# On description of blow-up behavior for ODEs with normally hyperbolic nature
in dynamics at infinity

Kaname Matsue , 111International Institute for Carbon-Neutral Energy Research
(WPI-I2CNER), Kyushu University, Fukuoka 819-0395, Japan Institute of
Mathematics for Industry, Kyushu University, Fukuoka 819-0395, Japan
<EMAIL_ADDRESS>
###### Abstract

We describe blow-up behavior for ODEs by means of dynamics at infinity with
complex asymptotic behavior in autonomous systems, as well as in nonautonomous
systems. Based on preceding studies, a variant of closed embeddings of phase
spaces and the time-scale transformation determined by the structure of vector
fields at infinity reduce our description of blow-ups to unravel the shadowing
property of (pre)compact trajectories on the horizon, the geometric object
expressing the infinity, with the specific convergence rates. Geometrically,
this description is organized by asymptotic phase of invariant sets on the
horizon. Blow-up solutions in nonautonomous systems can be described in a
similar way. As a corollary, normally, or partially hyperbolic invariant
manifolds on the horizon possessing asymptotic phase are shown to induce blow-
ups.

Keywords: blow-up solutions, embeddings, desingularization, normally
hyperbolic invariant manifolds, asymptotic phase, nonautonomous differential
equations

AMS subject classifications:  34A26, 34C08, 34C45, 35B44, 37C60, 37D10, 58K55

##  1 Introduction

Blow-up behavior of solutions in differential equations is widely studied in
decades from many aspects such as mathematical, numerical and physical ones.
Fundamental questions in blow-up phenomena are whether or not a solution blows
up at a finite time and, if it does, when, where and how it blows up. While
numerous studies involving blow-ups and related finite-time singularities such
as finite-time extinction, collapse and quenching are provided from various
viewpoints, the author and his collaborators have provided characterizations
of finite-time singularities involving blow-ups by means of dynamics at
infinity based on compactifications of phase spaces and appropriate time-scale
desingularizations [38, 39, 2, 30]. These machineries reduce the problem for
finding and characterizing blow-up solutions to that for characterizing
(center-)stable manifolds of equilibria/periodic orbits on the horizon, the
geometric object expressing infinity as the boundaries of compactified
manifolds or their upper-tangent spaces, for the transformed vector fields
called desingularized vector fields so that “dynamics at infinity” makes
sense. In particular, asymptotic theory in dynamical systems provides various
characterizations of blow-up solutions under appropriate setting and transform
in vector fields. Note that this approach is applied to studying features of
solutions of PDEs with blow-up and/or quenching behavior (e.g., [23, 24, 25,
26]), while the approach itself is also used to study bounded objects in
dynamical systems (e.g., [7, 8, 17, 22, 31]). Moreover, the proposed machinery
also provides various achievements in computer-assisted proofs of blow-up
solutions with well-established concepts in dynamical systems [35, 40, 41,
52].

In the previous studies, blow-up solutions are considered only for autonomous
vector fields ${\bf y}^{\prime}=f({\bf y})$ admitting, as already mentioned,
equilibria and/or periodic orbits at infinity. While various generalization
will be proposed, we shall pay our attention to the following generalizations
here:

  * •

blow-up behavior with more complex nature at infinity, and

  * •

characterization of blow-ups in nonautonomous systems.

One of appropriate candidates in the first direction would be blow-ups
shadowing normally hyperbolic invariant manifolds (NHIMs for short) at
infinity (cf. [34] for characterizing unbounded geometry by means of NHIMs).
In the case of nonautonomous systems

${\bf y}^{\prime}=f(t,{\bf y})$ (1.1)

under a suitable setting to $f$, an appropriate treatment of time variable $t$
is required. One of typical treatments is to regard the time variable $t$ as
an additional phase variable and the original system as the extended
autonomous system

$\frac{d{\bf y}}{d\eta}=f(t,{\bf y}),\quad\frac{dt}{d\eta}=1,$ (1.2)

in which case treatments of invariant sets becomes different from general
autonomous systems222 For example, equilibria do not make sense in (1.2). .
While useful machineries such as compactifications in the time variable (e.g.,
[53]) are established for studying global-in-time solutions in nonautonomous
systems, we will see that the above extended autonomous system (1.2) with an
appropriate rule of scaling in $t$ is suitable for characterizing blow-ups in
nonautonomous systems in the sense that machineries developed for autonomous
systems can be applied.

Our main aims here are to see that the following stuff essentially
characterize blow-ups both in autonomous and nonautonomous systems:

  * •

shadowing property of trajectories at infinity;

  * •

specific (and sufficiently fast) “converge rate” to them.

Geometrically, the first requirement is described by means of asymptotic phase
of invariant sets (e.g., [37] and references therein), which is manifested in
invariant manifolds with partial / normal hyperbolicity. An example is
invariant foliations of stable manifolds of NHIMs. The second requirement is
essentially related to “convergence rate”, which is stronger than usual
characterizations of the above hyperbolicity. In a special case, we refer to
the theory in linear differential systems, e.g., [45, 46, 47] (see also [29,
51]), where dichotomy spectrum characterize stability or asymptotic behavior
of solutions. We shall see that a special structure of the spectrum provides
our requirement of “convergence rate”, which includes the case of the presence
of hyperbolic equilibria/periodic orbits at infinity ([38]).

The rest of the present paper is organized as follows. In Section 2, we review
a fundamental concept of NHIMs and put a remark on asymptotic phase. In
Section 3, we summarize fundamental settings of vector fields we shall treat
throughout the present paper and machineries to characterize dynamics at
infinity including blow-up solutions; embeddings of phase spaces and time-
scale desingularizations. In Section 4, we pay our attention to autonomous
systems, and provide a characterization of blow-up solutions shadowing
trajectories at infinity. This characterization generalizes preceding works
where blow-up solutions shadowing hyperbolic equilibria and periodic orbits at
infinity are characterized (e.g., [38, 39]). We further describe blow-up
solutions with asymptotic rates automatically determined by the quasi-
homogeneity of vector fields, which are often referred to as type-I blow-ups,
under the specific “convergence rate” to shadowed trajectories. In Section 5,
we move to nonautonomous systems and derive a criterion of the existence of
blow-ups with their asymptotic behavior. In addition to a fundamental
description, a geometric treatment of NHIMs with boundaries towards our aims
is also presented to geometrically describe the blow-up criterion. In Section
6, we show several examples of blow-up solutions in nonautonomous systems. Our
examples shown there revisit to preceding works in various backgrounds to
characterize finite-time singularities, and will be shown that blow-ups are
described in a unified way.

Several technical details are collected in Appendix. In Appendix A, another
embedding frequently applied in practical problems is reviewed. In Appendix B,
a sufficient condition to satisfy the rate requirement discussed in Section 4
is provided by means of spectral theory in linear differential systems. In
Appendix C.3, a geometric technique to provide a sufficient condition for
blow-up rates of blow-up solutions in nonautonomous systems discussed in
Section 5 is presented.

######  Remark 1.1 (Notation on vector bundles. e.g., [19, 45]).

Let $\xi=(\pi,E,X)$ be an $n$-dimensional vector bundle over the base
(topological) space $X$, that is, the triple of $X$, the total space
$E=\bigcup_{{\bf x}\in X}F_{\bf x}$ with the fiber $F=F_{\bf
x}\cong\mathbb{R}^{n}$, and the continuous projection $\pi:E\to X$. For any
subset $M\subset X$ of the base space, let $\xi|_{M}=(\pi,E(M),M)$ with
$E(M)=\bigcup_{{\bf x}\in M}F_{\bf x}$ be the restriction of $\xi$ over $M$.

##  2 Preliminaries 1: normally hyperbolic invariant manifolds

Here we briefly summarize a fundamental concept in dynamical systems we shall
use later, normally hyperbolic invariant manifolds (NHIMs) followed by [10,
20], as well as a remark on a property which NHIMs admit, asymptotic phase.

Let $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a $C^{r}$-vector field with $r\geq
1$ and $\varphi=\varphi_{f}$ be the generated flow. For any $t\in\mathbb{R}$,
the associated time-$t$ map is denoted by $\varphi^{t}=\varphi_{f}^{t}$. For
any subset $M\subset\mathbb{R}^{n}$, let $T_{M}\mathbb{R}^{n}:=\pi^{-1}(M)$
associated with the vector bundle $(\pi,T\mathbb{R}^{n},\mathbb{R}^{n})$.

######  Remark 2.1.

Arguments in this subsection are valid for $C^{r}$-vector fields $f:Q\to TQ$,
where $Q$ is a $C^{r}$ Riemannian manifold. Nevertheless we pay our attention
to the case $Q=\mathbb{R}^{n}$ for simplicity. Indeed, our interests here are
submanifolds of $\mathbb{R}^{n}$ for some $n$ and associated vector fields.

######  Definition 2.2 (Normal hyperbolicity, cf. [20, 10]).

Let $M\subset\mathbb{R}^{n}$ be a compact, connected manifold with $\partial
M=\emptyset$ which is invariant for $\varphi$. We say that $M$ is $r$-normally
hyperbolic333 In [20], the above definition is referred to as $r$-eventually
relatively normal hyperbolicity. This property is sufficient in our purpose.
if there is a continuous splitting

$T_{M}\mathbb{R}^{n}=TM\oplus E^{u}\oplus E^{s}$ (2.1)

of $T_{M}\mathbb{R}^{n}$ such that, for any $t$, all subbundles $TM$, $E^{u}$
and $E^{s}$ are $D\varphi^{t}$-invariant and that there are constants $C>0$
and $\lambda_{s}<0<\lambda_{u}$ satisfying

$\|D\varphi^{t}|_{E^{s}_{p}}\|\leq
Ce^{\lambda_{s}t}m\left(D\varphi^{t}|_{T_{p}M}\right)^{i},\quad
m\left(D\varphi^{t}|_{E^{u}_{p}}\right)\geq\frac{e^{\lambda_{u}t}}{C}\|D\varphi^{t}|_{T_{p}M}\|^{i}$
(2.2)

for all $p\in M$, $t\geq 0$, and $0\leq i\leq r$. We shall call $M$ an
($r$-)normally hyperbolic invariant manifold, or ($r$-)NHIM for short. Now
$\oplus$ in (2.1) denotes the Whitney sum of vector bundles and

$m(A):=\inf\left\\{\|Av\|\mid\|v\|=1\right\\}$

is the minimum norm444 If $A$ in invertible, then $m(A)=\|A^{-1}\|^{-1}$.  of
a matrix $A$. We particularly say a NHIM $M$ being normally attracting ($M$
being a NAIM for short) if $E^{u}$ is the zero bundle; $E^{u}=\\{0\\}$.

Denote $n_{T},n_{u},n_{s}$ with $n=n_{T}+n_{u}+n_{s}$ by the rank ($=$
dimension of fibers) of $TM$, $E^{u}$ and $E^{s}$, respectively. In
particular, $TM$, $E^{u}$ and $E^{s}$ are vector bundles with the base space
$M$ and $n_{T}$-, $n_{u}$\- and $n_{s}$-dimensional fibers. We shall call
$E^{u}$ and $E^{s}$ the unstable bundle and the stable bundle of $M$,
respectively, while $TM$ is the tangent bundle of $M$. When $M$ is a NAIM,
then $n_{u}=0$. The vector bundle structure of $E^{s}$ is written by
$\xi^{s}=(\pi_{s},E^{s},F^{s})$ with

$E^{s}=\bigcup_{p\in M}F^{s}_{p},\quad F^{s}\cong
F^{s}_{p}\cong\mathbb{R}^{n_{s}}$

for all $p\in M$. A similar description holds for $E^{u}$. Several properties
which NHIMs possess are summarized below.

######  Proposition 2.3 (e.g., [20, 43, 44, 54]).

Let $M\subset\mathbb{R}^{n}$ be a boundaryless, compact NHIM for
$\varphi_{f}$.

  1. 1.

(Invariant foliation). There is a local stable manifold $W^{s}_{\rm loc}(M)$
tangent to the Whitney sum $TM\oplus E^{s}$ at $M$, which is a
$C^{r}$-embedded submanifold of $\mathbb{R}^{n}$ and invariantly fibered by
embedded ($n_{s}$-dimensional) disks $D_{p}^{s}$, denoted by $W^{s}_{\rm
loc}(p)$ for each $p\in M$, constructing fibers of the local stable invariant
foliation

$W^{s}_{\rm loc}(M)(=W^{s}_{\rm loc}(M;\varphi))=\bigcup_{p\in M}W^{s}_{\rm
loc}(p).$

In particular, $W^{s}_{\rm loc}(p)$ is an invariant manifold satisfying
$T_{p}W^{s}_{\rm loc}(p)=E_{p}^{s}$ for each $p\in M$.

  2. 2.

(Exponential decay along fibers). There exist $C_{s}>0$ and $\lambda_{s}<0$
such that, for each $p\in M$, the following holds for all $t\geq 0$ whenever
$q\in W^{s}_{\rm loc}(p)$:

$\|\varphi_{f}(t,q)-\varphi_{f}(t,p)\|<C_{s}e^{\lambda_{s}t}.$ (2.3)

  3. 3.

(Linearization). $\varphi_{f}$ is topologically conjugate to $N\varphi_{f}$
near $M$, namely there are neighborhoods $U,U^{\prime}$ of $M$ and the zero
section $Z(M)$ of the bundle $\mathcal{N}=E^{u}\oplus E^{s}$, respectively in
$\mathbb{R}^{n}$, and a homeomorphism $h:U\to U^{\prime}$ such that
$N\varphi_{f}^{t}\circ h=h\circ\varphi_{f}^{t}$ in $U$ for all
$t\in\mathbb{R}$ if trajectories are inside corresponding neighborhoods, where
$N\varphi_{f}^{t}=(\varphi_{f}^{t}|_{Z(M)},D\varphi_{f}^{t}|_{E^{u}\oplus
E^{s}})$ denotes the (linear skew-product) flow555 The definition is left to
Appendix B because it is argued only when technical details in our objectives
are discussed.  on $\mathcal{N}$. The flow $\varphi_{f}^{t}|_{Z(M)}$ is
identified with $\varphi_{f}^{t}|_{M}$ through the identity $Z(M)\cong M$.
Because $M$ is assumed to be compact, taking $U$ and $U^{\prime}$ smaller if
necessary, both $h$ and $h^{-1}$ may be assumed to be bounded on $U$ and
$U^{\prime}$, respectively.

The theory of NHIMs is also established for manifolds with nonempty boundary.
We collect basic notions of normal hyperbolicity in such a case.

######  Definition 2.4 (Inflowing/overflowing invariant manifolds, e.g.,
[54]).

Suppose that $M$ is a compact manifold in an ambient manifold $Q$ possibly
with nonempty boundary: $\partial M\not=\emptyset$. We say that $M$ is
inflowing invariant for the vector field $f$ if $\varphi_{f}^{t}(M)\subset M$
for all $t\geq 0$ and if $f$ points strictly inward at $\partial M$.
Similarly, we say that $M$ is overflowing invariant for the vector field $f$
if $\varphi_{f}^{t}(M)\subset M$ for all $t\leq 0$ and if $f$ points strictly
outward at $\partial M$. If $M$ is inflowing/overflowing invariant with the
structures (2.1) and (2.2), we call $M$ inflowing/overflowing NHIM,
respectively666 If $\partial M=\emptyset$ and $M$ is invariant, then $M$ is
both overflowing and inflowing invariant. But if $\partial M\not=\emptyset$,
$M$ being invariant does not imply that $M$ is both overflowing and inflowing
invariant. .

It is proved that inflowing invariant manifolds admit invariant foliations of
local stable manifolds, while overflowing invariant manifolds admit invariant
foliations of local unstable manifolds (e.g., [10, 54]).

Finally we shall make a remark on invariant foliations; shadowing property for
NHIMs. Even if an invariant manifold is not normally hyperbolic, there are
several cases which it admits the similar property.

######  Remark 2.5 (Asymptotic phase, cf. [36, 37]).

Let $X=(X,d)$ be a metric space and $\varphi$ be a flow on $X$. Suppose that
there is an attracting invariant set $A$ with a basin $B$:

$\lim_{t\to\infty}d(\varphi^{t}({\bf x}),A)=0,\quad\forall{\bf x}\in B.$

It is said that a mapping $t\mapsto\varphi^{t}({\bf x})$, ${\bf x}\in B$, has
an asymptotic phase if there is ${\bf x}_{\ast}\in A$ such that

$\lim_{t\to\infty}d(\varphi^{t}({\bf x}),\varphi^{t}({\bf x}_{\ast}))=0.$

Stable invariant foliation for a NHIM $M$ is a special case of asymptotic
phase: $A=M$ and $B=W^{s}_{\rm loc}(M)$. Asymptotic phase beyond normal
hyperbolicity is constructed in several studies. See e.g., [36, 37] for a
brief summary of historical backgrounds in asymptotic phase and its existence
in a generalized setting.

##  3 Preliminaries 2: dynamics at infinity

In this section we provide tools and methodologies to consider divergent
solutions in terms of global-in-time trajectories converging to invariant sets
for appropriately transformed vector fields as general as possible.

###  3.1 Asymptotically quasi-homogeneous systems

First of all, we collect our present settings precisely, which are based on
[38].

######  Definition 3.1 (Homogeneity index and admissible domain).

Let $\alpha=(\alpha_{1},\cdots,\alpha_{n})$ be a set of nonnegative integers.
We say the index set $I_{\alpha}=\\{i\in\\{1,\cdots,n\\}\mid\alpha_{i}>0\\}$
the set of homogeneity indices associated with
$\alpha=(\alpha_{1},\cdots,\alpha_{n})$. Let $U\subset\mathbb{R}^{n}$. We say
the domain $U\subset\mathbb{R}^{n}$ admissible with respect to the sequence
$\alpha$ if

$U=\left\\{{\bf x}=(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}\mid
x_{i}\in\mathbb{R}\text{ if }i\in I_{\alpha},\
(x_{j_{1}},\cdots,x_{j_{n-l}})\in\tilde{U}\right\\},$

where $\\{j_{1},\cdots,j_{n-l}\\}=\\{1,\cdots,n\\}\setminus I_{\alpha}$ and
$\tilde{U}$ is an open set in $\mathbb{R}^{n-l}$ spanning
$(x_{j_{1}},\cdots,x_{j_{n-l}})$ with
$\\{j_{1},\cdots,j_{n-l}\\}=\\{1,\cdots,n\\}\setminus I_{\alpha}$.

######  Definition 3.2 (Asymptotically quasi-homogeneous vector fields, cf.
[6, 38]).

Let $U$ be an admissible set $U\subset\mathbb{R}^{n}$ with respect to
$\alpha$. Also, let $f_{0}:U\to\mathbb{R}$ be a function. Let
$\alpha_{1},\ldots,\alpha_{n}$ be nonnegative integers with
$(\alpha_{1},\ldots,\alpha_{n})\not=(0,\ldots,0)$ and $k>0$. We say that
$f_{0}$ is a quasi-homogeneous function777 In preceding studies, all
$\alpha_{i}$’s and $k$ are typically assumed to be natural numbers. In the
present study, on the other hand, the above generalization is valid.  of type
$\alpha=(\alpha_{1},\ldots,\alpha_{n})$ and order $k$ if

$f_{0}(s^{\Lambda_{\alpha}}{\bf x})=s^{k}f_{0}({\bf x})\quad\text{ for all
}{\bf x}=(x_{1},\ldots,x_{n})^{T}\in U\text{ and }s>0,$

where888 Throughout the rest of this paper, the power of real positive numbers
or functions to matrices is described in the similar manner.

$\Lambda_{\alpha}={\rm diag}\left(\alpha_{1},\ldots,\alpha_{n}\right),\quad
s^{\Lambda_{\alpha}}{\bf
x}=(s^{\alpha_{1}}x_{1},\ldots,s^{\alpha_{n}}x_{n})^{T}.$

Next, let $X=\sum_{i=1}^{n}f_{i}({\bf x})\frac{\partial}{\partial x_{i}}$ be a
continuous vector field defined on $U$. We say that $X$, or simply
$f=(f_{1},\ldots,f_{n})^{T}$ is a quasi-homogeneous vector field of type
$\alpha=(\alpha_{1},\ldots,\alpha_{n})$ and order $k+1$ if each component
$f_{i}$ is a quasi-homogeneous function of type $\alpha$ and order
$k+\alpha_{i}$.

Finally, we say that $X=\sum_{i=1}^{n}f_{i}({\bf x})\frac{\partial}{\partial
x_{i}}$, or simply $f:U\to\mathbb{R}^{n}$ is an asymptotically quasi-
homogeneous vector field of type $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ and
order $k+1$ (at infinity) if there is a quasi-homogeneous vector field
$f_{\alpha,k}=(f_{i;\alpha,k})_{i=1}^{n}$ of type $\alpha$ and order $k+1$
such that

$f_{i}(s^{\Lambda_{\alpha}}{\bf x})-s^{k+\alpha_{i}}f_{i;\alpha,k}({\bf
x})=o(s^{k+\alpha_{i}}),\quad i\in\\{1,\ldots,n\\},$ (3.1)

as $s\to+\infty$ uniformly on $\left\\{{\bf x}\in U\mid\sum_{i\in
I_{\alpha}}x_{i}^{2}=1,(x_{j_{1}},\cdots,x_{j_{n-l}})\in\tilde{K}\right\\}$
for any compact subset $K\subset\tilde{U}$.

######  Remark 3.3.

In the above definition, non-polynomial-like functions such as $\sin x$ are
not included to characterize quasi-homogeneity. Indeed, such functions are
allowed to exist only in the residual terms characterized by the asymptotic
quasi-homogeneity (3.1). On the other hand, (3.1) is required for all
$i\in\\{1,\ldots,n\\}$.

A fundamental property of quasi-homogeneous functions and vector fields is
reviewed in e.g. [2]. Throughout the rest of this section, consider an
(autonomous) $C^{r}$ vector field (1.1) with $r\geq 1$, where
$f:U\to\mathbb{R}^{n}$ is asymptotically quasi-homogeneous of type
$\alpha=(\alpha_{1},\ldots,\alpha_{n})$ and order $k+1$ at infinity defined on
an admissible set $U\subset\mathbb{R}^{n}$ with respect to $\alpha$.

###  3.2 Embedding of phase spaces and dynamics at infinity

Our process to describe blow-up solutions from the viewpoint of dynamical
systems is summarized below.

  1. 1.

For given (asymptotically quasi-homogeneous) vector field $f$, determine its
type $\alpha$ and order $k+1$. See Section 5 for the treatment of $t$ in
nonautonomous systems.

  2. 2.

Apply an embedding of phase spaces with the type $\alpha$, precisely defined
below, and transform $f$ through this embedding.

  3. 3.

Introduce the time-scale transformation determined by the order $k+1$ of $f$
to desingularize the transformed vector field at infinity.

The above process is widely used in autonomous systems (cf. [21, 22, 23, 24,
25, 26, 35, 38, 39, 40, 52]) and we follow this strategy in the present
argument as well as nonautonomous systems later.

######  Remark 3.4.

In preceding studies, embeddings we would review below were referred to as
compactifications. On the other hand, because the embedded manifolds involving
our interests here are typically non-compact, including the case of
nonautonomous systems, we shall use the terminology “embeddings” instead of
“compactifications”.

In the following arguments in this section, we fix a type
$\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{Z}_{\geq
0}^{n}\setminus\\{(0,\ldots,0)\\}$, and let $I_{\alpha}$ be the associated set
of homogeneity indices. Also, let $U\subset\mathbb{R}^{n}$ be an admissible
set with respect to $\alpha$. A “global-type” embedding and the associated
vector field suitable for describing dynamics at infinity are introduced here.
Another example of embeddings is shown in Appendix A.

######  Definition 3.5 (Quasi-parabolic embedding, cf. [40]).

Let $\\{\beta_{i}\\}_{i\in I_{\alpha}}$ be the collection of natural numbers
so that

$\alpha_{i}\beta_{i}\equiv c\in\mathbb{N},\quad i\in I_{\alpha}$ (3.2)

is the least common multiplier. In particular, $\\{\beta_{i}\\}_{i\in
I_{\alpha}}$ is chosen to be the smallest among possible collections. Let
$p({\bf y})$ be a functional given by

$p({\bf y})\equiv\left(\sum_{i\in
I_{\alpha}}y_{i}^{2\beta_{i}}\right)^{1/2c}.$ (3.3)

Define the mapping $T_{{\rm para};\alpha}:U\to\mathbb{R}^{n}$ as the inverse
of

$S_{{\rm para};\alpha}({\bf x})={\bf y},\quad
y_{j}=\kappa^{\alpha_{j}}x_{j},\quad j=1,\ldots,n,$ (3.4)

where

$\kappa=\kappa({\bf x})=(1-p({\bf x})^{2c})^{-1}\equiv\left(1-\sum_{j\in
I_{\alpha}}x_{j}^{2\beta_{j}}\right)^{-1}.$

We say the mapping $T_{{\rm para};\alpha}$ the quasi-parabolic embedding (with
type $\alpha$).

######  Remark 3.6.

The functional $\kappa=\tilde{\kappa}({\bf y})$ as a functional determined by
${\bf y}$ is implicitly determined by $p({\bf y})$. Details of such a
characterization of $\kappa$ in terms of ${\bf y}$, and the bijectivity and
smoothness of $T_{{\rm para};\alpha}$ are shown in [40] with a general class
of embeddings including quasi-parabolic compactifications (embeddings).

The above definition is originally introduced in the case
$I_{\alpha}=\\{1,\ldots,n\\}$. It is nevertheless valid for the general case,
$I_{\alpha}\not=\\{1,\ldots,n\\}$, to characterize infinity suitable for the
behavior of $f$. It should be however noted that $T_{{\rm para};\alpha}$ and
its inverse $S_{{\rm para};\alpha}$ are defined for all
$j\in\\{1,\ldots,n\\}$, not only $j\in I_{\alpha}$.

As proved in [40], $T_{{\rm para};\alpha}$ maps $U$ one-to-one onto the set
$\mathcal{D}\equiv\\{{\bf x}\in U\mid p({\bf x})<1\\}$. Infinity in the
original coordinate then corresponds to a point on the level set of $p$:

$\mathcal{E}=\\{{\bf x}\in U\mid p({\bf x})=1\\}.$

######  Definition 3.7.

We call the set $\mathcal{E}$ the horizon.

Once we fix an embedding associated with the type
$\alpha=(\alpha_{1},\ldots,\alpha_{n})$ of the vector field $f$ with order
$k+1$, we can derive the vector field which makes sense including the horizon.
Then the dynamics at infinity makes sense through the appropriately
transformed vector field called the desingularized vector field, denoted by
$g$. The common approach is twofold. Firstly, we rewrite the vector field
(1.1) with respect to the new variable defined by embeddings. Secondly, we
introduce the time-scale transformation of the form $d\tau=q({\bf
x})\kappa({\bf x}(t))^{k}dt$ for some function $q({\bf x})$ which is bounded
including the horizon. We then obtain the vector field with respect to the new
time variable $\tau$, which is continuous, including the horizon.

######  Remark 3.8.

Continuity of the desingularized vector field $g$ including the horizon is
guaranteed by the smoothness of $f$ and asymptotic quasi-homogeneity ([38]).
In the case of parabolic-type embeddings introduced here, $g$ inherits the
smoothness of $f$ including the horizon, which is not always the case of other
embeddings in general. Details are discussed in [38].

######  Definition 3.9 (Time-scale desingularization).

Define the new time variable $\tau$ by

$d\tau=q({\bf x})^{-1}(1-p({\bf x})^{2c})^{-k}dt,\quad q({\bf
x})=1-\frac{2c-1}{2c}(1-p({\bf x})^{2c}),$ (3.5)

equivalently

$t-t_{0}=\int_{\tau_{0}}^{\tau}q({\bf x}(\tau))(1-p({\bf
x}(\tau))^{2c})^{k}d\tau,$

where $\tau_{0}$ and $t_{0}$ denote the correspondence of initial times, ${\bf
x}(\tau)=T_{{\rm para};\alpha}({\bf y}(\tau))$ and ${\bf y}(\tau)$ is a
solution ${\bf y}(t)$ under the parameter $\tau$. We shall call (3.5) the
time-scale desingularization of order $k+1$.

The change of coordinate and the above desingularization yield the following
vector field $g=(g_{1},\ldots,g_{n})^{T}$, which is continuous on
$\widetilde{\mathcal{D}}\equiv\mathcal{D}\cup\mathcal{E}$:

$\displaystyle\dot{x}_{i}\equiv\frac{dx_{i}}{d\tau}=g_{i}({\bf x})=q({\bf
x})\left\\{\tilde{f}_{i}({\bf
x})-\alpha_{i}x_{i}\sum_{j=1}^{n}(\nabla\kappa)_{j}\kappa^{\alpha_{j}-1}\tilde{f}_{j}({\bf
x})\right\\},$

where

$\tilde{f}_{j}({\bf
x}):=\kappa^{-(k+\alpha_{j})}f_{j}(\kappa^{\Lambda_{\alpha}}{\bf x}),\quad
j=1,\ldots,n,$ (3.6)

and $\nabla\kappa=\nabla_{\bf x}\kappa=((\nabla_{\bf
x}\kappa)_{1},\ldots,(\nabla_{\bf x}\kappa)_{n})^{T}$ is

$(\nabla_{\bf
x}\kappa)_{j}=\begin{cases}\frac{\kappa^{1-\alpha_{j}}x_{j}^{2\beta_{j}-1}}{\alpha_{j}q({\bf
x})}&j\in I_{\alpha},\\\ 0&\text{otherwise},\end{cases}$

as derived in [40]. In particular, the vector field $g$ and the differential
equation is simply written as follows:

$\displaystyle\dot{\bf x}=g({\bf x})=q({\bf x})\tilde{f}({\bf x})-G({\bf
x})\Lambda_{\alpha}{\bf x},$ (3.7)

where $\tilde{f}=(\tilde{f}_{1},\ldots,\tilde{f}_{n})^{T}$ and

$\displaystyle G({\bf x})$ $\displaystyle\equiv\sum_{j\in
I_{\alpha}}\frac{x_{j}^{2\beta_{j}-1}}{\alpha_{j}}\tilde{f}_{j}({\bf x}).$
(3.8)

Smoothness of $f$ and its asymptotic quasi-homogeneity guarantee the
smoothness of the right-hand side $g$ of (3.7) including the horizon
$\mathcal{E}\equiv\\{p({\bf x})=1\\}$. In particular, dynamics at infinity,
such as divergence of solutions to specific directions, is characterized
through dynamics generated by (3.7) around the horizon. See [38, 40] for
details.

######  Remark 3.10 (Invariant structure).

The horizon $\mathcal{E}$ is invariant in $\widetilde{\mathcal{D}}$. Indeed,
direct calculations yield that

$\left.\frac{d}{d\tau}p({\bf x}(\tau))^{2c}\right|_{\tau=0}=0\quad\text{
whenever }\quad{\bf x}(0)\in\mathcal{E}.$

In particular, $\mathcal{E}$ is a codimension-one invariant submanifold of
$\widetilde{\mathcal{D}}$ if $I_{\alpha}=\\{1,\ldots,n\\}$. See e.g. [38],
where detailed calculations are shown in a similar type of global embeddings.
We shall apply this invariant structure to extracting the detailed blow-up
structure later.

##  4 Normally hyperbolic blow-up

In the present and the next sections, we state the main results: description
of blow-up solutions by means of solutions shadowing “trajectories at
infinity”. Key features of blow-up description we shall show here are that,
for the desingularized vector field, the convergence to a trajectory on the
horizon at a suitably fast rate provides finite-time blow-up, and that the
asymptotic blow-up profile is described by the shadowed trajectory, which are
extensions of blow-up characterization derived in preceding works (e.g.,
[38]). The shadowing property is geometrically described by asymptotic phase,
which is observed in invariant manifolds with “hyperbolic” property. A
suitable class of such manifolds is NHIMs, which yields that NHIMs on the
horizon induce blow-up solutions.

Another issue in blow-up description is the magnitude of blow-ups referred to
as blow-up rates. In the present argument, we assume convergence to shadowed
trajectories with specific rates, which is stronger than typical convergence
properties observed in asymptotic phase. These convergence rates can relate to
spectral properties of associated linearized systems, which are left to
Appendix B, where the correspondence by means of dichotomy spectrum
established in the theory of linear differential systems (e.g., [45, 46]) is
discussed.

Now let $f:U\to\mathbb{R}^{n}$ be a smooth (at least $C^{1}$) asymptotically
quasi-homogeneous vector field of type $\alpha=(\alpha_{1},\ldots,\alpha_{n})$
associated with the set of homogeneity indices $I_{\alpha}$ with an admissible
domain $U\subset\mathbb{R}^{n}$, and order $k+1$. Also, let $g$ be the
desingularized vector field (3.7) associated with $f$ and the quasi-parabolic
embedding $T_{{\rm para};\alpha}$ of type $\alpha$, and $\varphi_{g}$ be the
flow generated by $g$. The result is also valid for directional embeddings
summarized in Appendix A, and possibly other ones with suitable assumptions999
As far as we understand, other possible choices of compactificaitons are
Poincaré-type ones (e.g., [38]) and Poincaré-Lyapunov disks (e.g., [7]).
Roughly speaking, the former is the embedding of $\mathbb{R}^{n}$ into the
hemisphere in $\mathbb{R}^{n+1}$, and the latter is the change of variables
with quasi-polar coordinates. .

First we provide a general description of blow-up solutions by means of
solutions shadowing “trajectories at infinity”.

######  Theorem 4.1 (Blow-up: a general description).

Suppose that $g$ admits a trajectory $\gamma=\\{{\bf
x}_{\gamma}(\tau)\\}_{\tau\in\mathbb{R}}\subset\mathcal{E}$ such that, for
some ${\bf y}_{0}\in U$ and the solution ${\bf y}(\tau)$ through ${\bf y}_{0}$
at $\tau=0$,

$\displaystyle\|T_{{\rm para};\alpha}({\bf y}(\tau))-{\bf
x}_{\gamma}(\tau)\|<Ce^{-\lambda\tau}$ (4.1)

for constants $C>0$ and $\lambda>0$. Then $t_{\max}<\infty$ holds; namely,
${\bf y}(t)$ is a blow-up solution.

If we further assume

$\lim_{\tau\to+\infty}e^{(\lambda+\epsilon)\tau}\|T_{{\rm para};\alpha}({\bf
y}(\tau))-{\bf x}_{M}(\tau)\|=\begin{cases}+\infty&\epsilon>0\\\
0&\epsilon<0\end{cases},$ (4.2)

then

$\displaystyle p({\bf y}(t))\sim
C_{0}(-\ln(t_{\max}-t))(t_{\max}-t)^{-1/k}\quad\text{ as }\quad t\to
t_{\max}-0$

for some function $C_{0}(-\ln(t_{\max}-t))$ satisfying

$C_{0}(\tau)=o(e^{\epsilon\tau}),\quad C_{0}(\tau)^{-1}=o(e^{\epsilon\tau})$

for any $\epsilon>0$ as $\tau\to\infty$, equivalently
$-\ln(t_{\max}-t)\to+\infty$. Moreover, we have

$y_{i}(t)\sim
C_{0}(-\ln(t_{\max}-t))^{\alpha_{i}}x_{\gamma,i}(-\bar{c}\ln(t_{\max}-t))(t_{\max}-t)^{-\alpha_{i}/k}\quad\text{
as }\quad t\to t_{\max}-0$

for some constant $\bar{c}>0$, provided $x_{\gamma,i}(\tau)\not\to 0$ as
$\tau\to\infty$.

######  Remark 4.2.

The asymptotic behavior (4.2) covers the cases

$\lim_{\tau\to+\infty}\frac{\|T_{{\rm para};\alpha}({\bf y}(\tau))-{\bf
x}_{M}(\tau)\|}{e^{-\lambda\tau}\tau^{m}}=c_{0}>0,$

for some integer $m$, or

$0\leq c_{1}\leq\frac{\|T_{{\rm para};\alpha}({\bf y}(\tau))-{\bf
x}_{M}(\tau)\|}{e^{-\lambda\tau}\tau^{m}}<c_{2}$

for all sufficiently large $\tau>0$.

###### Proof.

Let ${\bf x}(\tau)=T_{{\rm para};\alpha}({\bf y}(\tau))$ in the
$\tau$-timescale. The maximal existence time $t_{\max}$ for ${\bf y}(t)$ in
the $t$-timescale is then estimated through (3.5) as follows:

$\displaystyle t_{\max}$ $\displaystyle\equiv\int_{0}^{\infty}q({\bf
x}(\tau))(1-p({\bf x}(\tau))^{2c})^{k}d\tau\leq\int_{0}^{\infty}\\{1-p({\bf
x}(\tau))^{2c}\\}^{k}d\tau$

from the fact that $q({\bf x})\leq 1$. On the other hand, because ${\bf
x}(\tau)$ shadows ${\bf x}_{\gamma}(\tau)$ in the sense of (4.1), the quantity
$1-p({\bf x}(\tau))^{2c}$ can be rewritten as

$\displaystyle 1-p({\bf x}(\tau))^{2c}$ $\displaystyle=1-\sum_{i\in
I_{\alpha}}x_{i}(\tau)^{2\beta_{i}}$ $\displaystyle=\sum_{i\in
I_{\alpha}}x_{\gamma,i}(\tau)^{2\beta_{i}}-\sum_{i\in
I_{\alpha}}x_{i}(\tau)^{2\beta_{i}}$ $\displaystyle=\sum_{i\in
I_{\alpha}}(x_{\gamma,i}(\tau)^{2}-x_{i}(\tau)^{2})\left\\{\sum_{j=0}^{\beta_{i}-1}x_{\gamma,i}(\tau)^{2(\beta_{i}-1-j)}x_{i}(\tau)^{2j}\right\\}$
$\displaystyle=\sum_{i\in
I_{\alpha}}(x_{\gamma,i}(\tau)-x_{i}(\tau))(x_{\gamma,i}(\tau)+x_{i}(\tau))\left\\{\sum_{j=0}^{\beta_{i}-1}x_{\gamma,i}(\tau)^{2(\beta_{i}-1-j)}x_{i}(\tau)^{2j}\right\\}$
(4.3)

and, using the inequality

$\left\|(x_{\gamma,i}(\tau)-x_{i}(\tau))(x_{\gamma,i}(\tau)+x_{i}(\tau))\left\\{\sum_{j=0}^{\beta_{i}-1}x_{\gamma,i}(\tau)^{2(\beta_{i}-1-j)}x_{i}(\tau)^{2j}\right\\}\right\|\leq
C_{i}e^{\lambda\tau}$

obtained from (4.1) for some constant101010 Because only components $x_{i}$
and $x_{\gamma,i}$ with $i\in I_{\alpha}$ are considered, we have
$|x_{i}(\tau)|,|x_{\gamma,i}(\tau)|\leq 1$ for all $\tau\geq 0$.  $C_{i}>0$,
we have

$\displaystyle t_{\max}$ $\displaystyle\leq\int_{0}^{\infty}\\{1-p({\bf
x}(\tau))^{2c}\\}^{k}d\tau\leq\tilde{C}\int_{0}^{\infty}e^{k\lambda\tau}d\tau<\infty$

for some constant $\tilde{C}>0$. In particular, ${\bf y}(t)$ in the original
$t$-timescale is a blow-up solution.

Next we assume (4.2). For $t$ less than but sufficiently close to $t_{\max}$,
we have

$t_{\max}-t=\int_{\tau}^{\infty}q({\bf x}(\tilde{\tau}))(1-p({\bf
x}(\tilde{\tau}))^{2c})^{k}d\tilde{\tau}$

with sufficiently large $\tau$, and the asymptotic expression of the right-
hand side is our interest here. Here estimate the functional
$\kappa^{-1}=1-p({\bf x}(\tau))^{2c}$. Using (4.3), the functional is written
as follows:

$\displaystyle 1-p({\bf x}(\tau))^{2c}$ $\displaystyle=\sum_{i\in
I_{\alpha}}(x_{\gamma,i}(\tau)-x_{i}(\tau))(x_{\gamma,i}(\tau)+x_{i}(\tau))\left\\{\sum_{j=0}^{\beta_{i}-1}x_{\gamma,i}(\tau)^{2(\beta_{i}-1-j)}x_{i}(\tau)^{2j}\right\\}$
$\displaystyle=e^{-\lambda\tau}\bar{F}(\tau),$

where $\bar{F}(\tau)=o(e^{\epsilon\tau})$ and
$\bar{F}(\tau)^{-1}=o(e^{\epsilon\tau})$ for any $\epsilon>0$ as
$\tau\to\infty$. Therefore we have

$\frac{1}{k}\ln(t_{\max}-t)=-\lambda\tau+\tilde{F}(\tau)$

as $\tau\to\infty$, where $\tilde{F}(\tau)=o(\tau)$ and
$\tilde{F}(\tau)^{-1}=o(\tau)$ as $\tau\to\infty$. The above estimates are
combined to obtain

$p({\bf y}(t))=\left\\{1-\sum_{i\in
I_{\alpha}}x_{i}(t)^{2\beta_{i}}\right\\}^{-1}\left(\sum_{i\in
I_{\alpha}}x_{i}(t)^{2\beta_{i}}\right)^{1/2c}\sim
C_{0}(-\ln(t_{\max}-t))(t_{\max}-t)^{-1/k}$

as $t\to t_{\max}$, and the proof involving the asymptotic behavior of $p({\bf
y}(t))$ is completed. The last statement follows from the definition of
$T_{{\rm para};\alpha}$ (Definition 3.5) and asymptotic relations we have
obtained. ∎

A usual way to verify assumptions, in particular (4.1), is to construct a
collection of trajectories on the horizon guaranteeing the shadowing property
in advance. One specific characterization of such a property is asymptotic
phase. Once one can construct invariant sets admitting asymptotic phase with
appropriate convergence rate, assumptions involving (4.1) are satisfied and
blow-up solutions can be described by means of invariant manifolds for $g$ on
the horizon. For example, when NHIMs on the horizon are constructed, our
description is summarized as follows, which shall be called normally
hyperbolic blow-up. Note that other characterizations of asymptotic phase (cf.
Remark 2.5) yield the similar results.

######  Corollary 4.3 (Normally hyperbolic blow-up).

Suppose that $g$ admits a compact connected NHIM $M\subset\mathcal{E}$ with
$\partial M=\emptyset$. If the solution ${\bf y}(t)$ of (1.1) with a bounded
initial point ${\bf y}_{0}\in\mathbb{R}^{n}$ whose image ${\bf
x}(\tau)=T_{{\rm para};\alpha}({\bf y}(\tau))$ is on $W_{\rm
loc}^{s}(M;\varphi_{g})$, then $t_{\max}<\infty$ holds; namely, ${\bf y}(t)$
is a blow-up solution. Moreover, there is a trajectory $\gamma=\\{{\bf
x}_{\gamma}(\tau)\\}_{\tau\geq 0}$ for (3.7) on $M\subset\mathcal{E}$ such
that (4.1) holds.

###### Proof.

Thanks to the stable foliation of $W_{\rm loc}^{s}(M)\equiv W_{\rm
loc}^{s}(M;\varphi_{g})$, there is a trajectory $\gamma=\\{{\bf
x}_{\gamma}(\tau)\\}_{\tau\geq 0}\subset M$ such that ${\bf x}(0)=T_{{\rm
para};\alpha}({\bf y}_{0})\in W_{\rm loc}^{s}({\bf x}_{\gamma}(0))$ and ${\bf
x}(\tau)=T_{{\rm para};\alpha}({\bf y}(\tau))\in W_{\rm loc}^{s}({\bf
x}_{\gamma}(\tau))$ for all $\tau\geq 0$. The decay estimate (4.1) is the
direct consequence of (2.3). ∎

As mentioned in the beginning, description of blow-up rates requires more
precise information about convergence rates. Details are left to Appendix B.

##  5 Nonautonomous blow-up

Next we describe blow-up solutions in the nonautonomous system

${\bf y}^{\prime}=f(t,{\bf y}),\quad{\bf y}(t_{0})={\bf y}_{0}\in U$ (5.1)

with $t_{0}\in\mathbb{R}$, an open set $U\subset\mathbb{R}^{n}$ and a
$C^{r}$-mapping $f:\mathbb{R}\times U\to\mathbb{R}^{n}$ with $r\geq 1$. While
the essential idea to describe blow-up solutions is the same as the autonomous
case, namely embeddings of phase spaces, we need to treat nonautonomous terms
appropriately. First, we have to consider a “natural scaling of time $t$” so
that dynamics at infinity can be considered in the similar way to autonomous
cases. Second, a natural determination of “invariant sets at infinity” for
nonautonomous systems is necessary so that the similar ideas to autonomous
systems can be applied. We shall see later that NHIMs will be an appropriate
class to describe blow-up solutions in nonautonomous systems.

In the present section, we first propose a natural treatment of the time
variable $t$ in our consideration of dynamics at infinity based on extended
autonomous systems, and provide a description of blow-up solutions in a
general setting then. To apply the description to e.g., NHIMs, a geometric
technique is introduced so that invariant manifolds with boundary can be
treated within the framework of compact, boundaryless invariant manifolds, in
particular NHIMs. Details of the technique are stated in Section 5.3 and
Appendix C.3. With the help of this technique, blow-ups in the extended
autonomous systems can be characterized in the similar way to Section 4.

###  5.1 Desingularized vector fields for nonautonomous systems

One natural treatment to consider (5.1) is to regard the time variable $t$ as
another independent variable depending on an extra variable $\eta$, namely
regard (5.1) as the extended autonomous system:

$\frac{d{\bf y}}{d\eta}=f(t,{\bf
y}),\quad\frac{dt}{d\eta}=1,\quad(t(\eta_{0}),{\bf y}(\eta_{0}))=(t_{0},{\bf
y}_{0})\in\mathbb{R}\times U.$ (5.2)

We pay attention to the scaling of the “time” variable $t$ so that the
extended system is regarded as an asymptotically quasi-homogenenous system.
Because the motion of time does not originally change as solution evolve, it
is natural to regard the scaling of $t$ as

$t=\kappa^{0}\tilde{t}=\tilde{t},$ (5.3)

indicating that the time-variable is not scaled depending on the location of
trajectories. Once this regulation is determined, the machinery provided in
Section 3 is applied to (5.2) towards a blow-up description in a similar way
to autonomous systems. We shall describe the desingularized vector field
associated with (5.1) under the assertion (5.3) with the quasi-parabolic
embedding. Application of directional embeddings (Appendix A) is derived in
the similar way.

Throughout the rest of this section, given a system (5.1), assume that the
extended vector field is asymptotically quasi-homogeneous of type
$\alpha=(0,\alpha_{1},\ldots,\alpha_{n})$ and order $k+1$ such that the set
$\mathbb{R}\times U$ is admissible with respect to $\alpha$. Here note that
the exponent $0$ in the first component of $\alpha$ stems from the scaling
(5.3). In particular, the associated set of homogeneity indices $I_{\alpha}$
is necessarily a proper subset of $\\{0,1,\ldots,n\\}$. We then apply the
quasi-parabolic embedding in Definition 3.5. In the present case, the
functional $p$ in (3.3), written as $p(t,{\bf y})$, is independent of $t$ and
we shall identify $p(t,{\bf y})$ with $p({\bf y})$. Using this identification,
derive the desingularized vector field introducing

$d\tau=(1-p({\bf x})^{2c})^{-k}q({\bf x})^{-1}d\eta$ (5.4)

in a usual manner, where $q({\bf x})$ is given in (3.5). We then obtain the
corresponding desingularized vector field, which can be written by

$\frac{d}{d\tau}\begin{pmatrix}t\\\ {\bf x}\end{pmatrix}=g(t,{\bf x})\equiv
q({\bf x})\bar{f}(t,{\bf x})-G(t,{\bf
x})\bar{\Lambda}_{\alpha}\begin{pmatrix}t\\\ {\bf x}\end{pmatrix}$ (5.5)

with the following notations, which are consistent with the general derivation
of desingularized vector fields in autonomous systems:

$\displaystyle\bar{f}(t,{\bf x})$
$\displaystyle=\begin{pmatrix}\tilde{f}_{0},\tilde{f}_{1},\ldots,\tilde{f}_{n}\end{pmatrix}^{T}\quad\text{
with }\quad f_{0}(t,{\bf y})=1,$ $\displaystyle G(t,{\bf x})$
$\displaystyle\equiv\sum_{j\in
I_{\alpha}}\frac{x_{j}^{2\beta_{j}-1}}{\alpha_{j}}\tilde{f}_{j}(t,{\bf
x}),\quad\bar{\Lambda}_{\alpha}={\rm diag}(0,\alpha_{1},\ldots,\alpha_{n}).$

Note that, in the above notation, $\tilde{f}_{0}(t,{\bf x})=(1-p({\bf
x})^{2c})^{k}$ via (3.6). The above identifications are also consistent with
the evolution of $t$ followed by the time-scale desingularization (5.4).

###  5.2 Fundamental description of blow-ups

Like autonomous cases, geometric description of blow-ups for (5.2) is based on
the construction of invariant manifolds with “hyperbolic” structure on the
horizon for the desingularized vector field (5.5). On the other hand,
$dt/d\tau=0$ on the horizon $\mathcal{E}=\\{p({\bf x})=1\\}$, and invariant
sets on $\mathcal{E}$ in the present case will consist of slices of invariant
sets parameterized by the fixed variable $t$, the first component of phase
variables $(t,{\bf x})$. Therefore, even in the simplest case such as
“equilibria” on the horizon, they are not isolated. In this case, “hyperbolic
structure” will not be expected for invariant sets on the horizon. Instead,
normal, or partial hyperbolicity can be naturally considered regarding the
evolution in $t$-variable as the tangential direction. In particular, the
concept of normally hyperbolic blow-up stated in Section 4 will be naturally
applied in practical problems.

As in the previous section, we only consider quasi-parabolic embeddings. Fix
the corresponding embedding $T_{{\rm para};\alpha}$ associated with $f$. Let
$g$ be the desingularized vector field (5.5) associated with $f$ and $T_{{\rm
para};\alpha}$. Also, let $\varphi_{g}$ be the flow generated by $g$. Finally,
let $\pi_{\nu}$ be the projection onto the $\nu$-component, $\nu=t,{\bf x}$.

Here we describe blow-up solutions in a general setting. As seen in Section 4,
our blow-up description; existence and blow-up rates, relies on the existence
of asymptotic phase and precise exponential decay behavior of trajectory
shadowing invariant sets on the horizon for $g$. Because the desingularized
vector field $g(t,{\bf x})$ in (5.5) essentially has the same structure as $g$
in autonomous cases, (3.7), the similar calculations and estimates yield the
corresponding description of blow-ups.

######  Theorem 5.1 (Nonautonomous blow-up: a general description).

Suppose that $g$ admits a precompact trajectory $\gamma=\\{{\bf
x}_{\gamma}(\tau)\\}_{\tau\in\mathbb{R}}\subset\mathcal{E}$ such that, for
some $(t_{0},{\bf y}_{0})\in U$ and the solution ${\bf y}(\tau)$ through
$(t_{0},{\bf y}_{0})$ at $\tau=0$ corresponding $\tau=0$ to $t=t_{0}$,

$\displaystyle\lim_{\tau\to+\infty}\|\pi_{\bf x}T_{{\rm
para};\alpha}(t(\tau),{\bf y}(\tau))-{\bf x}_{\gamma}(\tau)\|=0.$ (5.6)

Then $t_{\max}<\infty$ holds; namely, ${\bf y}(t)$ is a blow-up solution.

If we further assume

$\displaystyle\lim_{\tau\to+\infty}e^{(\lambda+\epsilon)\tau}\|\pi_{\bf
x}T_{{\rm para};\alpha}(t(\tau),{\bf y}(\tau))-{\bf
x}_{\gamma}(\tau)\|=\begin{cases}+\infty&\epsilon>0\\\
0&\epsilon<0\end{cases}$ (5.7)

for a positive constant $\lambda>0$, then

$\displaystyle p({\bf y}(t))\sim
C_{0}(-\ln(t_{\max}-t))(t_{\max}-t)^{-1/k}\quad\text{ as }\quad t\to
t_{\max}-0$

for some function $C_{0}(-\ln(t_{\max}-t))$ satisfying

$C_{0}(\tau)=o(e^{\epsilon\tau}),\quad C_{0}(\tau)^{-1}=o(e^{\epsilon\tau})$

for any $\epsilon>0$ as $\tau\to\infty$, equivalently
$-\ln(t_{\max}-t)\to+\infty$. Moreover, we have

$y_{i}(t)\sim
C_{0}(-\ln(t_{\max}-t))^{\alpha_{i}}x_{\gamma,i}(-\bar{c}\ln(t_{\max}-t))(t_{\max}-t)^{-\alpha_{i}/k}\quad\text{
as }\quad t\to t_{\max}-0$

for some constant $\bar{c}>0$, provided $x_{\gamma,i}(\tau)\not\to 0$ as
$\tau\to\infty$.

###### Proof.

Observe that $dt/d\tau=0$ on the horizon $\mathcal{E}$. From the compactness
of the closure ${\rm cl}(\gamma)$ in $\mathcal{E}$, the trajectory $\gamma$ is
bounded. Hence $\gamma$ is included in $\mathcal{E}\cap\\{t=\bar{t}\\}$ for
some $\bar{t}$. Combining with the convergence (4.1), the maximal existence
time $t_{\max}$ of ${\bf y}(t)$ with ${\bf y}(t_{0})={\bf y}_{0}$ must be
equal to $\bar{t}<\infty$, in particular ${\bf y}(t)$ is a blow-up solution.
The remaining results follow from the same arguments as Theorem 4.1. ∎

In the nonautonomous setting, convergence rate of trajectories to shadowed
ones is not required because finiteness of $t_{\max}$ is automatically
achieved from the boundedness of $\gamma$ and the fact $dt/d\tau=0$ on
$\mathcal{E}$. The remaining requirements to describe blow-ups are similar to
those stated in e.g., Proposition B.7.

###  5.3 Blow-up descriptions for concrete problems

Here we address an intrinsic difficulty so that the first part of our result,
Theorem 5.1, can be applied and a technique to overcome it, which ensures the
application of the result to a wide range of nonautonomous systems. Theorem
5.1 essentially relies on the existence of asymptotic phase, which are
typically ensured for compact, boundaryless invariant manifolds. On the other
hand, as seen in simple cases (e.g., Section 6.1), constructed invariant
manifolds on $\mathcal{E}$ for the extended system is not compact in general.
Although several generalizations to noncompact invariant manifolds can be
applied (e.g., [9]), we notice that our attention is blow-up behavior occuring
in finite range of time $t$ (not $\tau$ !), and hence the restriction of
invariant manifolds to a compact segment of invariant manifolds is sufficient
to consider the concrete behavior of solutions.

We shall pay attention to NHIMs for simplicity, in which case a well-known
vector field modification technique on the boundaries can be applied to
constructing inflowing/overflowing invariant manifolds (see Definition 2.4 and
comments below). Its direct application provides invariant foliations,
yielding the existence of blow-up solutions in our problems. As provided
below, we can easily construct a modification to obtain both compact
inflowing/overflowing invariant manifolds depending on our requirements. We
shall only show a modification for constructing inflowing invariant manifolds.
Modification for constructing overflowing invariant manifolds is similar111111
Overflowing invariant manifolds on the horizon are applied to detecting blow-
up behavior in the reverse time direction. .

For any set $M\subset\widetilde{D}\equiv\mathcal{D}\cup\mathcal{E}$,
$\bar{t}\in\mathbb{R}$ and an interval $I\subset\mathbb{R}$, let

$M_{\bar{t}}\equiv M\cap\\{t=\bar{t}\\},\quad M_{I}\equiv M\cap\\{t\in I\\}$
(5.8)

be the slice and the tube of $M$ on $I$, respectively. First observe that,
when $M\subset\mathcal{E}$ is a compact invariant manifold for $g$ admitting
the normally hyperbolic structure, the slice $M_{\bar{t}}$ is a compact NHIM
in $\mathcal{E}\cap\\{t=\bar{t}\\}$ because the $dt/d\tau$ always vanishes on
$\mathcal{E}$ and hence $t$-evolution plays a role in (non-essential)
tangential evolution for $g$. Here we choose a collection of compact intervals
$I,I^{\prime}$ satisfying

$t_{0}\in I\subset{\rm int}\,I^{\prime}\subset I^{\prime},$ (5.9)

which are used to modify manifolds over a small interval in
$I^{\prime}\setminus I$ so that dynamics over $I$ is unchanged.

######  Assumption 5.2.

Fix an initial time $t_{0}\in\mathbb{R}$. There is a collection of compact
intervals $I,I^{\prime}$ satisfying (5.9) and an invariant manifold
$M\subset\mathcal{E}$, not necessarily $\partial M\not=\emptyset$, for $g$ in
(5.5) satisfying the following assertions:

  * •

$M_{I^{\prime}}$ admits the normally hyperbolic structure for $\varphi_{g}$,
in particular, there is a continuous splitting

$T_{M_{I^{\prime}}}\mathbb{R}^{n+1}=TM_{I^{\prime}}\oplus
E^{u}(M_{I^{\prime}})\oplus E^{s}(M_{I^{\prime}})$

and, for some constants $C\geq 1$ and $\mu<0<\lambda$,

$m(D\varphi_{g}|_{E^{u}(M_{I^{\prime}})})\geq\frac{e^{\lambda
t}}{C}\|D\varphi_{g}|_{TM_{I^{\prime}}}\|^{i},\quad\|D\varphi_{g}|_{E^{s}(M_{I^{\prime}})}\|\leq
Ce^{\mu t}m(D\varphi_{g}|_{TM_{I^{\prime}}})^{i}$

hold for all $t\geq 0$ and $0\leq i\leq r$.

  * •

For any $\bar{t}\in I^{\prime}$, the slice $M_{\bar{t}}$ is nonempty, compact,
connected and boundaryless. In particular, such a slice $M_{\bar{t}}$ is a
NHIM.

######  Remark 5.3.

Without the loss of generality, we may assume that $t_{0}\in I$ in the above
assumption. Indeed, our interest is asymptotic behavior of solutions ${\bf
u}(\tau)=(t(\tau),{\bf x}(\tau))$ for $g$ approaching to $\mathcal{E}$ as
$\tau\to\infty$. If $t_{0}=t(\tau_{0})\not\in I$, evolve the solution until
$\tau_{0}+\tilde{\tau}$ so that $t(\tau_{0}+\bar{\tau})\in I$ and replacing
$t_{0}$ by $t(\tau_{0}+\bar{\tau})$, which is the case of our assumption.

In Assumption 5.2, we have a compact (normally hyperbolic) invariant manifold
$M_{I^{\prime}}\subset\mathcal{E}$ with the (nontrivial) invariant boundary.
Modification of vector fields on the boundary (cf. [54]) is well used to
guarantee persistence and/or invariant foliations of invariant manifolds. In
our case, let $\rho_{I,I^{\prime}}:\mathbb{R}\to\mathbb{R}$ be a smooth,
nonnegative bump-type function such that

$\rho_{I,I^{\prime}}(t)=\begin{cases}1&t\not\in I^{\prime},\\\ 0&t\in
I.\end{cases}$

We then modify the desingularized vector field (5.5) to the following:

$\frac{d}{d\tau}\begin{pmatrix}t\\\ {\bf x}\end{pmatrix}=\tilde{g}(t,{\bf
x})\equiv g(t,{\bf
x})+\delta\rho_{I,I^{\prime}}(t)\begin{pmatrix}\nu_{I^{\prime}}\\\
0\end{pmatrix},$ (5.10)

where $\delta>0$, and $\nu_{I^{\prime}}$ is the inward unit normal
vector121212 For constructing overflowing invariant manifolds, modify
$\nu_{I^{\prime}}$ to be the outward unit normal vector, which ensures the
invariant foliation of unstable manifolds ([54]).  of $\partial I^{\prime}$.
Then we know that $\tilde{g}=g$ on $M_{I}$ and that trajectories on
$\mathcal{E}$ approach to $M_{I}$ in $t$-direction. In other words, for any
interval $\tilde{I}$ satisfying $I\subset{\rm int}\tilde{I}\subset
I^{\prime}$, the tube $M_{\tilde{I}}$ is inflowing invariant for the modified
vector field $\tilde{g}$. Therefore, according to the general theory of NHIMs,
the inflowing invariant manifold $M_{\tilde{I}}$ for $\tilde{g}$ admits the
local stable manifold $W^{s}_{\rm loc}(M_{\tilde{I}};\tilde{g})$ and its
invariant foliation (e.g., [54]) such that the restriction of the base space
of $W^{s}_{\rm loc}(M_{\tilde{I}};\tilde{g})$ to $M_{I}$ provides the
invariant foliation of $W^{s}_{\rm loc}(M_{I};g)$; for the original vector
field $g$ in $\\{t\in I\\}$. As a result, blow-up solutions for (5.2) stated
in Theorem 5.1 can be described by means of $M_{I}$.

######  Corollary 5.4 (Nonautonomous blow-up: the existence by means of
invariant manifolds).

Fix an initial time $t_{0}\in\mathbb{R}$. Suppose that $g$ admits an invariant
manifold $M\subset\mathcal{E}$ satisfying all requirements in Assumption 5.2
with compact intervals $I,I^{\prime}\subset\mathbb{R}$. Also assume that the
solution ${\bf y}(t)$ of (5.2) with a bounded initial point ${\bf y}_{0}(={\bf
y}(t_{0}))\in\mathbb{R}^{n}$ whose image ${\bf x}(\tau)=T_{{\rm
para};\alpha}({\bf y}(\tau))$ for $T_{{\rm para};\alpha}$ is on $W_{\rm
loc}^{s}(M_{I};g)$, where $\tau_{0}$ is the corresponding time of $t_{0}$ for
the time-scale desingularization. Then $t_{\max}<\infty$ and (5.6) holds with
a trajectory $\gamma\subset M_{I}$.

We leave the treatment of blow-up rates in Appendix C because several
additional techniques stemming from differential topology are required, in
addition to techniques discussed in Appendix B. Here we only provide a special
case, which can be simply applied to various nonautonomous systems such as
examples in Section 6.

######  Corollary 5.5 (Blow-up rates for nonautonomous blow-up: a special
case).

In addition to settings in Assumption 5.2, suppose that

$\displaystyle M_{I^{\prime}}=\\{(t,{\bf x}_{\ast}(t))\mid t\in
I^{\prime},\,{\bf x}_{\ast}(t)\text{ is an equilibrium satisfying (\ref{NH-
spec-nonaut-stationary}) below}\\}$ $\displaystyle\sharp\\{{\rm
Spec}(Dg(t,{\bf x}_{\ast}(t)))\cap i\mathbb{R}\\}=1\quad\text{ for all }\quad
t\in I^{\prime}$ (5.11)

where ${\rm Spec}(A)$ denotes the set of eigenvalues of a squared matrix $A$,
that is, $M_{I^{\prime}}$ is a curve of hyperbolic equilibria parameterized by
$t$. Then the asymptotic estimate (5.7) holds true with ${\bf
x}_{\gamma}(\tau)\equiv{\bf x}_{\ast}(t)$ for each $t\in I$. In particular,
asymptotic behavior for blow-up solutions stated in Theorem 5.1 is obtained.

###### Proof.

See Section C.2. ∎

##  6 Examples

Examples showing the applicability of our results are collected. In addition
to the applicability, we aim at seeing that the present methodology unifies
the approach to finite-time singularities in nonautonomous problems. The
examples we shall show are highlighted as follows:

  * •

Section 6.1 addresses the first Painlevé equation, which the existence of
singular solutions is well-known in the treatment of regular singular points.

  * •

Section 6.2 relates to a regularized system for exhibiting singular shocks in
(non-)hyperbolic systems of conservation laws.

  * •

Section 6.3 addresses self-similar profiles of nonlinear diffusion equations
with finite-time singularity.

  * •

Section 6.4 addresses radially symmetric profiles of a reaction-diffusion
equation with negative power nonlinearity.

All examples were investigated individually in preceding studies, and
independent techniques were required to unravel the comprehensive morphology
of solutions. When we pay our attention to blow-ups, the proposed methodology
unravels all these features in a systematic way, indicating that a series of
machineries involving embeddings of phase spaces as well as time-scale
desingularizations can unify descriptions of finite-time singularities.

######  Remark 6.1.

In all examples, invariant manifolds in our interest are collections of
equilibria. Therefore Corollary 5.5 are mainly applied.

######  Remark 6.2.

As we shall see in Section 6.1, as well as commented in [35], arguments
through global (parabolic-type or Poincaré-type, say [38]) embeddings require
lengthy and tedious calculations, although we do not suffer from any change of
charts131313 Advantages that quasi-parabolic embeddings are used as our
central machinery are discussed in [30, 35]. . This is the reason why we apply
another embeddings, called directional embeddings briefly summarized in
Appendix A, to the remaining examples. Note that all results in Sections 4 and
5 are essentially valid. See e.g. [38] for the correspondence of arguments.

###  6.1 The first Painlevé equation

First we consider the first Painlevé equation

$u^{\prime\prime}=6u^{2}+t,\quad^{\prime}=\frac{d}{dt}$ (6.1)

from the viewpoint of blow-up description. It is well-known that (6.1)
possesses the following solution: for fixed $t_{0}\in\mathbb{R}$,

$u(t)\sim 6(t_{\max}-t)^{-2}\quad\text{ as }\quad t\to t_{\max}-0$ (6.2)

for some $t_{\max}=t_{\max}(t_{0})>t_{0}$ and sufficiently large initial point
$u_{0}=u(t_{0})$. Although this asymptotic behavior can be derived via
substitution of a formal (Frobenius-type) power series solution into the
equation (e.g., [33]), we shall derive the existence of such a solution
through dynamics at infinity.

Rewrite (6.1) as the system of the first order ODEs:

$\begin{cases}\chi^{\prime}=1,&\\\ u^{\prime}=v,&\\\
v^{\prime}=6u^{2}+\chi,&\end{cases}\quad^{\prime}=\frac{d}{dt}.$ (6.3)

We immediately have the following property.

######  Lemma 6.3.

The system (6.3) is asymptotically quasi-homogeneous of type $(0,2,3)$ and
order $k+1=2$.

In the present example, we apply the quasi-parabolic embedding

$\chi=\chi,\quad u=\frac{x_{1}}{(1-p({\bf x})^{6})^{2}},\quad
v=\frac{x_{2}}{(1-p({\bf x})^{6})^{3}},$

where

${\bf x}=(x_{1},x_{2})^{T},\quad p({\bf x})^{6}=x_{1}^{6}+x_{2}^{4}$

with $(\beta_{1},\beta_{2})=(3,2)$ and $c=6$. Following the general derivation
(Section 3), let

$\displaystyle\kappa:=(1-p({\bf x})^{6})^{-1},\quad\tilde{f}_{1}(\chi,{\bf
x}):=x_{2},\quad\tilde{f}_{2}(\chi,{\bf x}):=6x_{1}^{2}+\kappa^{-4}\chi,$
$\displaystyle G(\chi,{\bf x}):=\frac{x_{1}^{5}}{2}\tilde{f}_{1}(\chi,{\bf
x})+\frac{x_{2}^{3}}{3}\tilde{f}_{2}(\chi,{\bf
x})=\frac{x_{1}^{5}}{2}x_{2}+\frac{x_{2}^{3}}{3}(6x_{1}^{2}+\kappa^{-4}\chi).$

Under the time-scale desingularization

$\frac{d\tau}{dt}=\kappa^{-1},$

we have the desingularized vector field (5.5) with $c=6$, $k=1$ and
$\bar{\Lambda}_{\alpha}={\rm diag}(0,2,3)$.

The next step is to find invariant sets on the horizon $\mathcal{E}=\\{p({\bf
x})=1\\}$. On $\mathcal{E}$, equilibria satisfy $g_{1}(\chi,{\bf
x})=g_{2}(\chi,{\bf x})=0$, where

$g_{1}(\chi,{\bf x})=x_{2}(1-x_{1}^{6}-4x_{1}^{3}x_{2}^{2}),\quad
g_{2}(\chi,{\bf x})=\frac{3}{2}x_{1}^{2}(4-x_{1}^{3}x_{2}^{2}-4x_{2}^{4})$

are the second and the third components of the desingularized vector field,
respectively. Note that $\dot{\chi}=0$ is automatically satisfied on
$\mathcal{E}$. We find, for each $\chi\in\mathbb{R}$, the points $(\chi,{\bf
x}_{\ast})$, where

$\displaystyle{\bf
x}_{\ast}=(x_{1;+},x_{2;+}),\quad(x_{1;+},x_{2;-}),\quad(x_{1;-},x_{2;+}),\,\text{
or }\,(x_{1;-},x_{2;-}),$ $\displaystyle x_{1;\pm}=\pm\frac{1}{17^{1/6}},\quad
x_{2;\pm}=\pm\frac{2}{17^{1/4}}$

are equilibria on $\mathcal{E}$.

Now we pay attention to the family of equilibria
$M_{++}=\\{(\chi,x_{1;+},x_{2;+})\\}$. The Jacobian matrix of (5.5) at each
point on $M_{++}$ is

$\begin{pmatrix}0&\ast\\\
O&\tilde{A}\end{pmatrix},\quad\tilde{A}=\begin{pmatrix}\frac{11}{2}x_{1}^{5}\tilde{f}_{1}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{1}}\right|_{\mathcal{E}}-2\left.\left(G+x_{1}\frac{\partial G}{\partial
x_{1}}\right)\right|_{\mathcal{E}}&\frac{11}{3}x_{2}^{3}\tilde{f}_{1}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{2}}\right|_{\mathcal{E}}-\left.2x_{1}\frac{\partial G}{\partial
x_{2}}\right|_{\mathcal{E}}\\\
\frac{11}{2}x_{1}^{5}\tilde{f}_{2}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{1}}\right|_{\mathcal{E}}-\left.3x_{2}\frac{\partial G}{\partial
x_{1}}\right|_{\mathcal{E}}&\frac{11}{3}x_{2}^{3}\tilde{f}_{2}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{2}}\right|_{\mathcal{E}}-3\left.\left(G+x_{2}\frac{\partial G}{\partial
x_{2}}\right)\right|_{\mathcal{E}}\end{pmatrix},$

where

$\displaystyle\tilde{f}_{1}|_{\mathcal{E}}=x_{2}|_{\mathcal{E}},\quad\tilde{f}_{2}|_{\mathcal{E}}=6x_{1}^{2}|_{\mathcal{E}},\quad\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{1}}\right|_{\mathcal{E}}=0,\quad\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{2}}\right|_{\mathcal{E}}=1,\quad\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{1}}\right|_{\mathcal{E}}=12x_{1},\quad\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{2}}\right|_{\mathcal{E}}=0,$ $\displaystyle
G|_{\mathcal{E}}=\frac{x_{1}^{5}}{2}\tilde{f}_{1}|_{\mathcal{E}}+\frac{x_{2}^{3}}{3}\tilde{f}_{2}|_{\mathcal{E}}=\left.\left(\frac{1}{2}x_{1}^{5}x_{2}+2x_{1}^{2}x_{2}^{3}\right)\right|_{\mathcal{E}},$
$\displaystyle\left.\frac{\partial G}{\partial
x_{1}}\right|_{\mathcal{E}}=\frac{5}{2}x_{1}^{4}\tilde{f}_{1}|_{\mathcal{E}}+\frac{x_{1}^{5}}{2}\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{1}}\right|_{\mathcal{E}}+\frac{x_{2}^{3}}{3}\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{1}}\right|_{\mathcal{E}}=\left.\left(\frac{5}{2}x_{1}^{4}x_{2}+4x_{1}x_{2}^{3}\right)\right|_{\mathcal{E}},$
$\displaystyle\left.\frac{\partial G}{\partial
x_{2}}\right|_{\mathcal{E}}=\frac{x_{1}^{5}}{2}\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{2}}\right|_{\mathcal{E}}+x_{2}^{2}\tilde{f}_{2}|_{\mathcal{E}}+\frac{x_{2}^{3}}{3}\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{2}}\right|_{\mathcal{E}}=\left.\left(\frac{1}{2}x_{1}^{5}+6x_{1}^{2}x_{2}^{2}\right)\right|_{\mathcal{E}},$

and several terms being identically zero on $\mathcal{E}$ are omitted. Note
that the above matrix is independent of $\chi$, namely the flow and its
linearization are autonomous on $\mathcal{E}$. Substituting
$(x_{1},x_{2})=(x_{1;+},x_{2;+})$ into the above matrix, we have

$\displaystyle\frac{11}{2}x_{1}^{5}\tilde{f}_{1}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{1}}\right|_{\mathcal{E}}-2\left.\left(G+x_{1}\frac{\partial G}{\partial
x_{1}}\right)\right|_{\mathcal{E}}$
$\displaystyle=\frac{11}{17^{13/12}}-\frac{108}{17^{13/12}}=-\frac{97}{17^{13/12}},$
$\displaystyle\frac{11}{3}x_{2}^{3}\tilde{f}_{1}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{1}}{\partial
x_{2}}\right|_{\mathcal{E}}-\left.2x_{1}\frac{\partial G}{\partial
x_{2}}\right|_{\mathcal{E}}$ $\displaystyle=\frac{11\cdot
16}{51}+1-\frac{49}{17}=\frac{80}{51},$
$\displaystyle\frac{11}{2}x_{1}^{5}\tilde{f}_{2}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{1}}\right|_{\mathcal{E}}-\left.3x_{2}\frac{\partial G}{\partial
x_{1}}\right|_{\mathcal{E}}$ $\displaystyle=\frac{33}{17^{7/6}}+\frac{12\cdot
17}{17^{7/6}}-\frac{6}{17^{7/6}}\left(5+32\right)=\frac{15}{17^{7/6}},$
$\displaystyle\frac{11}{3}x_{2}^{3}\tilde{f}_{2}|_{\mathcal{E}}+\left.\frac{\partial\tilde{f}_{2}}{\partial
x_{2}}\right|_{\mathcal{E}}-3\left.\left(G+x_{2}\frac{\partial G}{\partial
x_{2}}\right)\right|_{\mathcal{E}}=\frac{176}{17^{13/12}}-\frac{3}{17^{13/12}}\left(1+16+1+48\right)=-\frac{22}{17^{13/12}}$

and hence the matrix $\tilde{A}$ is

$\tilde{A}=\begin{pmatrix}-97/17^{13/12}&80/51\\\
15/17^{7/6}&-22/17^{13/12}\end{pmatrix}\equiv\begin{pmatrix}a&b\\\
c&d\end{pmatrix}.$

Now

$\displaystyle
a+d=-\frac{119}{17^{13/12}},\quad(a+d)^{2}=\frac{119^{2}}{17^{13/6}},\quad ad-
bc=\frac{97\cdot 22-400}{17^{13/6}},$ $\displaystyle(a+d)^{2}-4(ad-
bc)=\frac{119^{2}-4(97\cdot 22-400)}{17^{13/6}}=\frac{4025}{17^{13/6}}$

and hence the eigenvalues of $\tilde{A}$ are

$\lambda=\frac{1}{2}\left\\{(a+d)\pm\sqrt{(a+d)^{2}-4(ad-
bc)}\right\\}=\frac{-119\pm\sqrt{7225}}{2\cdot 17^{13/12}}=\frac{-119\pm
85}{2\cdot 17^{13/12}}=\frac{-1}{17^{1/12}},\quad\frac{-6}{17^{1/12}},$

indicating that $(x_{1;+},x_{2;+})$ is attracting (sink-type), and hence the
invariant manifold $M_{++}$ is normally attracting. In particular, for any
compact interval $I\subset\mathbb{R}$, the restriction $M_{++;I}\equiv
M_{++}\cap\\{t\in I\\}$ is a NAIM satisfying all requirement in Corollary 5.5.

We therefore see that, for particular choice of an interval
$I\subset\mathbb{R}$, solutions ${\bf x}(\tau)\subset W^{s}_{\rm
loc}(M_{++;I})$ characterize blow-up solutions of (6.3) with the asymptotic
behavior $O((t_{\max}-t)^{-2})$ as $t\to t_{\max}-0$. This result confirms and
guarantees of the existence of blow-up solutions of the form (6.2) without any
algebraic trial-and-error arguments.

######  Theorem 6.4 (cf. [33]).

Consider (6.1) with the initial time $t_{0}\in\mathbb{R}$ and
$u(t_{0})=u_{0}\in\mathbb{R}$. If $u_{0}$ is sufficiently large, the solution
$u(t)=u(t;t_{0},u_{0})$ with $u(t_{0})=u_{0}$ blows up at a time
$t_{\max}=t_{\max}(t_{0})<\infty$ with

$u(t)\sim c_{u}(t_{\max}-t)^{-2}\quad\text{ as }\quad t\to t_{\max}-0$

for some constant $c_{u}>0$.

###  6.2 A problem in Dafermos profiles for singular shocks

The next example is the following system with $0\leq\epsilon\ll 1$ as a
parameter controlling the multi-scale structure:

$\begin{cases}\chi^{\prime}=\epsilon,&\\\ u_{1}^{\prime}=u_{1}^{2}-u_{2}-\chi
u_{1}-w_{1},&\\\ u_{2}^{\prime}=\frac{1}{3}u_{1}^{3}-u_{1}-\chi
u_{2}-w_{2},&\\\ w_{1}^{\prime}=-\epsilon u_{1},&\\\ w_{2}^{\prime}=-\epsilon
u_{2},&\\\ \end{cases}\quad^{\prime}=\frac{d}{dt}.$ (6.4)

######  Remark 6.5.

The origin of (6.4) is the system of conservation laws

$(u_{1})_{t}+\left(u_{1}^{2}-u_{2}\right)_{x}=0,\quad(u_{2})_{t}+\left(\frac{1}{3}u_{1}^{3}-u_{1}\right)_{x}=0\quad\Leftrightarrow\quad
U_{t}+F(U)_{x}=0,$ (6.5)

proposed by Keyfitz and Kranser (e.g., [28, 32, 27]. See also [48, 50]) with
identification

$U(t,x)=(u_{1}(t,x),u_{2}(t,x))^{T},\quad
F(U)=\left(u_{1}^{2}-u_{2},\frac{1}{3}u_{1}^{3}-u_{1}\right)^{T}.$

Applying the Dafermos regularization (e.g., [49]) to (6.5), we have

$U_{t}+F(U)_{x}=\epsilon tU_{xx},$ (6.6)

and applying the ansatz of self-similar profiles, called the Dafermos
profiles: $U(t,x)=U^{\epsilon}(x/t)$, we obtain the extended autonomous system

$\chi^{\prime}=\epsilon,\quad U^{\prime}=V,\quad V^{\prime}=\\{DF(U)-\chi
I\\}V,\quad^{\prime}=\frac{d}{dt},$

where $U=(u_{1},u_{2})^{T}$ and $V=(v_{1},v_{2})^{T}$. An alternative form is
derived through the change of variables (e.g., [49])

$(U,V,\chi)\mapsto(U,W,\chi)=(U,F(U)-\chi U-V,\chi),$

which is exactly (6.4), where $W=(w_{1},w_{2})^{T}$. In [49], this system is
considered to construct the Dafermos profile of singular shocks, namely shock
fronts with Dirac’s delta-type singularity on the front. A novel observation
in singular shocks is that these shocks can be constructed as the singular
limit of singularly perturbed solutions admitting arbitrarily large
magnitudes. The corresponding trajectories are constructed by the connecting
orbits of equilibria at infinity under technical change of coordinates. The
present example as well as preceding works by the author (e.g., [38]) is
inspired by reconstructing such singular objects within standard and unified
arguments of dynamics at infinity.

Based on arguments in preceding works, we have the following observation.

######  Lemma 6.6 (cf. [38, 49]).

The system (6.4) is asymptotically quasi-homogeneous of type $(0,1,2,1,2)$ and
order $k+1=2$.

We apply the directional embedding
$T_{d;\alpha}:(\chi,u_{1},u_{2},w_{1},w_{2})\mapsto(\chi,x_{1},s,x_{3},x_{4})$
defined by

$\chi=\chi,\quad u_{1}=\frac{x_{1}}{s},\quad u_{2}=\frac{1}{s^{2}},\quad
w_{1}=\frac{x_{3}}{s},\quad w_{2}=\frac{x_{4}}{s^{2}}.$ (6.7)

######  Remark 6.7.

The present choice of the localizing direction, namely $u_{2}=s^{-2}$, stems
from the previous arguments for deriving blow-up behavior of dynamical systems
with nonlinearity extracted from (6.5) (cf. [35, 38, 40]):

$u_{1}^{\prime}=u_{1}^{2}-u_{2},\quad
u_{2}^{\prime}=\frac{1}{3}u_{1}^{3}-u_{1}.$

It is observed in the above references that the corresponding desingularized
vector fields through global embeddings like quasi-parabolic ones admit
totally four equilibria on the horizon, all of which correspond to blow-up
solutions (in the positive or negative time direction) such that $v$ goes to
“positive” infinity. See references mentioned for details. This is the reason
why we have chosen the embedding in the direction specified as (6.7).

Using the change of coordinates (6.7) as well as the time-scale
desingularization

$\frac{d\tau}{dt}=s^{k}=s,$

we obtain the associated desingularized vector field:

$\begin{cases}\dot{\chi}=\epsilon s,&\\\
\dot{x}_{1}=x_{1}\tilde{g}_{0}+(x_{1}^{2}-1)-s(\chi x_{1}+x_{3}),\\\
\dot{s}=s\tilde{g}_{0},&\\\ \dot{x}_{3}=x_{3}\tilde{g}_{0}-\epsilon x_{1}s,\\\
\dot{x}_{4}=2x_{4}\tilde{g}_{0}-\epsilon s,\\\
\end{cases}\quad\dot{}=\frac{d}{d\tau},\quad\tilde{g}_{0}=-\frac{1}{6}x_{1}^{3}+\frac{x_{1}}{2}s^{2}+\frac{s}{2}(\chi+x_{4}).$
(6.8)

We easily see that the horizon $\\{s=0\\}$ admits four families of equilibria

$(\chi,x_{1},s,x_{3},x_{4})=\left(\chi,\pm\sqrt{3\pm\sqrt{3}},0,0,0\right).$

Let

$p_{1,\pm}:=\pm\sqrt{3+\sqrt{3}},\quad p_{2,\pm}:=\pm\sqrt{3-\sqrt{3}}.$ (6.9)

The Jacobian matrices on the above family of equilibria are

$\begin{pmatrix}0&0&\epsilon&0&0\\\
0&-\frac{2}{3}p_{i,\pm}^{3}+2p_{i,\pm}&-\frac{1}{2}\chi p_{i,\pm}&0&0\\\
0&0&-\frac{1}{6}p_{i,\pm}^{3}&0&0\\\ 0&0&-\epsilon
p_{i,\pm}&-\frac{1}{6}p_{i,\pm}^{3}&0\\\
0&0&-\epsilon&0&-\frac{1}{6}p_{i,\pm}^{3}\\\ \end{pmatrix}$

For simplicity, we further assume that $\epsilon=0$ for a while. Then, in the
singular fast limit system, namely (6.8) with $\epsilon=0$, four families of
equilibria are invariant with normally hyperbolic structure. Among four points
(6.9), $p_{1,+}$ and $p_{2,+}$ are those where the flow is attracting towards
the horizon. Moreover, $p_{1,+}$ is a saddle, which is our main issue141414
According to the mechanism of singular shocks mentioned in Remark 6.5, saddle-
like invariant sets would be considered to play a key role in constructing
these shocks. . From our construction and the above observation, the family

$M_{1,+}=\\{(\chi,p_{1,+},0,0,0)\mid\chi\in\mathbb{R}\\}$

determines a (noncompact) NHIM. Because $\chi$ is just a a parameter on the
horizon $\\{s=0\\}$, for any compact interval $I\subset\mathbb{R}$, the
restriction

$M_{1,+;I}=\\{(\chi,p_{1,+},0,0,0)\mid\chi\in I\\}$

becomes a compact NHIM (with boundary). In particular, all requirements in
Corollary 5.5 are satisfied. By the geometric singular perturbation theory
(cf. [14]) (with boundary modification), the above conclusion also holds for
all $0\leq\epsilon\ll 1$.

We therefore conclude that, for sufficiently large $I$ depending on the
initial time $t_{0}$, the local stable manifold $W_{\rm loc}^{s}(M_{1,+;I})$
induce a blow-up solution. More precisely, there is an equilibrium
$p_{\ast}=(\chi_{\ast},p_{1,+},0,0,0)\in M_{1,+;I}$ for some
$\chi_{\ast}=\chi_{\ast}(t_{0})$ such that the trajectory on $W_{\rm
loc}^{s}(p_{\ast})$ induces a blow-up solution with the asymptotic behavior

$u_{1}(t)=\frac{x_{1}(t)}{s(t)}\sim c(t_{\max}-t)^{-1}\quad\text{ as }\quad
t\to t_{\max}-0$

for some constant $c>0$.

######  Theorem 6.8 (cf. [28, 49]).

Consider (6.4) and a solution $(\chi,u_{1},u_{2},w_{1},w_{2})(t)$ with
$0\leq\epsilon\ll 1$. Let $g$ be the desingularized vector field (6.8). Assume
that, under the directional embedding $T_{d;\alpha}$ given in (6.7),
$T_{d;\alpha}(\chi,u_{1},u_{2},w_{1},w_{2})(\bar{\tau})\in W_{\rm
loc}^{s}(p_{\ast})$ holds for some $\bar{\tau}$. Then
$(\chi,u_{1},u_{2},w_{1},w_{2})(t)$ is a blow-up solution with

$u_{1}(t)\sim c_{1}(t_{\max}-t)^{-1},\quad u_{2}(t)\sim
c_{2}(t_{\max}-t)^{-2}\quad\text{ as }\quad t\to t_{\max}-0$

for some constants $c_{1},c_{2}>0$.

In the above theorem we do not mention the asymptotic behavior of $w_{1}$ and
$w_{2}$ because the corresponding $x$-component of $p_{\ast}$ is $0$, in which
case further terms have to be calculated for these asymptotic behavior within
the present approach (cf. [2]).

###  6.3 A system associated with self-similarity

The next example we shall consider is the following system:

$(u^{m-1}u^{\prime})^{\prime}+\beta\chi u^{\prime}+\alpha
u=0,\quad^{\prime}=\frac{d}{d\chi},\quad\chi\in\mathbb{R},$ (6.10)

where $\alpha,\beta\in\mathbb{R}$ are parameters. The system (6.10) originates
from the diffusion equation (e.g., [15])

$U_{t}=(U^{m-1}U_{x})_{x}.$

The parameter $m$ controls the strength of nonlinear diffusion. Paying our
attention to self-similar solutions of the form151515 The ansatz (6.11)
represents the backward self-similarity. Although there are another types of
self-similarity; forward and exponential-type ones, the governing equation
becomes the same one, (6.10).

$U(t,x)=(T-t)^{\alpha}u(x(T-t)^{\beta}),\quad t<T$ (6.11)

for some $T>0$, the system is reduced to (6.10). We concentrate on the very
fast diffusion case, $m<0$, and assume that $\beta<0$. In what follows we
investigate blow-up solutions following our proposed machinery. First rewrite
(6.10) as the first order extended autonomous system:

$\begin{cases}\chi^{\prime}=1,&\\\ u^{\prime}=u^{1-m}v,&\\\
v^{\prime}=-\beta\chi u^{1-m}v-\alpha u,&\\\
\end{cases}\quad^{\prime}=\frac{d}{dt}.$ (6.12)

Direct calculations yield the following observation.

######  Lemma 6.9.

The system (6.12) is asymptotically quasi-homogeneous of type $(0,1,1)$ and
order $k+1=2-m$.

We then apply the directional embedding of type $(0,1,1)$:

$\chi=\chi,\quad u=\frac{1}{s},\quad v=\frac{x}{s}.$ (6.13)

Using this change of coordinates as well as the time-scale desingularization

$\frac{d\tau}{dt}=s^{1-m},$

we obtain the associated desingularized vector field:

$\begin{cases}\dot{\chi}=s^{1-m},&\\\ \dot{s}=-sx,&\\\
\dot{x}=-x^{2}-\beta\chi x-\alpha s^{1-m},&\\\
\end{cases}\quad\dot{}=\frac{d}{d\tau}.$ (6.14)

We easily see that the horizon $\\{s=0\\}$ admits a family of equilibria

$\\{(\chi,0,0)\\},\quad\text{ and }\quad\\{(\chi,0,-\beta\chi)\\}.$

We further investigate the dynamics around the invariant manifold
$M_{\geq\chi_{0}}=\\{(\chi,0,-\beta\chi)\in\mathbb{R}^{3}\mid\chi\geq\chi_{0}\\}$
with $\chi_{0}>0$. Thanks to the assumption $m<0$, the linearized matrix at
each point on $M_{\geq\chi_{0}}$ becomes very simple:

$\begin{pmatrix}0&0&0\\\ 0&\beta\chi&0\\\
\beta^{2}\chi&0&\beta\chi\end{pmatrix},$

which indicates that, for any fixed $\chi_{1}\in(\chi_{0},\infty)$, the
invariant manifold $M_{[\chi_{0},\chi_{1}]}\equiv
M_{\geq\chi_{0}}\cap\\{\chi\in[\chi_{0},\chi_{1}]\\}$ is invariant with
normally attracting structure. Therefore Corollary 5.5 can be applied to
(6.12), yielding the following result. The rate we have obtained coincides
with the preceding result shown in [15].

######  Theorem 6.10 (cf. [15]).

Assume that $m<0$ and $\beta<0$. Then the solution $(\chi,u(\chi),v(\chi))$
through the initial point $(\chi_{0},u_{0},v_{0})$ with $\chi_{0}>0$ whose
image under the directional embedding (6.13) is on the (local) stable manifold
of $M_{\geq\chi_{0}}$ for (6.14) is a blow-up solution with the blow-up rate
$u(\chi)\sim c_{u}(\chi_{\max}-\chi)^{-1/(1-m)}$ and $v(\chi)\sim
c_{v}(\chi_{\max}-\chi)^{-1/(1-m)}$ for some constants $c_{u},c_{v}>0$.

######  Remark 6.11.

In [15], dynamics of self-similar profiles is studied under a technical change
of variables, which changes (6.10) to an autonomous system. More precisely,
(6.10) with $u(0)=1$ and $u^{\prime}(0)=a\in\mathbb{R}$ is transformed via

$\chi=e^{r},\quad X(r)=\frac{\chi u^{\prime}}{u},\quad Y(r)=\chi^{2}u^{1-m}$

into

$\begin{cases}\dot{X}=X-mX^{2}-(\alpha+\beta X)Y,&\\\
\dot{Y}=(2+(1-m)X)Y,&\end{cases}\quad\dot{X}=dX/dr.$

In our methodology, on the other hand, no additional techniques are introduced
to obtain the same result when we pay our attention to blow-up solutions.

We have assumed $\beta<0$ in the present study. We can easily check that the
family of equilibria $\\{(\chi,0,-\beta\chi)\\}$ for (6.14) on the horizon
undergoes a transcritical bifurcation at $\beta=0$. In particular, when
$\beta>0$, the invariant manifold $M_{\geq\chi_{0}}$ becomes normally
repelling and hence blow-up associated with $M_{\geq\chi_{0}}$ never occurs,
which is consistent with observations in [15]. One of the original aims in
[15] was to construct self-similar profiles $U$ mentioned in (6.11). Another
parameter $\alpha$ should have the constraint for the backward self-
similarity: $2\beta=(1-m)\alpha-1$, which can be determined once $\beta$ is
fixed, and $\alpha$ is not explicitly appeared in characterization of blow-up
profiles in Theorem 6.10.

######  Remark 6.12.

Calculations of the Jacobian matrices imply that another branch of equilibria
on the horizon; $\\{(\chi,0,0)\mid\chi\in I\\}$ for any compact interval
$I\subset\mathbb{R}$, is not normally hyperbolic. Indeed, this branch contains
a center direction in the normal component. In this case, the center manifold
reduction in the corresponding direction will be required to determine the
precise blow-up behavior (cf. [21, 23, 24, 39]).

###  6.4 Radially symmetric solutions in a MEMS-type equation

The final example is the following system:

$u^{\prime\prime}+\frac{n-1}{r}u^{\prime}-r^{q}u^{-p}=0,\quad^{\prime}=\frac{d}{dr},$
(6.15)

where $n\in\mathbb{N}$, $p\in\mathbb{N}$, $q\in\mathbb{R}$ and
$r\in(0,\infty)$.

######  Remark 6.13.

The equation (6.15) originates from the following PDE, which is known as a
MEMS (micro-electro mechanical system) type equation:

$U_{t}=\Delta U-|x|^{q}U^{-p},\quad t>0,\quad x\in\mathbb{R}^{n}.$ (6.16)

Restricting our attention to radially symmetric stationary solutions
$U(t,x)=u(r)$ with $r=|x|$, then $u$ must satisfy (6.15). In [25], radially
symmetric stationary solutions $u$ with compact supports and derivatives
diverging as $r\to r_{+}-0$ and/or $r\to r_{-}+0$ for some
$0<r_{-}<r_{+}<\infty$, namely quenching solutions, are constructed. In
particular, all solutions (with $p\in 2\mathbb{N}$) $u(r)$ are negative on
these supports. This work reflects our selection of the directional embedding
in (6.19) below.

Keeping assertions in Remark 6.13 in mind, we pay attention to $w=u^{-1}$
instead of $u$ itself:

$2w^{-1}(w^{\prime})^{2}-w^{\prime\prime}-\frac{n-1}{r}w^{\prime}-r^{q}w^{p+2}=0,\quad^{\prime}=\frac{d}{dr},$
(6.17)

equivalently

$\frac{d}{d\rho}\begin{pmatrix}r\\\ w\\\ v\end{pmatrix}=\begin{pmatrix}1\\\
v\\\ -\frac{n-1}{r}v-r^{q}w^{p+2}+2w^{-1}v^{2}\end{pmatrix}.$ (6.18)

######  Lemma 6.14.

The system (6.18) is asymptotically quasi-homogeneous of type $(0,2,p+3)$ and
order $p+2$ when $p$ is even, while type $(0,1,(p+3)/2)$ and order $(p+3)/2$
when $p$ is odd.

In the case of $p$ being even, introducing the directional embedding

$r=r,\quad w=-\frac{1}{s^{2}},\quad v=\frac{x}{s^{p+3}}$ (6.19)

and the time-scale desingularization

$\frac{d\tau}{d\rho}=s^{-(p+1)},$

we have the following desingularized vector field:

$\frac{d}{d\tau}\begin{pmatrix}r\\\ s\\\
x\end{pmatrix}=\begin{pmatrix}s^{p+1}\\\ \frac{1}{2}sx\\\
\frac{p-1}{2}x^{2}-\frac{n-1}{r}s^{p+1}x-r^{q}\end{pmatrix}.$ (6.20)

Equilibria on the horizon $\mathcal{E}=\\{s=0\\}$ exist because $p$ is
even161616 When $p$ is odd, the equation requiring the existence of equilibria
on $\mathcal{E}$ is essentially the same as the one associated with (6.20),
which yields that there are no equilibria on $\mathcal{E}$, because the sign
of $r^{q}$ in (6.20) becomes “$+$”. , in which case these are

$p_{\pm}(r)=\left(r,0,\pm\frac{r^{q/2}}{\sqrt{\frac{p-1}{2}}}\right),$

parameterized by $r>0$. The Jacobian matrix on the horizon with $r>0$ is

$\begin{pmatrix}0&0&0\\\ 0&\frac{1}{2}x&0\\\
-qr^{q-1}&0&2x\sqrt{\frac{p-1}{2}}\end{pmatrix},$

implying that, for any fixed $r_{0}>0$, the invariant set
$M_{+}=\\{p_{+}(r)\mid r\geq r_{0}\\}$ is normally repelling, while
$M_{-}=\\{p_{-}(r)\mid r\geq r_{0}\\}$ is normally attracting. In particular,
$M_{-}$ satisfies all requirements in Corollary 5.5. As a consequence, we have
the following theorem.

######  Theorem 6.15.

Consider (6.17) with $p\in 2\mathbb{N}$. Then the solution $(r,w(r),v(r))$
through the initial point $(r_{0},w_{0},v_{0})$ with $r_{0}>0$ whose image
under the directional embedding (6.19) is on the (local) stable manifold of
$M_{\geq r_{0}}$ for (6.20) is a blow-up solution with the blow-up rate

$w(r)\sim-c_{w}(r_{\max}-r)^{-2/(p+1)},\quad v(r)\sim-
c_{v}(r_{\max}-r)^{-(p+3)/(p+1)}\quad\text{ as }\quad r\to r_{\max}-0$

for some constants $r_{\max}>r_{0}$ and $c_{w},c_{v}>0$. The corresponding
solution $u(r)$ of (6.15) has the following asymptotic behavior:

$u(r)\sim-c_{u}(r_{\max}-r)^{2/(p+1)},\quad
u^{\prime}(r)\sim\tilde{c}_{u}(r_{\max}-r)^{-(p-1)/(p+1)}\quad\text{ as }\quad
r\to r_{\max}-0$

for some constants $c_{u},\tilde{c}_{u}>0$. In particular, $u(r)$ is a
quenching solution of (6.15).

The family of blow-up solutions we have obtained is different from those
discussed in [25]. This can be because dominant terms of the vector field at
infinity we pay attention to, namely the basic scalings, are different.

## Conclusion

In this paper, we have developed methodologies to characterize blow-up
solutions shadowing “trajectories at infinity”, and those in nonautonomous
systems. A unified description of blow-up solutions is organized by means of
shadowing trajectories, or geometrically asymptotic phase of invariant
manifolds, and specific exponential decay rates, which generalizes
methodologies developed in preceding works [38, 39]. Indeed, as investigated
in examples (Section 6), various finite-time singularities involving blow-ups
are extracted by a common machinery. We strongly believe that the present
investigation will be the basis on characterizing blow-up solutions from the
viewpoint of dynamical systems. We end this paper by providing several
directions as future works.

### Asymptotic expansions and applications to computer-assisted proofs

One direction of future works would be multi-order asymptotic expansions of
blow-ups, as developed for stationary blow-ups, namely the case where blow-ups
are described by equilibria on the horizon when the original vector field is
autonomous [2, 30]. A natural question in this theme towards the
generalization is the development of the similar methodology to blow-ups
induced by e.g., periodic orbits, NHIMs or others such as chaotic invariant
sets on the horizon, and in nonautonomous systems. In [2, 30], not only a
systematic methodology to calculate asymptotic expansions of blow-up solutions
in arbitrary orders, but also a natural correspondence between the essential
information characterizing asymptotic expansions and linearized information
around equilibria on the horizon are shown. The latter correspondence enables
us to extract blow-up structure in a simple way. Indeed, as seen in Section
6.1, globally defined compactifications or embeddings require tedious
calculations towards the conclusion, which affect the success of applications,
such as computer-assisted proofs, equivalently rigorous numerics (e.g., [35,
40, 41, 52]). In general, global embeddings drastically raise the complexity
of vector fields, such as types and orders of polynomial, or general
asymptotically quasi-homogeneous vector fields, and require quite large
computational costs, even for typical numerical computations of blow-ups. The
achievement of the correspondence to asymptotic expansions can reduce the
computational costs for blow-ups, in particular for them induced by periodic
orbits and NHIMs on the horizon and for nonautonomous systems once the
corresponding derivation for these setting is achieved, because the systems
involving asymptotic expansions are much simpler than those involving
embeddings. Because applications involving computer-assisted proofs of
periodic orbits and NHIMs themselves are widely achieved (e.g., [18]),
combination with the present and future machineries will immediately
contribute to develop computer-assisted proofs of blow-ups with the above
asymptotic behavior at infinity and their global extensions established in
[35] for stationary blow-ups.

### Blow-up rates, spectral intervals and unique ergodicity

Another natural direction in studies of blow-up solutions is to determine
blow-up rates, the principal terms of blow-up solutions, where the leading
term is our interest unlike the multi-order asymptotic expansions. This is one
of central issues in blow-up studies for decades, in particular for PDEs,
while all blow-ups mentioned in the present paper are referred to as type-I,
namely the case where the blow-up rates are automatically and uniquely
determined by type and order of asymptotically quasi-homogeneous vector
fields. Note that examples exhibiting rates other than type-I are collected in
[39], where nonhyperbolicity of invariant sets on the horizon plays a central
role in determining special blow-up rates, including ones referred to as type-
II in the field of PDEs. Although investigations of such blow-up rates with
complex asymptotic behavior and in nonautonomous systems are ones of natural
direction of future investigations (cf. [34]), we would like to leave another
direction involving blow-up rates.

One essential condition to determine blow-up rates in the present arguments is
the precise exponential rate of convergence to trajectories on the horizon. It
will be shown in Appendix B that discrete distribution of dichotomy spectra of
linear skew-product flows provides the required convergence rate in the case
of NHIMs. In special cases mentioned in Proposition B.4, dichotomy spectra (or
specially eigenvalues and Floquet exponents for constant and periodic cases,
respectively) determine the unique exponential decay rates. It would be also
true from comments in [47], that unique ergodicity of flows on the base
spaces; the hulls of trajectories on NHIMs being minimal sets, could yield the
same conclusion. On the other hand, there are several examples that dichotomy
spectra include continuous spectral intervals, namely intervals with positive
Lebesgue measure in the spectra, which are observed when the flow on the base
space is non-uniquely ergodic. Our question here is the following:

  * •

Is the blow-up rate uniquely determined even when the associated base sets in
NHIMs admit more than one ergodic measures, or (4.2) is violated ?

If uniquely determined, its determination with the mechanism is a central
issue. Otherwise, the concrete description of blow-up rate in such a case
would be the issue.

## Acknowledgements

KM was partially supported by World Premier International Research Center
Initiative (WPI), Ministry of Education, Culture, Sports, Science and
Technology (MEXT), Japan, and JSPS Grant-in-Aid for Scientists (B) (No.
JP23K20813). He would like also to thank Professors Jordi-Lluís Figueras and
Akitoshi Takayasu for giving him inspiring suggestions to the present study.

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##  Appendix A Directional embeddings

Here a locally defined embedding is introduced, which shall be called a
directional embedding.

######  Definition A.1 (Directional embedding, cf. [8, 38]).

A directional embedding of type $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ is
defined as

$\displaystyle y=(y_{1},\ldots,y_{n})\mapsto
T_{d;\alpha}(y)=(s,\hat{x})\equiv(s,\hat{x}_{1},\ldots,\hat{x}_{i_{0}-1},\hat{x}_{i_{0}+1},\ldots,\hat{x}_{n}),$
$\displaystyle
y_{i}:=\frac{\hat{x}_{i}}{s^{\alpha_{i}}}\quad(i\not=i_{0}),\quad
y_{i_{0}}:=\pm\frac{1}{s^{\alpha_{i_{0}}}}$ (A.1)

with given direction $i_{0}$ and the signature $\pm$, provided $i_{0}\in
I_{\alpha}$. This embedding is bijective in $U\cap\\{\pm y_{i_{0}}>0\\}$, in
which sense directional embeddings are local ones. In particular, this
embedding is available when we are interested in solutions of (1.1) such that
the $i_{0}$-th component has the identical sign during time evolution. The
image of $T_{d;\alpha}$ is

$\mathcal{D}=\\{(s,\hat{x}_{1},\ldots,\hat{x}_{i_{0}-1},\hat{x}_{i_{0}+1},\ldots,\hat{x}_{n})\mid
s>0,\,\hat{x}_{i}\in\mathbb{R}\,(i\in
I_{\alpha}\setminus\\{i_{0}\\}),\,(\hat{x}_{j_{1}},\ldots,\hat{x}_{j_{n-l}})\in\tilde{U}\\},$
(A.2)

where $\\{j_{1},\cdots,j_{n-l}\\}=\\{1,\cdots,n\\}\setminus I_{\alpha}$. The
set $\mathcal{E}=\\{s=0\\}$ corresponds to the infinity in the original
coordinate, which shall be called the horizon.

For simplicity, fix $i_{0}=1$ in (A.1) in the following arguments. Next
transform (1.1) via (A.1), which is straightforward:

$\displaystyle\frac{ds}{dt}$
$\displaystyle=-\frac{1}{\alpha_{1}}s^{-k+1}\hat{f}_{1}(s,\hat{x}_{2},\ldots,\hat{x}_{n}),$
$\displaystyle\frac{d\hat{x}_{i}}{dt}$
$\displaystyle=s^{-k}\left\\{\hat{f}_{i}(s,\hat{x}_{2},\ldots,\hat{x}_{n})-\frac{\alpha_{i}}{\alpha_{1}}x_{i}\hat{f}_{1}(s,\hat{x}_{2},\ldots,\hat{x}_{n})\right\\}\quad(i=2,\ldots,n),$

where

$\hat{f}_{i}(s,\hat{x}_{2},\ldots,\hat{x}_{n})\equiv
s^{k+\alpha_{i}}f_{i}(s^{-\alpha_{1}},s^{-\alpha_{2}}\hat{x}_{2},\ldots,s^{-\alpha_{n}}\hat{x}_{n}),\quad
i=1,\ldots,n.$ (A.3)

The resulting vector field is still singular near the horizon, but it turns
out that the order of divergence of vector field as $s\to+0$ is $O(s^{-k})$,
and hence the following time-scale transformation is available.

######  Definition A.2 (Time-variable desingularization: the directional
version).

Define the new time variable $\tau_{d}$ by

$d\tau_{d}=s(t)^{-k}dt$ (A.4)

equivalently,

$t=t_{0}+\int_{\tau_{0}}^{\tau}s(\tau_{d})^{k}d\tau_{d},$ (A.5)

where $\tau_{0}$ and $t_{0}$ denote the correspondence of initial times, and
$s(\tau_{d})$ is the solution trajectory $s(t)$ under the parameter
$\tau_{d}$. We shall call (A.4) the time-variable desingularization (of order
$k+1$).

The vector field $g=g_{d}$ in $\tau_{d}$-time-scale is171717 The existence of
$B$ follows by cyclic permutations and the fact that $\alpha_{1}>0$. See also
[38].

$\begin{pmatrix}\frac{ds}{d\tau_{d}}\\\ \frac{dx_{2}}{d\tau_{d}}\\\ \vdots\\\
\frac{dx_{n}}{d\tau_{d}}\end{pmatrix}=g_{d}(s,\hat{x}_{2},\ldots,\hat{x}_{n})\equiv\begin{pmatrix}-s&0&\cdots&0\\\
0&1&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&1\end{pmatrix}B\begin{pmatrix}\hat{f}_{1}\\\ \hat{f}_{2}\\\
\vdots\\\ \hat{f}_{n}\end{pmatrix},\quad
B=\begin{pmatrix}\alpha_{1}&0&\cdots&0&0\\\
\alpha_{2}\hat{x}_{2}&1&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
\alpha_{n-1}\hat{x}_{n-1}&0&\cdots&1&0\\\
\alpha_{n}\hat{x}_{n}&0&\cdots&0&1\end{pmatrix}.$ (A.6)

The componentwise expression is

$\displaystyle\frac{ds}{d\tau_{d}}$
$\displaystyle=g_{d,1}(s,\hat{x}_{2},\ldots,\hat{x}_{n})\equiv-\frac{1}{\alpha_{1}}s\hat{f}_{1}(s,\hat{x}_{2},\ldots,\hat{x}_{n}),$
$\displaystyle\frac{d\hat{x}_{i}}{d\tau_{d}}$
$\displaystyle=g_{d,i}(s,\hat{x}_{2},\ldots,\hat{x}_{n})\equiv\hat{f}_{i}(s,\hat{x}_{2},\ldots,\hat{x}_{n})-\frac{\alpha_{i}}{\alpha_{1}}x_{i}\hat{f}_{1}(s,\hat{x}_{2},\ldots,\hat{x}_{n})\quad(i=2,\ldots,n).$

Note that the above derivation makes sense even if
$I_{\alpha}\not=\\{1,\ldots,n\\}$ provided $1\in I_{\alpha}$, namely
$\alpha_{1}>0$. The resulting vector field is as smooth as $f$ including $s=0$
and hence dynamics at infinity makes sense through dynamics generated by (A.6)
around the horizon $\mathcal{E}=\\{s=0\\}$. Finally note that a correspondence
between global/directional embeddings is provided in [38].

##  Appendix B Discussion 1: Blow-up rates and spectral properties of
invariant manifolds

Here we discuss the precise nature of blow-up behavior stated in Theorem 4.1
by means of spectral theory for linear differential systems (e.g., [45, 46, 3,
4, 5]).

An essence to describe (type-I) blow-up rates is the exponential decay with
the uniquely determined rate (4.2). One of the next issue will be when (4.2)
is achieved. Because the usual decay estimates in characterizing normal
hyperbolicity as shown in Proposition 2.3, or general hyperbolicity describing
asymptotic phase, provide only upper bounds of decay rates, which are not
sufficient to obtain (4.2) in general. Fortunately, compact boundaryless NHIMs
admits a local topological conjugacy to the linearized flow and the decay
rates are characterized by spectral properties of the linearized systems on
NHIMs. In particular, our interests are reduced to the asymptotic behavior in
systems of the form

${\bf v}^{\prime}=A(t){\bf v},$ (B.1)

where $A:\mathbb{R}\to\mathbb{R}^{n\times n}$ denotes a bounded, continuous
matrix-valued function. The spectral theory for (B.1) is back to Sacker and
Sell (e.g., [45]). Recall that the linear skew-product flow $\theta$ on the
vector bundle $\xi=(\pi,E,X)$ is the pair of continuous maps
$\theta=(\varphi,\Phi)$ given by $\theta(t,{\bf v},{\bf x})=(\varphi(t,{\bf
x}),\Phi(t,{\bf x}){\bf v})$, where $\varphi$ is a flow on the base space $X$
of $\xi$, and $\Phi(t,{\bf x})$ is linear on the fiber $F_{\varphi(t,{\bf
x})}$.

Let $\theta=(\varphi,\Phi)$ be a linear skew-product flow on a vector bundle
$\xi=(\pi,E,X)$ with a compact Hausdorff base space $X$. For each
$\lambda\in\mathbb{R}$, define the linear skew-product flow $\theta_{\lambda}$
by

$\theta_{\lambda}(t,{\bf v},{\bf x})\equiv(\varphi(t,{\bf
x}),\Phi_{\lambda}(t,{\bf x}){\bf v}):=(\varphi(t,{\bf x}),e^{-\lambda
t}\Phi(t,{\bf x}){\bf v}).$

Stability of solutions for linear skew-product flows is discussed in terms of
dichotomy given below, which is a generalization of hyperbolicity for
equilibria and periodic orbits.

######  Definition B.1 (Exponential dichotomy. e.g., [46]).

Let $M$ be a subset of the base space $X$. We say that the linear skew-product
flow $\theta_{\lambda}$ admits an exponential dichotomy over $M$ if there is a
projector $P:E(M)\to E(M)$ and positive constants $K,a>0$ such that

$\displaystyle\left|\Phi_{\lambda}(t,{\bf x})P({\bf
x})\Phi_{\lambda}^{-1}(s,{\bf x})\right|$ $\displaystyle\leq
Ke^{-a(t-s)},\quad s\leq t,$ $\displaystyle\left|\Phi_{\lambda}(t,{\bf
x})[I-P({\bf x})]\Phi_{\lambda}^{-1}(s,{\bf x})\right|$ $\displaystyle\leq
Ke^{-a(s-t)},\quad t\leq s,$

for all ${\bf x}\in M$.

The dichotomy determines the spectrum for linear skew-product flows on vector
bundles. Let $(\xi,\theta)$ be the pair of a vector bundle $\xi$ and a linear
skew-product flow $\theta$ on $\xi$.

######  Definition B.2 (Dichotomy spectrum, cf. [45, 46]).

For a subset $M\subset X$, define

$\displaystyle\rho({\bf x})$
$\displaystyle:=\\{\lambda\in\mathbb{R}\mid\theta_{\lambda}\text{ has the
exponential dichotomy}\\}\quad({\bf x}\in X),$ $\displaystyle\rho(M)$
$\displaystyle:=\bigcap_{{\bf x}\in M}\rho({\bf
x}),\quad\Sigma(M):=\mathbb{R}\setminus\rho(M).$

We call $\Sigma(M)$ the dichotomy spectrum of $(\xi,\theta)$ over $M$. By
$\Sigma({\bf x})$ we mean $\Sigma(\\{{\bf x}\\})$ for each ${\bf x}\in X$.

One of benefits to apply dichotomy spectra is that decomposition of solutions
to (B.1) into invariant subbundles is applied. It is known that the dichotomy
spectrum $\Sigma(M)$ for a compact invariant set $M$ consists of at most $n$
mutually disjoint compact intervals, which is partially characterized as
follows.

######  Lemma B.3 (cf. [46], Lemma 6).

Under the setting in Definition B.2, let $M$ be a compact invariant set in
$X$. Then the dichotomy spectrum $\Sigma(M)$ is compact. More precisely, we
have

$\Sigma(M)=[a_{1},b_{1}]\cup\cdots\cup[a_{k},b_{k}]$ (B.2)

with181818 The notation in the present subsection of $k$ is independent of the
order of asymptotically quasi-homogeneous vector fields.  $k\leq n=\dim E(M)$
and $a_{1}\leq b_{1}<a_{2}\leq\cdots\leq b_{k-1}<a_{k}\leq b_{k}$.

######  Proposition B.4 (Invariant subbundles, e.g., [46]).

Let $M$ be a compact invariant set in $X$ and $\Sigma(M)$ be the dichotomy
spectrum of $(\xi|_{M},\theta)$ given by (B.2). Then the vector bundle $E(M)$
over $M$ is decomposed into

$E(M)=E_{1}\oplus\cdots\oplus E_{k}$

as the Whitney sum, such that each subbundle $E_{i}$ over $M$ is associated
with the spectral interval $[a_{i},b_{i}]$ and invariant under $\theta$. When
$\lambda\in(b_{i},a_{i+1})$ for some $i\in\\{1,\ldots,k-1\\}$, we have

$\displaystyle\lim_{t\to+\infty}\left\|e^{-\lambda t}\Phi(t,{\bf x}){\bf
v}\right\|\to 0\quad\text{ if }\quad({\bf v},{\bf x})\in
E_{1}\oplus\cdots\oplus E_{i},$
$\displaystyle\lim_{t\to-\infty}\left\|e^{-\lambda t}\Phi(t,{\bf x}){\bf
v}\right\|\to 0\quad\text{ if }\quad({\bf v},{\bf x})\in
E_{i+1}\oplus\cdots\oplus E_{k}.$

Once we detect the location of trajectories for the linear skew-product flow
on $(\xi|_{M},\theta)$, its spectral property is extracted by means of
invariant spectral subbundles, which can vary in time-evolution within the
specified spectral intervals. The simplest case will be the discrete
distribution of spectrum.

######  Proposition B.5 (Discrete distribution of spectrum. e.g., [46]).

For a pair $(\xi,\theta)$ mentioned with $\dim\xi=n$, let $X$ be the compact
Hausdorff base space of $\xi$ and $\varphi$ be the flow on $X$ determining
$\theta$. If one of the following for the trajectory ${\bf x}(t)$ on $X$ for
$\varphi$ holds true, then $\Sigma(M)$ consists of finite points, were
$M=\overline{\\{{\bf x}(t)\\}}\equiv H({\bf x}(t))$ denotes the hull of ${\bf
x}(t)$;

  * •

${\bf x}(t)$ is an equilibrium;

  * •

${\bf x}(t)$ is a time-periodic solution;

  * •

${\bf x}(t)$ is almost periodic (cf. [51, 16]) and the spectrum $\Sigma(M)$
consists of $n$-mutually disjoint closed intervals

It is also mentioned in [46] that this claim is also true for linear skew-
product flows on the vector bundle $\xi$ over the base space $X$ being
compact, minimal and a uniquely ergodic measure $\mu$ (unique ergodicity of
sets). Indeed, [46] indicates that it is possible to have a nontrivial
dichotomy spectral interval when one does not have unique ergodicity. We shall
refer to e.g., [42, 51] for ergodicity, because we will not discuss ergodicity
of dynamical systems in the present paper, while it is left to future works.
See e.g., [51] for detailed arguments of minimal sets.

######  Remark B.6 (Lyapunov exponents and dichotomy spectrum).

Lyapunov exponents are another well-known quantities to characterize
exponential behavior of solutions to (B.1). See e.g., [1] for details.
Lyapunov exponents determine the Lyapunov spectrum, the collection of
intervals between upper and lower Lyapunov exponents. It is well known that
Lyapunov spectrum is generally a proper subset of dichotomy spectrum,
indicating that Lyapunov exponents are not suitable for characterizing
decaying/expanding properties of solutions to (B.1). See e.g., [3, 4, 5]. On
the other hand, it is shown in e.g., [46] that, if a spectral interval
$[a_{i},b_{i}]$ in the dichotomy spectrum consists of one point;
$a_{i}=b_{i}$, then the quantity coincides with the corresponding Lyapunov
exponent, namely the exponent included in $[a_{i},b_{i}]$.

######  Proposition B.7.

Under the setting of Corollary 4.3, let $N\varphi_{g;{\bf x}_{\gamma}}$ be the
linear skew-product flow on the algebraic normal bundle
$\mathcal{N}=E^{u}(V_{{\bf x}_{\gamma}})\oplus E^{s}(V_{{\bf x}_{\gamma}})$
over the hull $V_{{\bf x}_{\gamma}}=H({\bf x}_{\gamma}(\tau))$ induced by the
normally hyperbolic structure of $M$ for $g$. Assume that the dichotomy
spectrum $\Sigma_{E^{s}}(V_{{\bf x}_{\gamma}})$ of $(E^{s}(V_{{\bf
x}_{\gamma}}),N\varphi_{g;{\bf x}_{\gamma}})$ restricted to the stable
subbundle $E^{s}(V_{{\bf x}_{\gamma}})$ is discrete:

$\Sigma_{E^{s}}(V_{{\bf
x}_{\gamma}})=\\{\lambda_{1}^{s},\ldots,\lambda_{m_{s}}^{s}\\}\subset(-\infty,0),$

where $m_{s}\leq n_{s}$. Then the exponential estimate (4.2) holds.

###### Proof.

Let ${\bf x}(\tau)=T_{{\rm para};\alpha}({\bf y}(\tau))$ and denote
$E^{s}=E^{s}(V_{{\bf x}_{\gamma}})$ for simplicity. Our assumption implies
that the bundle $\mathcal{N}=E^{s}\oplus E^{u}$ admits the following Whitney
sum decomposition into spectral invariant subbundles (Proposition B.4):

$E^{s}=E_{1}^{s}\oplus\ldots\oplus E_{m_{s}}^{s},$ (B.3)

$E_{i}^{s}$ is the spectral invariant subbundle associated with the point
spectrum $\\{\lambda_{i}^{s}\\}$, $i=1,\ldots,m_{s}$. Invariant structure of
the spectral decomposition indicates that the fiber component ${\bf v}(\tau)$
of a trajectory $({\bf v}(\tau),{\bf x}_{\gamma}(\tau))=h({\bf x}(\tau))$ for
the linear skew-product flow $N\varphi_{g;{\bf x}_{\gamma}}$ with ${\bf
x}(\tau)=T_{{\rm para};\alpha}({\bf y}(\tau))$ and the topological conjugacy
$h$ (cf. Proposition 2.3) is located on the subbundle

$E^{s}_{i_{1}}\oplus\cdots\oplus E^{s}_{i_{p}}\quad\text{ with }\quad
i_{1},\ldots,i_{p}\in\\{1,\ldots,m_{s}\\}\quad\text{ and }\quad
i_{1}<\cdots<i_{p}$ (B.4)

for all $\tau\in\mathbb{R}$. In particular, ${\bf v}(\tau)$ is uniquely
decomposed into

$\displaystyle{\bf
v}(\tau)=v_{i_{1}}(\tau)+\cdots+v_{i_{p}}(\tau),\quad\text{where}\quad(v_{i_{l}}(\tau),{\bf
x}_{\gamma}(\tau))\in E_{i_{l}}^{s}\,\text{ with }\quad
v_{i_{l}}(\tau)\not=0\,\text{ for all }\quad l=1,\ldots,p.$

Without the loss of generality, we may assume that
$\lambda_{1}<\cdots<\lambda_{m_{s}}<0$. By the characterization of exponential
dichotomy, each component $v_{i_{l}}(\tau)$ satisfies

$e^{(-\lambda_{i_{l}}+\epsilon)\tau}|v_{i_{l}}(\tau)|=\begin{cases}+\infty&\epsilon>0\\\
0&\epsilon<0\end{cases}\quad\text{ as }\quad\tau\to+\infty.$

The original ${\bf x}(\tau)$ as the image of such a trajectory through the
conjugacy $h$ is written by

${\bf x}(\tau)\equiv h^{-1}({\bf v}(\tau),{\bf x}_{\gamma}(\tau))={\bf
x}_{\gamma}(\tau)+{\bf v}(\tau)+\tilde{h}^{-1}({\bf v}(\tau),{\bf
x}_{\gamma}(\tau))$

via the identity $\tilde{h}^{-1}({\bf x})=h^{-1}({\bf x})-{\bf x}$ being
continuous and bounded (choosing a neighborhood of $M$ in $W_{\rm loc}^{s}(M)$
smaller so that it is included in a compact neighborhood of $M$ in $W_{\rm
loc}^{s}(M)$ ensuring the existence of $h$). Because the conjugacy $h$
preserves $M$, namely $h({\bf x})={\bf x}$ for all ${\bf x}\in M$, exponential
decay of ${\bf v}(\tau)$, continuity of $h$ with the mentioned property as
well as the estimate (2.3) yield

$\displaystyle e^{(\lambda_{i}+\epsilon)\tau}\left|(h^{-1})_{i}({\bf
v}(\tau),{\bf x}_{\gamma}(\tau))-x_{i;\gamma}(\tau)\right|$
$\displaystyle=e^{(\lambda_{i}+\epsilon)\tau}\left|v_{i}(\tau)+(\tilde{h}^{-1})_{i}({\bf
v}(\tau),{\bf x}_{\gamma}(\tau))\right|$
$\displaystyle=\begin{cases}+\infty&\epsilon>0\\\
0&\epsilon<0\end{cases}\quad\text{ as }\quad\tau\to+\infty$

with some $-\lambda_{i}\leq\lambda_{s}<0$ for all $i=1,\ldots,n$, where
$\lambda_{s}\in\\{\lambda_{i}\\}_{i=1}^{m_{s}}$ is uniformly determined. ∎

As a consequence, for NHIMs admitting the spectral information mentioned in
the proposition, Theorem 4.1 can be applied. In particular, combining
Proposition B.5, we can rephrase results in the preceding paper [38].

######  Corollary B.8 (Stationary and periodic blow-ups, [38]).

In Theorem 4.3, assume that $M$ is either a hyperbolic equilibrium or a
hyperbolic periodic orbit. Then all statements in Theorem 4.1 hold.

Blow-ups in backward time direction can be described in the similar way, in
which case trajectories on the local unstable manifold $W_{\rm
loc}^{u}(M;\varphi_{g})$ determine the behavior.

##  Appendix C Discussion 2: Blow-up rates in nonautonomous blow-ups

This appendix addresses several practical ideas to apply Theorem 5.1 involving
blow-up rates in concrete nonautonomous systems, and their validity to prove
the concrete blow-up behavior.

###  C.1 Blow-up rates

One approach is to use the idea stated in Proposition B.7 in autonomous cases;
the spectral information of $M_{I}$, the tube of an invariant manifold over
$I\subset\mathbb{R}$ given in (5.8), for the linearized systems. In
Proposition B.7, the spectral information is applied through the local
topological conjugacy, which is generally admitted for boundaryless NHIMs (cf.
[20, 44]). Because our invariant manifolds (for the modified vector field
$\tilde{g}$ given in (5.10)) of the form $M_{I^{\prime}}$ have nontrivial
boundary, some modifications are necessary if we apply the above conjugacy.
Here we refer to a result in [10], where global linearization of (normally
attracting) invariant manifolds is provided, no matter when invariant
manifolds have boundaries. The technique relies on several boundaryless
treatments in differential topology (e.g., [19]) and the uniformity
lemma191919 If we try to provide the similar result to invariant manifolds
which are not normally hyperbolic, the corresponding estimate will be
necessary.  stemming from normally hyperbolic structure of manifolds (e.g.,
[11, 12, 54]) to ensure persistence of normally hyperbolic structure in
perturbed vector fields.

######  Proposition C.1.

Consider the desingularized vector field $g$ in (5.5) satisfying requirements
in Assumption 5.2. Then there are a Riemannian manifold $\hat{Q}$ and a vector
field $\hat{g}$ on $\hat{Q}$ such that

  * •

$M_{I}$ is embedded into a compact, boundaryless, normally attracting
invariant manifold $\hat{N}\subset\hat{Q}$,

  * •

$M_{I}$ is normally attracting for $\hat{g}$ as the submanifold of $Q$, and
that

  * •

$\hat{g}=g$ on $W^{s}(M_{I};\varphi_{\hat{g}})$ as the submanifold of
$W^{s}(\hat{N};\varphi_{\hat{g}})$, where $\varphi_{\hat{g}}$ is the flow
generated by $\hat{g}$. In particular, the stable foliation of
$W^{s}(M_{I};\varphi_{\hat{g}})$ is constructed, which is consequently
regarded as the foliation of $W^{s}(M_{I};\varphi_{g})$.

###### Proof.

Let $\tilde{g}$ be the modified desingularized vector field given in (5.10)
admitting a sequence of normally hyperbolic manifolds $M_{I}\subset
M_{\tilde{I}}\subset M_{I^{\prime}}$ such that $M_{I}$ is invariant, while
$M_{\tilde{I}}$ and $M_{I^{\prime}}$ are inflowing invariant. Let
$Q:=W^{s}(M_{I^{\prime}};\varphi_{\tilde{g}})$, the global stable manifold of
$M_{I^{\prime}}$ for $\tilde{g}$. We then apply Proposition C.4 below to
$N=M_{\tilde{I}}$ and $M=M_{\hat{I}}$, which directly yields the result. ∎

Once the treatment of invariant manifolds on $\mathcal{E}$ admitting normally
hyperbolic structure is understood, blow-up rates are evaluated in the similar
way to autonomous cases.

######  Corollary C.2 (Nonautonomous blow-up: blow-up rates).

Under the setting in Corollary 5.4, if all assumptions in Proposition B.7 are
satisfied for the hull $V_{{\bf x}_{\gamma}}\equiv{\rm cl}(\gamma)$, then the
blow-up solution ${\bf y}(t)$ admits the asymptotic behavior

$\displaystyle p({\bf y}(t))\sim
C_{0}(-\ln(t_{\max}-t))(t_{\max}-t)^{-1/k}\quad\text{ as }\quad t\to
t_{\max}-0$

for some function $C_{0}(-\ln(t_{\max}-t))$ satisfying

$C_{0}(\tau)=o(e^{\epsilon\tau}),\quad C_{0}(\tau)^{-1}=o(e^{\epsilon\tau})$

for any $\epsilon>0$ as $\tau\to\infty$, equivalently
$-\ln(t_{\max}-t)\to+\infty$, and

$y_{i}(t)\sim
C_{0}(-\ln(t_{\max}-t))^{\alpha_{i}}x_{\gamma,i}(-\bar{c}\ln(t_{\max}-t))(t_{\max}-t)^{-\alpha_{i}/k}\quad\text{
as }\quad t\to t_{\max}-0$

for some constant $\bar{c}>0$, provided $x_{M,i}(\tau)\not\to 0$ as
$\tau\to\infty$.

###### Proof.

It follows from Proposition C.1 that $M_{I}$ as the submanifold of
$Q=W^{s}(M_{I^{\prime}};\varphi_{g})$ admits the stable foliation of $W_{\rm
loc}^{s}(M_{I};\varphi_{\hat{g}})=W_{\rm loc}^{s}(M_{I};\varphi_{g})$. The
statement then follows from the distribution of $M_{I}\subset\mathcal{E}$, in
particular of $\pi_{t}M_{I}$ in $\mathbb{R}$. Results involving asymptotic
behavior of $p({\bf y}(t))$ and $y_{i}(t)$ are the direct consequence of
Proposition B.7 replacing $M$ with $\hat{N}$ stated in Proposition C.1. ∎

######  Remark C.3 (Beyond normal hyperbolicity).

Here we have restricted our attention to NHIMs because our geometric
description of blow-ups relies on embeddings of compact invariant manifolds
with boundary to compact “boundaryless” manifolds with the same normal
hyperbolicity restricted to the original manifolds. This technique is based on
[10] and its validity beyond normal hyperbolicity (like situations in [37])
remains nontrivial. If the similar embedding technique is constructed to
general invariant manifolds (admitting asymptotic phase), the same conclusion
for blow-up rates will be provided.

###  C.2 Proof of Corollary 5.5

The proof is a direct consequence of combinations of Proposition B.5 and
Corollary C.2.

First, thanks to Proposition C.1, the manifold $M_{I}$ is embedded into a
compact, boundaryless NAIMs in a manifold $\hat{Q}$ (mentioned in Proposition
C.4 for $Q=W^{s}(M_{I^{\prime}};\varphi_{\tilde{g}})$), and invariant
foliation of $W^{s}_{\rm loc}(M_{I};g)$ is admitted. In particular, $g$ admits
a local topological conjugacy on $M_{I}$ to $Dg|_{\hat{Q}}$, where we have
used the identification of $g$ and the modified vector field $\hat{g}$ on
$W^{s}_{\rm loc}(M_{I};g)$. Next, because $M_{I}$ is assumed to be a
collection of equilibria, the dichotomy spectrum of $Dg$ on each hull $V_{{\bf
x}_{\gamma}}$ of trajectories on $M_{I}$, which is exactly a point ${\bf
x}(t)\in M_{I}$ itself for each $t\in I$, are discrete by Proposition B.5.
Therefore, Corollary C.2 can be applied to obtaining the conclusion.

###  C.3 Reduction to boundaryless inflowing NAIMs

Finally we review a technique to modify inflowing invariant manifolds with
normally hyperbolic structure with non-trivial boundary to boundaryless ones
used in the argument of Proposition C.1. This was provided in [10], where the
reduction of inflowing normally attracting invariant manifolds (NAIMs) with
boundaries to boundaryless ones was discussed. The main aim of the discussion
was to apply global linearization theorem for boundaryless NHIMs stated in
preceding works (e.g. [20, 44]) to inflowing NAIMs. See e.g., [19] for
fundamental concepts in differential geometry and differential topology. Let
$Q$ be a $C^{r\geq 1}$ Riemannian manifold.

######  Proposition C.4 (Reduction to boundaryless NAIM, [10], Proposition
B.1).

Let $M,N\subset Q$ be compact inflowing $r$-NAIMs, with $M\subset{\rm
int}_{Q}N$, for the $C^{r\geq 1}$ flow $\varphi$ generated by the $C^{r\geq
1}$ vector field $f$ on $Q$. Let $U_{0}$ be an arbitrarily small tubular
neighborhood of $\partial N$, having smooth boundary $\partial U_{0}$ and
disjoint from $W^{s}_{\rm loc}(M;\varphi)$. Define $\hat{Q}$ to be the double
of $Q\setminus U_{0}$.

Then there is a $C^{\infty}$ differential structure on $\hat{Q}$ and a $C^{r}$
vector field $\hat{f}:\hat{Q}\to T\hat{Q}$ satisfying the following
properties:

  1. 1.

$\hat{f}$ is equal to $f$ on each copy of $Q\setminus U_{0}$ except on an
arbitrarily small neighborhood of $\partial U_{0}$.

  2. 2.

There is a compact and boundaryless $r$-NAIM $\hat{N}$ for $\hat{f}$, with
$\hat{N}$ equal to $N$ on each copy of $Q\setminus U_{0}$, except on an
arbitrarily small neighborhood of $\partial U_{0}$.

  3. 3.

The global stable foliation of $M$ for $f$ does not intersect $U_{0}$, and it
coincides with the global stable foliation of $M$ for $\hat{f}$, where $M$ and
$W^{s}(M)$ are identified via inclusion with subset of a copy of $Q\setminus
U_{0}$ in $\hat{Q}$.

Furthermore, let $\hat{\varphi}$ be the $C^{r}$-flow generated by $\hat{f}$,
and let $\hat{E}^{s}$ be the $D\hat{\varphi}^{t}|_{\hat{N}}$-invariant stable
vector bundle for the NAIM $\hat{N}$. If, additionally, there are constants
$C>0$ and $\alpha<0$ such that, for all $m\in M$, $t\geq 0$ and $0\leq i\leq
k$, the $k$-center bunching condition202020 This condition implies that
$W^{s}(M;\varphi)$ and $E^{s}$ as fiber or vector bundles are $C^{k}$,
according to [13].

$\|D\varphi^{t}|_{T_{m}M}\|^{i}\|D\varphi^{t}|_{E^{s}_{m}}\|\leq Ce^{\alpha
t}m\left(D\varphi^{t}|_{T_{m}M}\right)$ (C.1)

is satisfied for the original system on $Q$, then (C.1) is also satisfied
replacing $M$, $E^{s}$ and $\varphi^{t}$ with $\hat{N}$, $\hat{E}^{s}$ and
$\hat{\varphi}^{t}$, respectively, and $\alpha$ with some different constant
$\hat{\alpha}<0$.

Similarly, if additionally there are constants $0<\delta<-\alpha<-\beta$ and
$C\geq 1$ such that, for all $t\geq 0$,

$\begin{cases}C^{-1}e^{-\delta t}\leq
m\left(D\varphi^{t}|_{TM}\right)\leq\|D\varphi^{t}|_{TM}\|\leq Ce^{\delta
t},&\\\ C^{-1}e^{-\delta t}\leq
m\left(\left(D\varphi^{t}|_{TM}\right)^{-1}\right)\leq\|\left(D\varphi^{t}|_{TM}\right)^{-1}\|\leq
Ce^{\delta t},&\\\ C^{-1}e^{\beta t}\leq
m\left(D\varphi^{t}|_{E^{s}}\right)\leq\|D\varphi^{t}|_{E^{s}}\|\leq
Ce^{\alpha t}&\end{cases}$ (C.2)

uniformly on $TM$ and $E^{s}$, then we can choose $\hat{f}$ appropriately such
that the same conclusion holds true for $\hat{\varphi}^{t}$, $T\hat{N}$ and
$\hat{E}^{s}$ with modified constants
$0<\hat{\delta}<-\hat{\alpha}<-\hat{\beta}$ arbitrarily close to
$\delta,\alpha,\beta$, respectively.

The essence of this result is that the differential structure of manifolds and
vector fields are modified only in a neighborhood of $\partial U_{0}$,
although the ambient manifold $Q$ is implicitly assumed to be boundaryless.
Even when $Q$ has a nontrivial boundary $\partial Q$, the above technique can
be applied as long as $(Q,\partial Q)$ is a Riemannian manifold and $\partial
Q$ is invariant for the vector field $f:Q\to TQ$. In this result, normally
attracting, or normally hyperbolic properties of $N$ and $M$ is assumed. In
this situation, the uniformity lemma212121 If we try to provide the similar
result to invariant manifolds which are not normally hyperbolic, the
corresponding estimate will be necessary.  can be applied (e.g., [11, 12, 54])
to making uniform estimates in (C.2).

In Appendix C.1, this proposition is used to provide linearization of dynamics
at infinity for nonautonomous systems. More precisely, this technique is used
to characterize spectral properties for invariant manifolds in nonautonomous
systems by means of compact, boundaryless invariant manifolds, where local
topological conjugacy can be applied. Although our interests are NHIMs, all
arguments involving NAIMs can be applied by restricting our attention to the
(global) stable manifold: $Q=W^{s}(M;\varphi)$ with a compact, connected
(inflowing) invariant manifold $M$ with a normally hyperbolic structure, then
$M$ is normally attracting for the flow restricted to $Q$, and hence the
theory of NAIMs can be directly applied222222 This is possible because the
stable manifold $W^{s}(M)$ of an invariant manifold $M$ for a given flow is
also invariant. .