}}(N_{mj}r_{j})(N_{ls}r_{s}),\\\
[\textup{D}f\textup{D}N]_{i}(Nr,r)&=\sum_{l,m=1}^{n}\sum_{j,s=1}^{n-k}\frac{\partial
f_{i}}{\partial z_{l}}\frac{\partial N_{lj}}{\partial
z_{m}}(N_{ms}r_{s})r_{j},\end{split}$

for each $i=1,\ldots,n-k$. The definition extends to submanifolds, i.e. a
submanifold $\mathcal{C}_{r}\subset S$ is called a regular contact
(sub)manifold if every $z\in\mathcal{C}_{r}$ is a regular contact point.

The case of regular contact along a 1-dimensional submanifold
$\mathcal{C}_{r}$ of a 2-dimensional critical manifold $S$ in $\mathbb{R}^{3}$
is sketched in Figure 6. The rank condition (49) implies that there is a
single non-trivial multiplier, which we may denote by $\mu_{1}$, which
satisfies $\mu_{1}(z)=1$ along a $(k-1)$-dimensional regular contact manifold
$\mathcal{C}^{r}\subset S$. The algebraic multiplicity of the zero eigenvalue
of the matrix $N\textrm{D}f$ is $k+1$ at a regular contact point, which is one
greater than the geometric multiplicity $k$. Geometrically, this corresponds
to a tangency between $S$ and the curve(s) containing orbits of the layer
problem (8). More precisely, if $z\in S$ is a regular contact point, then
$\textup{range}(N(z))$ and $\textup{T}_{z}S$ are tangent along the
$1$-dimensional line given by the span of the (generalised) null vector
$\mathfrak{N}_{1}(z)$ of the matrix $\textrm{D}fN(z)$ which corresponds to the
critical multiplier $\mu_{1}(z)$. The transversality condition implies that
$S$ is a regularly embedded submanifold of $\mathbb{R}^{n}$, the nondegeneracy
condition implies that the tangency described above is quadratic when
considered on the nonlinear level, and the slow regularity condition implies
that orbits of the reduced vector field $\Pi^{S}_{\mathcal{N}}G(z,0)|_{z\in
S}$, which locally approximates the slow dynamics (recall Theorem 2.13),
extend to transversal intersections with $\mathcal{C}^{r}$; see again Figure
6.

Figure 6: Singular geometry and dynamics near a $2$-dimensional critical
manifold $S$ in $\mathbb{R}^{3}$ with a normally hyperbolic attracting and
repelling submanifolds $S^{a}$ and $S^{r}$, shown in shaded blue and purple
respectively, separated by a $1$-dimensional regular contact submanifold
$\mathcal{C}^{r}\subset S$ shown in red. We also sketch a number of
representative trajectories of the reduced flow which approximates the
dynamics on $S$ (shaded blue and purple on $S^{a}$ and $S^{r}$ respectively),
as well as sample iterates of the layer map (green with arrows to indicate
forward iteration), which lie on invariant curves which contain orbits of the
layer map (shaded grey). These curves are tangent to $S$ along
$\mathcal{C}^{r}$ due to the contact condition (49) in Definition 5.1, and the
tangency is locally quadratic due to the nondegeneracy condition. The slow
regularity condition ensures that orbits of the reduced problem extend to a
transversal intersection with $\mathcal{C}^{r}$.

######  Remark 5.2.

The slow regularity condition appearing in [49, Def. 4.4] is

$N\textup{adj}(\textup{D}fN)\textup{D}fG(z,0)\neq\mathbb{O}_{n,1},$ (50)

where $\textup{adj}(\textup{D}fN)(z)$ denotes the adjoint/adjugate matrix,
which is defined by the relation $\textup{adj}(A)=\det(A)A$. The slow
regularity condition in Definition 5.1 is equivalent to (50) as long as the
rank condition (49) is satisfied, since

$\textup{rk}(\textup{D}fN)=n-k-1\quad\implies\quad\textup{rk}\left(\textup{adj}(\textup{D}fN)\right)=1\quad\implies\quad\textup{adj}(\textup{D}fN)=\alpha
rl,$

where $l$ and $r$ are null vectors of $\textup{D}fN$ and $\alpha\neq 0$ is a
constant.

######  Remark 5.3.

For fast-slow maps in standard form (7), Definition 5.1 reduces to a
definition which closely resembles that the well-known definition of a regular
fold point in continuous-time GSPT (see e.g. [48]). More precisely, a regular
contact point for the map (7) is a point $z\in S$ such that

$\textup{rk}\ (\textup{D}_{y}f(z))=n-k-1,\qquad
l\cdot\textup{D}_{x}f(z)\neq\mathbb{O}_{k,k},\qquad
l\cdot\textup{D}^{2}_{yy}f(r,r)\neq 0,$

where $l$ and $r$ are left and right null vectors of $\textup{D}_{y}f(z)$
respectively, and $\textup{D}^{2}_{yy}f(r,r)$ is the bilinear form given
component-wise by

$[\textup{D}_{yy}^{2}]_{i}(r,r)=\sum_{j,l=1}^{n-k}\frac{\partial^{2}f_{i}}{\partial
y_{j}\partial y_{l}}\bigg{|}_{z}r_{j}r_{l},\qquad i=1,\ldots,n-k.$

######  Remark 5.4.

The transversality condition $\textup{rk}(\textrm{D}f)(z)=n-k$ is included in
Definition 5.1 in order to clarify the comparison to classical bifurcation
theory, but it is automatically satisfied due to Assumption 1. Assumption 2 is
also not necessary because the definition is local; we refer again to the
discussion following the statement of Assumption 2. We opt to keep it in order
to simplify the formulation.

In order to understand the dynamics near a regular contact point (or
submanifold), we would like to apply Theorem 3.5. Unfortunately, Theorem 3.5
does not apply directly if $\textup{codim}(S)\geq 2$, in which case the
linearisation in (9) is not unipotent due to the presence of a non-trivial
multiplier $\mu_{2}(z)$ satisfying $|\mu_{2}(z)|\neq 1$. Nevertheless, we can
apply Theorem 3.5 to a restricted map after applying the center manifold
theorem along $\mathcal{C}^{r}$.

###  5.1 Center manifold reduction

It was shown in [48, 49] that the mathematical analysis near a regular fold
submanifold $\mathcal{C}^{r}\subset S$ in fast-slow ODE systems can be
simplified after a preliminary reduction to a $(k+1)$-dimensional local local
center manifold along $\mathcal{C}^{r}$. A similar feature appears in the
context of regular fold submanifolds in fast-slow maps in general form (5).

We consider maps in general form (5) under Assumptions 1-2, which are
$C^{r}$-smooth with $r\geq 3$, $\textup{codim}(S)\geq 2$, and a regular
contact point at $z=0\in S$. Using Assumption 1, in particular the assumption
that $S$ is a regularly embedded submanifold of $\mathbb{R}^{n}$, we may
choose local coordinates $z=(x,y)\in\mathbb{R}^{k}\times\mathbb{R}^{n-k}$ such
that the matrix $D_{y}f$ is regular (full rank and invertible) in a
neighbourhood of $z=0$. We write

$\begin{pmatrix}x\\\ y\end{pmatrix}\mapsto
H(x,y,\varepsilon)=\begin{pmatrix}x\\\
y\end{pmatrix}+\begin{pmatrix}N^{x}(x,y)\\\
N^{y}(x,y)\end{pmatrix}f(x,y)+\varepsilon\begin{pmatrix}G^{x}(x,y,\varepsilon)\\\
G^{y}(x,y,\varepsilon)\end{pmatrix},$ (51)

where $N^{x}(x,y)$ and $N^{y}(x,y)$ are matrices of size $k\times(n-k)$ and
$(n-k)\times(n-k)$ respectively, and $G^{x}(x,y,\varepsilon)$ and
$G^{y}(x,y,\varepsilon)$ are column vectors of size $k\times 1$ and
$(n-k)\times 1$ respectively. Before we apply the center manifold theorem, it
is helpful to isolate the $(k+1)$-dimensional center eigenspace
$E^{\textup{c}}(0)$. This can be achieved in a two step procedure which is
directly analogous to the corresponding derivation in the context of fast-slow
ODEs in [49, Ch. 4.4].

######  Lemma 5.5.

The map (51) is locally $C^{r}$-conjugate to the map

$\widehat{H}(x,u,w,\varepsilon)=\begin{pmatrix}x\\\ u\\\
w\end{pmatrix}+\begin{pmatrix}\widehat{N}^{x}(x,u,w)\\\
\widehat{N}^{u}(x,u,w)\\\
\widehat{N}^{w}(x,u,w)\end{pmatrix}\begin{pmatrix}u\\\
w\end{pmatrix}+\varepsilon\begin{pmatrix}\widehat{G}^{x}(x,u,w,\varepsilon)\\\
\widehat{G}^{u}(x,u,w,\varepsilon)\\\
\widehat{G}^{w}(x,u,w,\varepsilon)\end{pmatrix}+R(x,u,w,\varepsilon),$ (52)

where $(x,u,w)\in\mathbb{R}^{k}\times\mathbb{R}\times\mathbb{R}^{n-k-1}$, and
the remainder term satisfies

$R(x,u,w,\varepsilon)=\begin{pmatrix}\mathbb{O}_{k,1}\\\
O(|(u,w,\varepsilon)|^{2})\end{pmatrix}.$

The critical manifold of the map (52) is contained within the hyperplane
defined by $u=0$, $w=0$, and the generalised eigenspace $E^{\textup{c}}(0)$ is
spanned by the $(x,u)$-coordinates.

###### Proof.

This can be proven with minor adaptations to the proof of [49, Thm. 4.1]. The
first step is to rectify $S$ with a nonlinear coordinate transformation

$v=f(x,y),\qquad y=K(x,v).$

Note that a local inverse exists due to the implicit function theorem, since
$D_{y}f$ is regular. In particular, we have
$K(x,v)=(D_{y}f)^{-1}\left(v-D_{x}fx\right)+O(|(x,v)|^{2})$. Taylor expansion
about $v=0$ and $\varepsilon=0$ in the new coordinates leads to the map

$\widetilde{H}(x,v,\varepsilon)=\begin{pmatrix}x\\\
v\end{pmatrix}+\begin{pmatrix}N^{x}(x,K(x,v))\\\
\textrm{D}fN(x,K(x,v))\end{pmatrix}v+\varepsilon\begin{pmatrix}G^{x}(x,K(x,v),\varepsilon)\\\
\textrm{D}fG(x,K(x,v),\varepsilon)\end{pmatrix}+R(v,\varepsilon),$ (53)

where the remainder function satisfies
$R(v,\varepsilon)=(\mathbb{O}_{k,1},O(|(v,\varepsilon)|^{2}))^{\textnormal{T}}$.
The advantage is that $S$ has been rectified along $v=0$. In particular, the
tangent space $\textup{T}_{0}S$ is now spanned by the $x$-coordinates, since
the Jacobian matrix has the form

$\textrm{D}\widetilde{H}(0,0,0)=\mathbb{I}_{n}+\begin{pmatrix}\mathbb{O}_{k,k}&N^{x}(x,K(x,v))\\\
\mathbb{O}_{n-k,k}&\textrm{D}fN(x,K(x,v))\end{pmatrix}.$

The second step is to extract the fast direction along which contact occurs.
This can be achieved with a linear transformation of the form

$\begin{pmatrix}u\\\ w\end{pmatrix}=\begin{pmatrix}l\\\ Q\end{pmatrix}v,\qquad
v=\begin{pmatrix}r&P\end{pmatrix}\begin{pmatrix}u\\\ w\end{pmatrix}=ru+Pw,$

where $u\in\mathbb{R}$, $w\in\mathbb{R}^{n-k-1}$, $l$ and $r$ are left and
right null vectors of $\textrm{D}fN$ at $z=0$, and $Q$, $P$ are matrices of
size $(n-k-1)\times(n-k)$, $(n-k)\times(n-k-1)$ such that

$\begin{pmatrix}r&P\end{pmatrix}\begin{pmatrix}l\\\
Q\end{pmatrix}=\begin{pmatrix}l\\\
Q\end{pmatrix}\begin{pmatrix}r&P\end{pmatrix}=\mathbb{I}_{n-k}.$

Expressing (51) in $(x,u,w)$ coordinates, we obtain

$\widehat{H}(x,u,w,\varepsilon)=\begin{pmatrix}x\\\ u\\\
w\end{pmatrix}+\begin{pmatrix}N^{x}\\\ l\textrm{D}fN\\\
Q\textrm{D}fN\end{pmatrix}\begin{pmatrix}r&P\end{pmatrix}\begin{pmatrix}u\\\
w\end{pmatrix}+\varepsilon\begin{pmatrix}G^{x}\\\ l\textrm{D}fG\\\
Q\textrm{D}fG\end{pmatrix}+R(x,u,w,\varepsilon),$ (54)

where
$R(u,w,\varepsilon)=(\mathbb{O}_{k,1},O(|(u,w,\varepsilon)|^{2}))^{\textnormal{T}}$
and we have omitted the argument notation in the right-hand side for
notational simplicity. The map (54) is in the desired form (52).

Finally, the Jacobian matrix is upper block triangular of the form

$\textrm{D}\widehat{H}(0,0,0,0)=\mathbb{I}_{n}+\begin{pmatrix}\mathbb{O}_{k,k}&N^{x}r&N^{x}P\\\
\mathbb{O}_{1,k}&0&\mathbb{O}_{1,n-k-1}\\\
\mathbb{O}_{n-k-1,k}&\mathbb{O}_{n-k-1,1}&Q\textrm{D}fNP\end{pmatrix},$ (55)

which shows that the $(k+1)$-dimensional generalised center eigenspace
$E^{\textup{c}}(0)$ is spanned by the $(x,u)$-coordinates. ∎

We now state the center manifold theorem which applies near a regular contact
point of the map (5), in the local coordinates given by (52). This result
closely resembles the continuous-time counterpart for fast-slow ODEs in [49,
Thm. 4.1].

######  Lemma 5.6.

Consider the map (52) with a regular contact point
$0\in\mathcal{C}_{r}\subseteq S$. There exists a $C^{r}$-smooth,
$(k+1)$-dimensional local center manifold $W_{\textup{loc}}^{\textup{c}}(0)$
which is tangent to the center subspace $E^{\textup{c}}(0)$. There is an
$\varepsilon_{0}>0$ such that for each $\varepsilon\in[0,\varepsilon_{0})$,
$W_{\textup{loc}}^{\textup{c}}(0)$ has the local graph representation

$W_{\textup{loc}}^{\textup{c}}(0)=\left\\{(x,u,W(x,u,\varepsilon)):(x,u)\in\mathcal{W}\right\\},\qquad
W(x,u,\varepsilon)=uW_{0}(x,u)+\varepsilon W_{\textup{rem}}(x,u,\varepsilon),$
(56)

where $\mathcal{W}$ is a neighbourhood of $0$ in $\mathbb{R}^{k+1}$, and the
$C^{r}$-smooth function
$W:\mathcal{W}\times[0,\varepsilon_{0})\to\mathbb{R}^{n-k-1}$ satisfies
$W(x,0,0)=0$. The restricted map
$H|_{W_{\textup{loc}}^{\textup{c}}(0)}:\mathcal{W}\times[0,\varepsilon_{0})\to\mathbb{R}^{k+1}$
defines an $\varepsilon$-family of local diffeomorphism of the form

$\begin{pmatrix}x\\\
u\end{pmatrix}\mapsto\widehat{H}|_{W_{\textup{loc}}^{\textup{c}}(0)}(x,u,\varepsilon)=\begin{pmatrix}x\\\
u\end{pmatrix}+\begin{pmatrix}\widetilde{N}^{x}(x,u)\\\
\widetilde{N}^{u}(x,u)\end{pmatrix}\widetilde{f}(x,u)+\varepsilon\begin{pmatrix}\widetilde{G}^{x}(x,u,\varepsilon)\\\
\widetilde{G}^{u}(x,u,\varepsilon)\end{pmatrix},$ (57)

where $\widetilde{N}:=(\widetilde{N}^{x},\widetilde{N}^{u})^{\textnormal{T}}$
is a $(k+1)\times 1$ column vector given by

$\begin{pmatrix}\widetilde{N}^{x}\\\
\widetilde{N}^{u}\end{pmatrix}=\begin{pmatrix}N^{x}\left(r+PW_{0}(x,u)\right)\\\
l\textrm{D}fN\left(r+PW_{0}(x,u)\right)\end{pmatrix},$ (58)

the function $\widetilde{f}:\mathcal{W}\to\mathbb{R}$ is $C^{r}$-smooth and
satisfies

$\widetilde{f}(x,0)=0,\qquad\frac{\partial\widetilde{f}}{\partial u}(x,0)=1,$
(59)

i.e. $\widetilde{f}(x,u)=u+u(1+O(|x|,u))$, and
$\widetilde{G}:=(\widetilde{G}^{x},\widetilde{G}^{u})^{\textnormal{T}}$ is a
$(k+1)\times 1$ column vector such that

$\widetilde{G}(0,0,0)=\begin{pmatrix}\widetilde{G}^{x}(0,0,0,0)\\\
\widetilde{G}^{u}(0,0,0,0)\end{pmatrix}=\begin{pmatrix}G^{x}(0,0,0,0)\\\
l\textrm{D}fG(0,0,0,0)\end{pmatrix}.$ (60)

The restricted map (57) is fast-slow with a $k$-dimensional critical manifold
$S$ and a unipotent regular contact point at $(x,u)=(0,0)\in\mathbb{R}^{k+1}$.

###### Proof.

The existence of a $(k+1)$-dimensional $C^{r}$-smooth center manifold follows
from the center manifold theorem; see e.g. [34]. The $(n-k-1)\times(n-k-1)$
matrix $Q\textrm{D}fNP$ appearing in the bottom-right block of the Jacobian in
(55) is regular; its eigenvalues $\lambda_{j}(z)$ have non-zero real part
because the matrix $\mathbb{I}_{n-k-1}+Q\textrm{D}fNP$ encodes the non-trivial
multipliers $\mu_{j}(z)=1+\lambda_{j}(z)$ for $j=2,\ldots,n-k$ which are not
on the unit circle (and therefore not equal to $1$). Thus, the implicit
function theorem implies that $W_{\textrm{loc}}^{c}(0)$ can be written as a
graph over the $(x,u)$-coordinates. The particular form of
$W(x,u,\varepsilon)$ in (56) follows after Taylor expansion in $\varepsilon$,
and the factorisation $W(x,u,0)=uW_{0}(x,u)$ follows from the fact that $S$
lies in the hyperplane defined by $u=0$ and $w=W(x,0,0)=0$.

Directly restricting (54) (which is equivalent to (52)) to
$W_{\textrm{loc}}^{\textup{c}}(0)$ leads to the $(k+1)$-dimensional map in
(57), including the properties and expressions in (58), (59) and (60), after
further Taylor expansion in $\varepsilon$.

It remains to show that the restricted map (57) is a diffeomorphism with a
unipotent regular contact point at $(0,0)\in\mathbb{R}^{k+1}$. Evaluating the
Jacobian matrix when $\varepsilon=0$ at $(0,0)$, we obtain

$\textrm{D}\widehat{H}|_{W_{\textrm{loc}}^{\textup{c}}(0)}(0,0,0)=\mathbb{I}_{k+1}+\widetilde{N}(0,0)\textrm{D}\widetilde{f}(0,0)=\mathbb{I}_{k+1}+\begin{pmatrix}\mathbb{O}_{k,k}&\widehat{N}^{x}(0,0)\\\
\mathbb{O}_{1,k}&\widehat{N}^{u}(0,0)\end{pmatrix},$ (61)

which has $k$ trivial multipliers equal to $1$ and a single non-trivial
multiplier satisfying

$\mu_{1}(0,0)=1+\widetilde{N}^{u}(0,0)=1+l\textrm{D}fN\left(r+PW_{0}(0,0)\right)(0,0)=1,$
(62)

where we used the fact that $l\textrm{D}fN(0,0)=0$ ($l$ is a left null vector
of $\textrm{D}fN$ at $z=0$). Thus $(0,0)$ is a contact point for the map (14).
Since the matrix appearing on the right-hand side of (61) is nilpotent (its
square is the zero matrix), $(0,0)$ is a unipotent contact point for (57). The
fact that $(0,0)$ is a regular contact point follows after verifying the
genericity conditions in Definition 5.1. The transversality condition can be
checked directly, while the nondegeneracy and slow regularity conditions can
be shown to follow from the satisfaction of the nondegeneracy and slow
regularity conditions in the original map (52) on $\mathbb{R}^{n}$; the
details are standard, and omitted for brevity.

Finally, equation (62) is also sufficient to apply the inverse function
theorem. This implies that $\mathcal{W}$ and $\varepsilon_{0}>0$ can be fixed
sufficiently small for $H|_{W_{\textrm{loc}}^{\textup{c}}(0)}$ to be a
diffeomorphism on $\mathcal{W}\times[0,\varepsilon_{0})$. ∎

It is significant to note that the restricted map (57) describing dynamics on
the local center manifold $W^{\textup{c}}_{\textup{loc}}(0)$ is a
diffeomorphism, and that the Jacobian matrix (61) is unipotent. Thus, Theorem
3.5 applies, even though it does not apply directly in the original map on
$\mathbb{R}^{n}$ if $\textup{codim}(S)\geq 2$, i.e. prior to restriction to
$W_{\textrm{loc}}^{\textup{c}}(0)$.

###  5.2 Formal embedding and dynamics on the center manifold

Applying Theorem 3.5, we obtain a formal embedding result which can be used to
approximate the dynamics of the map (57) on
$W_{\textrm{loc}}^{\textup{c}}(0)$.

######  Theorem 5.7.

Consider the restricted map (57) on $W_{\textrm{loc}}^{\textup{c}}(0)$, as
described in Lemma 5.6. There exists a neighbourhood $\mathcal{W}\ni 0$ in
$\mathbb{R}^{k+1}$ and an $\varepsilon_{0}>0$ such that for all
$(x,u,\varepsilon)\in\mathcal{W}\times[0,\varepsilon_{0})$ and
$m\in\\{1,\ldots,r\\}$, where $r\geq 3$, we have

$j^{m}\widehat{H}|_{W_{\textup{loc}}^{\textup{c}}(0)}(x,u,\varepsilon)=j^{m}\Phi_{V}^{1}(x,u,\varepsilon),$
(63)

where $\Phi_{V}^{1}(x,u,\varepsilon)$ denotes the time-$1$ flow of an
$\varepsilon$-family of vector fields $V(x,u,\varepsilon)$ on
$\mathbb{R}^{k+1}$ satisfying

$j^{r}V(x,u,\varepsilon)=\mathfrak{N}(x,u)j^{r}\widetilde{f}(x,u)+\varepsilon\mathfrak{G}(x,u,\varepsilon),$
(64)

where $\mathfrak{N}$ and $\mathfrak{G}$ have the same dimensions, regularity
and smoothness as $(\widetilde{N}^{x},\widetilde{N}^{u})^{\textnormal{T}}$ and
$(\widetilde{G}^{x},\widetilde{G}^{u})^{\textnormal{T}}$ respectively, and
$j^{r}\widetilde{f}$ is the $r$-jet associated to the Taylor expansion of
$\widetilde{f}(x,u)$ about $(x,u)=(0,0)$. Moreover,

$\mathfrak{N}(0,0)=\begin{pmatrix}\widetilde{N}^{x}(0,0)\\\
\widetilde{N}^{u}(0,0)\end{pmatrix},\qquad
j^{r}\widetilde{f}(x,u)=u\left(1+O(|x|,u)\right),\qquad\mathfrak{G}(0,0,0)=\begin{pmatrix}\widetilde{G}^{x}(0,0,0)\\\
\widetilde{G}^{u}(0,0,0)\end{pmatrix}.$ (65)

The vector field (64) is fast-slow with a $k$-dimensional critical manifold

$j^{r}S:=\left\\{(x,u)\in\mathbb{R}^{k+1}:j^{r}\widetilde{f}(x,u)=0\right\\},$
(66)

and a regular contact point at $(0,0)\in j^{r}S$.

###### Proof.

The proof is similar to the proof of Proposition 4.5. Since Assumptions 1-2
can be checked directly, and $(0,0)$ is a unipotent contact point by Lemma
5.6, we can apply Theorem 3.5 directly. Doing so yields the approximation
property (63) and the expressions in (64), (65) and (66).

It remains to show that $(0,0)$ is a regular contact point for the truncated
vector field (64). It suffices to verify the defining conditions from
Definition 5.1 at $(0,0)$, which are analogous for fast-slow ODE systems; we
refer again to [49, Sec. 4.1 and 4.2] for definitions. The rank condition (49)
is satisfied since

$D(j^{r}\widetilde{f})\mathfrak{N}(0,0)=D\widetilde{f}\widetilde{N}(0,0)=0\quad\implies\quad\textup{rk}\
\left(D(j^{r}\widetilde{f})\mathfrak{N}(0,0)\right)=(k+1)-k-1=0,$

where we used the left-most expression in (65) and the fact that
$D\widetilde{f}(0,0)=D(j^{r}\widetilde{f})(0,0)$, since $r\geq 3>1$.

The transversality condition can also be checked directly. Using the
expression for $j^{r}\widetilde{f}(x,u)$ in (65) we obtain

$\textup{rk}(D(j^{r}\widetilde{f})(0,0))=\textup{rk}\begin{pmatrix}\mathbb{O}_{1,k}&1\end{pmatrix}=1,$

as required.

The nondegeneracy condition reduces to the requirement that

$\sum_{l,m=1}^{k+1}\left(\frac{\partial^{2}(j^{r}\tilde{f})}{\partial\xi_{l}\partial\xi_{m}}(0,0)\mathfrak{N}_{m}(0,0)\mathfrak{N}_{l}(0,0)+\frac{\partial(j^{r}\widetilde{f})}{\partial\xi_{l}}(0,0)\frac{\partial\mathfrak{N}_{l}}{\partial\xi_{m}}(0,0)\mathfrak{N}_{m}(0,0)\right)\neq
0,$ (67)

where we introduced the componentwise notation
$\xi=(\xi_{1},\ldots,\xi_{k},\xi_{k+1})=(x_{1},\ldots,x_{k},u)$ and
$\mathfrak{N}=(\mathfrak{N}^{x}_{1},\ldots,\mathfrak{N}^{x}_{k},\mathfrak{N}^{u})^{\textnormal{T}}$.
It follows from the form of $j^{r}\widetilde{f}(x,u)$ in (65) and the fact
that $\mathfrak{N}^{u}(0,0)=\widetilde{N}^{u}(0,0)=0$ that

$\frac{\partial^{2}(j^{r}\tilde{f})}{\partial\xi_{l}\partial\xi_{m}}(0,0)\mathfrak{N}_{m}(0,0)\mathfrak{N}_{l}(0,0)=0$

for all $l,m=1,\ldots,k+1$. Moreover, the terms on the right can be rewritten
as

$\frac{\partial(j^{r}\widetilde{f})}{\partial\xi_{l}}(0,0)\frac{\partial\mathfrak{N}_{l}}{\partial\xi_{m}}(0,0)\mathfrak{N}_{m}(0,0)=\frac{\partial\widetilde{f}}{\partial\xi_{l}}(0,0)\frac{\partial\mathfrak{N}_{l}}{\partial\xi_{m}}(0,0)\widetilde{N}_{m}(0,0),$

for all $l,m=1,\ldots,k+1$. Thus, the nondegeneracy condition in (67) reduces
to

$\sum_{l,m=1}^{k+1}\frac{\partial\widetilde{f}}{\partial\xi_{l}}(0,0)\frac{\partial\mathfrak{N}_{l}}{\partial\xi_{m}}(0,0)\widetilde{N}_{m}(0,0)\neq
0.$ (68)

Finally, one can show that

$\frac{\partial\mathfrak{N}_{l}}{\partial\xi_{m}}(0,0)=\frac{\partial\widetilde{N}_{l}}{\partial\xi_{m}}(0,0),$
(69)

after which (68) reduces to the nondegeneracy condition for the restricted map
(57), which is satisfied. The equality in (69) can be obtained by matching
coefficients in truncated Taylor expansions of (57) and the time-1 map in
(63). This calculation is standard but lengthy, so we defer it to Appendix D
for expository reasons.

It remains to check the slow regularity condition. Since
$D(j^{r}\widetilde{f})\widehat{N}(0,0)$ is a scalar we have that $r=l=1$.
Using this together with (65), the slow regularity condition reduces on the
vector field (64) reduces to the slow regularity condition on the map (57),
and we obtain

$\mathfrak{N}D(j^{r}\widetilde{f})\mathfrak{G}(0,0,0)=\widetilde{N}D\widetilde{f}\widetilde{G}(0,0,0)\neq
0,$

as required. ∎

Theorem 5.7 shows that the dynamics on the center manifold
$W_{\textrm{loc}}^{\textup{c}}(0)$ can be locally approximated by the time-$1$
map induced by a $(k+1)$-dimensional fast-slow ODE system with a regular fold
point at $(0,0)$. It is significant to note that Theorem 5.7 also allows for
an approximation of the dynamics of the original map (54) in $\mathbb{R}^{n}$,
if the non-critical non-trivial multipliers $\mu_{j}(z)$ satisfy

$|\mu_{j}(0)|<1,\qquad\forall j=2,\ldots,n-k,$

due to the local exponential attractivity of
$W_{\textrm{loc}}^{\textup{c}}(0)$ in this case. The dynamics near regular
fold points in general $(k+1)$-dimensional fast-slow ODE systems in canonical
local normal forms have been described in [48, 49] (the case $k=2$ was already
treated already in [47]). Although we do not consider the geometry and
dynamics in detail here, we conjecture that the approximation result in
Theorem 5.7 can be combined with the results in [48, 49] in order to describe
certain properties of the geometry and dynamics close to the contact point. In
particular, we expect that the extension of the attracting slow manifold can
be described using arguments which are similar to those applied in the context
of regular fold, transcritical and pitchfork points in Section 4; recall in
particular the proofs for Corollaries 4.9, 4.11 and 4.14.

##  6 Summary and Outlook

The following three features have been fundamental to the success of
continuous-time GSPT as an approach to the study of fast-slow ODE systems:

  * (I)

A systematic geometric formalism for identifying and analysing simpler
limiting problems associated to each time-scale;

  * (II)

A set of perturbation theorems in the normally hyperbolic regime, as provided
by Fenichel theory;

  * (III)

A set of independent results which describe the local dynamics close to non-
normally hyperbolic singularities.

The primary aim of this manuscript has been to continue the development of
DGSPT, based on the principle that (I)-(III) should also be viewed as
cornerstones for a geometric approach to the study of discrete fast-slow
systems induced by repeated iteration of a map. One can also view this work as
a natural continuation of the work in [26], which focused on (I)-(II) via the
development of DGSPT for general fast-slow maps (5) in the normally hyperbolic
regime.

In Section 2.2 we presented two new results in the normally hyperbolic regime,
namely Theorems 2.12 and 2.13. Both of these results can be used in order to
approximate the dynamics on slow manifolds via the time-$1$ map induced by the
flow of a vector field in the same dimension. Theorem 2.13 in particular
describes a close relationship to the reduced vector field which appears as
the limiting problem for the dynamics on the critical manifold in the
continuous-time setting. Not only does this elucidate a close connection to
continuous-time theory on the slow time-scale, but, for many purposes, it
provides a means of reducing the study of the map to the study of a vector
field which is known.

Our primary focus, however, was to contribute to the understanding of dynamics
near non-normally hyperbolic points, i.e. we focused primarily on (III). As we
saw in Section 3.1, codimension-$1$ non-normally hyperbolic singularities can
be split into three important types: fold/contact points in
$\mathcal{C}\subset S$, flip/period-doubling points in
$\mathcal{O}_{\textup{f}}\subset S$, and torus/Neimark-Sacker points in
$\mathcal{O}_{\textup{ns}}\subset S$. These correspond to points at which the
matrix $\mathbb{I}_{n-k}+\textrm{D}fN$ has a single multiplier at $+1$, $-1$,
or a pair of complex conjugate multipliers on $S^{1}\setminus\\{\pm 1\\}$,
respectively. The so-called ‘oscillatory singularities’ in
$\mathcal{O}_{\textup{f}}\cup\mathcal{O}_{\textup{ns}}$ can often be analysed
using techniques which rely on the invertibility of the matrix $\textrm{D}fN$,
we refer again to [3, 4, 17, 18, 41, 42] and the references therein. We
therefore chose to focus on the loss of normal hyperbolicity near
singularities in $\mathcal{C}$, where the matrix $\textrm{D}fN$ is singular.

Our most important theoretical result on the dynamics near singularities in
$\mathcal{C}$ is Theorem 3.5, which shows that the dynamics near unipotent
singularities, which form an important subset of $\mathcal{C}$, can be
approximated by the time-$1$ map of a formal vector field in the same
dimension. This vector field is fast-slow with a nilpotent singularity and a
critical manifold $j^{r}S$ that is $C^{r}$-close to the critical manifold of
the original map. The proof of Theorem 3.5 relied on direct arguments and a
suitable application of the Takens embedding theorem.

In order to demonstrate the applicability of Theorem 3.5, we used it in order
to generate results on the geometry and dynamics near non-normally hyperbolic
points of regular fold, transcritical and pitchfork type in planar fast-slow
maps in standard form; recall Section 4. We showed in Proposition 4.5 that
$2$-dimensional fast-slow maps in standard form (35) can be approximated by
the time-$1$ map of a planar fast-slow ODE system in standard form, and that
the approximating vector field has a non-normally hyperbolic point of regular
fold, transcritical or pitchfork type if the original map has a non-normally
hyperbolic point of the corresponding type. Using this result and the
established results from continuous-time theory in [30, 31], we characterised
the extension of attracting slow manifolds through a neighbourhood of regular
fold, transcritical and pitchfork points in the map (35) in Corollaries 4.9,
4.11 and 4.14 respectively. It is significant to note that the established
results in the continuous-time setting, i.e. those in [30, 31], were derived
using the geometric blow-up method, which is either not applicable or not yet
developed for the study of singularities in general fast-slow maps (with the
exception of discretized systems, recall Remarks 3.2 and 4.7). This points to
a major advantage of approximation results like Proposition 4.5, namely, that
they provide a means for analysing certain features of the map using methods
that are only developed or applicable in the continuous-time setting.

Finally in Section 5, we showed that Theorem 3.5 could be also used to
approximate the dynamics near regular contact points of fast-slow maps in
general nonstandard form (5) in $\mathbb{R}^{n}$, even though contact points
in $\mathcal{C}$ are not generically unipotent if $\textup{codim}(S)\geq 2$,
i.e. if $n-k\geq 2$. The key observation is that fast-slow maps with a
codimension-1 regular contact point admit of a local center manifold
reduction. The center manifold $W_{\textrm{loc}}^{\textup{c}}$ is
$(k+1)$-dimensional, and $\textup{codim}(S)=1$ in the restricted map
$H|_{W_{\textrm{loc}}^{\textup{c}}}$ which governs dynamics on
$W_{\textrm{loc}}^{\textup{c}}$; this was shown in Lemma 5.6. The map
$H|_{W_{\textrm{loc}}^{\textup{c}}}$ is also fast-slow with a regular contact
point, but in this case, the contact point is also unipotent. This allowed for
the application of Theorem 3.5, which in turn allowed us to show that the
dynamics of $H|_{W_{\textrm{loc}}^{\textup{c}}}$ can be approximated by the
time-$1$ map of a $(k+1)$-dimensional fast-slow system with a regular contact
point. This is summarised in Theorem 5.7.

There are many open questions remaining. For example, Corollaries 4.9, 4.11
and 4.14 describe the extension of attracting slow manifolds through a
neighbourhood of different types of non-normally hyperbolic points, but we did
not present results on the dynamics of larger sets of initial conditions.
Another open question pertains to the extent to which the approximation error
appearing in all of the formal embedding theorems presented herein can be
improved. In this regard it is worthy to reiterate that exponentially small
errors may be unavoidable, recall Remark 4.8, which suggests that the details
of exponentially small separations which are typical in canard theory, for
example, cannot be simply described with formal approximations of the kind
presented herein.

More generally, the development of DGSPT is closely tied to Guckenheimer’s
program for the development of a ‘global’ GSPT [21] which is general enough to
apply to the study of oscillatory fast-slow systems, i.e. fast-slow ODE
systems featuring a manifold or manifolds of limit cycles in the layer
problem. The connection arises because DGSPT provides a geometric framework
for studying the fast-slow Poincaré maps which arise naturally in this
setting. We believe that [26] and the present manuscript constitute
significant progress towards a rigorous geometric theory for global
oscillatory fast-slow systems, which has implications for the geometric study
of models of complex oscillatory dynamics in a wide range of areas including
mathematical neuroscience, biology and chemistry. The exact nature of this
connection is left for future work.

### Acknowledgements

SJ and CK acknowledge funding from the SFB/TRR 109 Discretization and Geometry
in Dynamics, i.e. Deutsche Forschungsgemeinschaft (DFG - German Research
Foundation) - Project-ID 195170736 - TRR109. CK thanks the VolkswagenStiftung
for support via a Lichtenberg Professorship.

## References

  * [1] L. Arcidiacono, M. Engel, and C. Kuehn, Discretized fast–slow systems near pitchfork singularities, Journal of Difference Equations and Applications, 25 (2019), pp. 1024–1051.
  * [2] S. M. Baer, T. Erneux, and J. Rinzel, The slow passage through a Hopf bifurcation: Delay, memory effects, and resonance, SIAM Journal on Applied mathematics, 49 (1989), pp. 55–71.
  * [3] C. Baesens, Slow sweep through a period-doubling cascade: Delayed bifurcations and renormalisation, Physica D: Nonlinear Phenomena, 53 (1991), pp. 319–375.
  * [4] , Gevrey series and dynamic bifurcations for analytic slow-fast mappings, Nonlinearity, 8 (1995), p. 179.
  * [5] C. L. Bouten, Bulletin of the American Mathematical Society, 23 (1916), p. 76.
  * [6] K.-T. Chen, Local diffeomorphisms – $C^{\infty}$-realization of formal properties, American Journal of Mathematics, 87 (1965), pp. 140–157.
  * [7] S.-N. Chow, C. Li, and D. Wang, Normal forms and bifurcation of planar vector fields, Cambridge University Press, 1994.
  * [8] P. De Maesschalck, F. Dumortier, and R. Roussarie, Canard Cycles, vol. 73, Springer, 2021.
  * [9] F. Dumortier, A flow box theorem for 2d slow-fast vector fields and diffeomorphisms and the slow log-determinant integral, Journal of Differential Equations, 333 (2022), pp. 361–406.
  * [10] F. Dumortier and R. Roussarie, Canard cycles and center manifolds, no. 577 in Memoirs of the American Mathematical Society, American Mathematical Society, 1996.
  * [11] M. Engel and G. A. Gottwald, Canards in modified equations for Euler discretizations, arXiv preprint arXiv:2304.08797, (2023).
  * [12] M. Engel and H. Jardón-Kojakhmetov, Extended and symmetric loss of stability for canards in planar fast-slow maps, SIAM Journal on Applied Dynamical Systems, 19 (2020), pp. 2530–2566.
  * [13] , Extended and symmetric loss of stability for canards in planar fast-slow maps, SIAM Journal on Applied Dynamical Systems, 19 (2020), pp. 2530–2566.
  * [14] M. Engel and C. Kuehn, Discretized fast-slow systems near transcritical singularities, Nonlinearity, 32 (2019), p. 2365.
  * [15] M. Engel, C. Kuehn, M. Petrera, and Y. Suris, Discretized fast–slow systems with canards in two dimensions, Journal of Nonlinear Science, 32 (2022), p. 19.
  * [16] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), pp. 53–98.
  * [17] A. Fruchard and R. Schäfke, Bifurcation delay and difference equations, Nonlinearity, 16 (2003), p. 2199.
  * [18] , A survey of some results on overstability and bifurcation delay, Discrete & Continuous Dynamical Systems-S, 2 (2009), p. 931.
  * [19] A. Goeke and S. Walcher, A constructive approach to quasi-steady state reductions, Journal of Mathematical Chemistry, 52 (2014), pp. 2596–2626.
  * [20] T. Gramchev and S. Walcher, Normal forms of maps: Formal and algebraic aspects, Acta Applicandae Mathematica, 87 (2005), pp. 123–146.
  * [21] J. Guckenheimer, Towards a global theory of singularly perturbed dynamical systems, in Nonlinear Dynamical Systems and Chaos, Springer, 1996, pp. 213–225.
  * [22] M. G. Hayes, T. J. Kaper, P. Szmolyan, and M. Wechselberger, Geometric desingularization of degenerate singularities in the presence of fast rotation: A new proof of known results for slow passage through Hopf bifurcations, Indagationes Mathematicae, 27 (2016), pp. 1184–1203.
  * [23] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Bulletin of the American Mathematical Society, 76 (1970), pp. 1015–1019.
  * [24] Y. S. Il’Yashenko and S. Y. Yakovenko, Nonlinear stokes phenomena in smooth classification problems, Advances in Soviet Mathematics, 14 (1993), pp. 235–287.
  * [25] H. Jardón-Kojakhmetov and C. Kuehn, A survey on the blow-up method for fast-slow systems, Contemporary Mathematics, 775 (2021), pp. 115–160.
  * [26] S. Jelbart and C. Kuehn, Discrete geometric singular perturbation theory, Discrete and Continuous Dynamical Systems, 43 (2023), pp. 57–120.
  * [27] S. Jelbart, C. Kuehn, and S.-V. Kuntz, Geometric blow-up for folded limit cycle manifolds in three time-scale systems, arXiv preprint arXiv:2208.01361, (2022).
  * [28] C. K. Jones, Geometric singular perturbation theory, in Dynamical systems, vol. 1609 of Lecture Notes in Mathematics, Springer, 1995, pp. 44–118.
  * [29] N. Kruff and S. Walcher, Coordinate-independent singular perturbation reduction for systems with three time scales, Mathematical Biosciences and Engineering, 16 (2019), pp. 5062–5091.
  * [30] M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions, SIAM Journal on Mathematical Analysis, 33 (2001), pp. 286–314.
  * [31] , Extending slow manifolds near transcritical and pitchfork singularities, Nonlinearity, 14 (2001), p. 1473.
  * [32] , Relaxation oscillation and canard explosion, Journal of Differential Equations, 174 (2001), pp. 312–368.
  * [33] C. Kuehn, Multiple time scale dynamics, vol. 191 of Applied Mathematical Sciences, Springer, 2015.
  * [34] Y. A. Kuznetsov, Elements of applied bifurcation theory, vol. 112 of Applied Mathematical Sciences, Springer Science & Business Media, 2013.