# Noise-induced chaos: a conditioned random dynamics perspective

Bernat Bassols-Cornudella<EMAIL_ADDRESS>Department of Mathematics, Imperial College London, London SW7 2AZ, UK  Jeroen
S.W. Lamb<EMAIL_ADDRESS>Department of Mathematics, Imperial
College London, London SW7 2AZ, UK  International Research Center for
Neurointelligence, The University of Tokyo, Tokyo 113-0033, Japan  Centre for
Applied Mathematics and Bioinformatics, Department of Mathematics and Natural
Sciences, Gulf University for Science and Technology, Halwally 32093, Kuwait

###### Abstract

We consider transitions to chaos in random dynamical systems induced by an
increase of noise amplitude. We show how the emergence of chaos (indicated by
a positive Lyapunov exponent) in a logistic map with bounded additive noise
can be analyzed in the framework of conditioned random dynamics through
expected escape times and conditioned Lyapunov exponents for a compartmental
model representing the competition between contracting and expanding behavior.
In contrast to the existing literature, our approach does not rely on small
noise assumptions, nor refers to deterministic paradigms. We find that the
noise-induced transition to chaos is caused by a rapid decay of the expected
escape time from the contracting compartment, while all other order parameters
remain approximately constant.

Random dynamical systems, random logistic map, noise-induced chaos, Lyapunov
exponent, conditioned Lyapunov exponent, non-linear dynamics, conditioned
random dynamics.

††preprint: CHA/123-QED

> It is well-known that adding noise to a nonlinear system with a non-chaotic
> attractor can cause the attractor to become chaotic. This phenomenon is
> known as noise-induced chaos. In the literature, studies of such
> bifurcations have focused on asymptotic properties of order parameters near
> criticality under small noise assumptions, with reference to deterministic
> paradigms. We consider noise-induced chaos from a novel perspective,
> employing concepts from the theory of conditioned random dynamical systems,
> such as the recently proposed conditioned Lyapunov exponent, that exist
> independently of the strength of the noise. In particular, we find that in a
> prototypical random logistic map, the noise-induced transition to chaos is
> caused by a noise-induced (rapid) decrease of the average escape time from
> the original attractor. This paper successfully pilots conditioned dynamics
> as a mathematical framework to study bifurcations in random systems beyond
> the small noise setting.

##  I Introduction

Chaos theory is considered to be one of the scientific revolutions of the 20th
century, providing new paradigms for the understanding of predictability and
stability with broad repercussions, from fundamental science to applied
engineering.

A key question in this theory addresses the mechanisms through which chaos
arises. In low-dimensional deterministic systems, various universal routes to
chaos have been identified, e.g. via period-doubling cascades[1],
intermittency[2], torus bifurcations[3] and homoclinic bifurcations[4].

There is recently a growing recognition of the importance to study dynamics in
the presence of noise, for instance in the context of quantum mechanics[5],
turbulence[6] and ecology[7]. Such random dynamical systems display behaviors
that are different from their deterministic counterparts.[8, 9] The
corresponding theory, however, remains in its early stages of development. In
particular, noise-induced transitions such as noise-induced order[10, 11]
(where adding noise to a deterministic chaotic systems suppresses chaos) and
noise-induced chaos[12] (where adding noise to a non-chaotic deterministic
system causes chaos) have been long recognized but remain poorly
understood.[13]

In this paper, we study noise-induced chaos using concepts from conditioned
random dynamical systems, providing the first practical application of the
recently introduced conditioned Lyapunov exponent.[14, 15]

Previous studies on noise-induced chaos[12, 16, 17, 18, 19, 20, 13] have
primarily focused on asymptotic properties of order parameters near
criticality, relying on certain insights into long chaotic transients in
deterministic systems and relating escape times to fractal properties of the
repeller.[20] Consequently, these studies assume a perturbative small noise
setting. Nonetheless, noise-induced chaos arises only beyond a determined
finite noise strength, with markedly different dynamics at smaller noise
levels. The interplay between noise and underlying deterministic equations of
motion lies at the heart of random dynamical systems theory. While in the
limit of small noise the resulting dynamics is almost deterministic, it is
beyond this limit where truly novel dynamical phenomena, like noise-induced
chaos, emerge.

This pilot study provides motivation for further research into the development
of effective compartmental models for random dynamical systems. Despite
progress in the theory of conditioned random dynamics[14, 21, 15, 22], further
theoretical progress is necessary to gain deeper insights. Similarly,
efficient computational tools for the analysis of concrete examples remain
only in their early stages of development.

##  II Random logistic map

As a paradigmatic prototypical model for noise-induced chaos, we consider a
logistic map with uniform bounded additive noise:

$\begin{split}x_{n+1}&=f(x_{n})+\omega_{n}\eqqcolon f_{\omega_{n}}(x_{n}),\\\
\omega_{n}&\sim\mathbb{P}\coloneqq\text{Unif}(-\varepsilon,\varepsilon),\end{split}$
(1)

where $f(x)\coloneqq 3.83x(1-x)$ is the logistic map with growth factor $3.83$
and $\varepsilon>0$ is the noise amplitude. In the absence of noise
($\varepsilon=0$) almost all initial conditions in $[0,1]$ are attracted to a
period-3 orbit while the complement of the domain of attraction contains a
chaotic repeller. At each iteration, $\omega_{n}$ is chosen i.i.d. from
$[-\varepsilon,\varepsilon]$ with uniform probability $\mathbb{P}$. We refer
to the semi-infinite sequence $\omega=\omega_{0}\omega_{1}\dots$ as a noise
realization and write $f^{n}_{\omega}(x_{0})\coloneqq
f_{\omega_{n-1}}\circ\ldots\circ f_{\omega_{0}}(x_{0}).$

Eq. (1) was considered in Ref. 23 with unbounded Gaussian noise. In many
modeling settings, bounded noise is natural[24] and unbounded noise may have
unintended dynamical consequences. For instance, with Gaussian unbounded
noise, trajectories of Eq. (1) almost surely tend to $-\infty$, whereas the
bounded noise version has a bounded attractor for modest noise amplitudes. As
a result, numerical results in Ref. 23 concern empirical statistics of finite
time observations, which are transient.

Eq. (1) is known to exhibit three dynamical regimes depending on the value of
the noise amplitude[25], $\varepsilon$, which are characterized by the support
of the (unique) stationary density, $p$, and the sign of the Lyapunov
exponent, $\Lambda$. Recall that the Lyapunov exponent is an indicator of a
system’s stability in the sense that it measures the exponential rate of
expansion (when positive) or contraction (when negative) of the distance
between forward orbits of two infinitesimally close points[26]. For the random
logistic map with bounded additive noise in Eq. (1), the Lyapunov exponent is
given by[25]:

$\begin{split}\Lambda&\coloneqq\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\ln|a(1-2x_{n})|\\\
&=\int_{A}\ln|a(1-2y)|p(y)\mathrm{d}y,\end{split}$

where the latter equality holds for $p$-almost every $x_{0}$ by Birkhoff’s
ergodic theorem. Here, $A=\mathrm{supp}\ p$ is the unique attractor of the
random logistic map identified in Ref. 27.

Figure 1: On the left, bifurcation diagram for the deterministic logistic map
$x_{n+1}=ax_{n}(1-x_{n})$, with $a\in[3.6,4]$. The value $a=3.83$ shows to be
inside the three periodic window, where almost all orbits converge towards a
three periodic orbit. On the right, bifurcation diagram for the random
logistic map $x_{n+1}=3.83x_{n}(1-x_{n})+\omega_{n}$, together with the
system’s Lyapunov exponent, $\Lambda$, with
$\omega_{n}\sim\text{Unif}(-\varepsilon,\varepsilon)$ and
$\varepsilon\in[0,0.0025]$. The three phases are separated by two dashed
vertical lines at the bifurcation points $\varepsilon_{D}$ and
$\varepsilon_{L}$.

The limiting dynamics of this system are sketched in FIG. 1. We briefly
summarize the main features of interest from Refs. 27, 25. In the absence of
noise, $\varepsilon=0$, almost all orbits converge towards a period three
attractor. If noise is small, $\varepsilon<\varepsilon_{D}\approx 0.000251$,
this attracting cycle fattens into a random periodic attractor $A$, consisting
of three disjoint intervals that are cyclically permuted by the dynamics.
Orbits in this attractor uniformly converge to each other if and only if they
start in the same interval.

As the noise level $\varepsilon$ increases, at $\varepsilon_{D}$, a
topological bifurcation merges these three intervals into one connected
attractor, after which any two orbits in this attractor almost surely converge
to a single random point attractor. This topological bifurcation is naturally
viewed from a set-valued perspective[28] and associated with a saddle-node
bifurcation of a map with extreme noise realization, cf. Ref. 25 in the
context of Eq. (1). In this regime, noise-induced synchronization (in the
sense of almost sure convergence of trajectories) is ensured[29] by the
negativity of the Lyapunov exponent $\Lambda$ although the convergence to the
random fixed point is no longer uniform.[30] At
$\varepsilon=\varepsilon_{L}\approx 0.00146$, the Lyapunov exponent becomes
positive and the random point attractor bifurcates into a random chaotic
attractor.

Following Ref. 25, we refer to the three corresponding dynamical regimes as
phase I ($\varepsilon<\varepsilon_{D}$), phase II
($\varepsilon_{D}<\varepsilon<\varepsilon_{L}$) and phase III
($\varepsilon_{L}<\varepsilon$).

In this paper we focus on the transition from (non-uniform) synchronization in
phase II ($\varepsilon_{D}<\varepsilon<\varepsilon_{L}$) to chaos in phase III
$(\varepsilon_{L}<\varepsilon)$. To reveal the mechanism driving this
transition, we consider the two-point motion of pairs of initial conditions
$(x_{0},y_{0})$ under the random dynamics with identical noise realization:

$\begin{array}[]{ccccc}x_{n+1}&=&ax_{n}(1-x_{n})&+&\omega_{n},\\\
y_{n+1}&=&ay_{n}(1-y_{n})&+&\omega_{n}.\end{array}$

To track the relative exponential expansion and contraction we introduce the
coordinates

$u_{n}\coloneqq\ln|x_{n}-y_{n}|\quad\text{and}\quad
v_{n}\coloneqq\frac{x_{n}+y_{n}}{2},$

in terms of which we rewrite the two-point motion as

$\begin{array}[]{ccl}u_{n+1}&=&u_{n}+\ln|a(1-2v_{n})|,\\\
v_{n+1}&=&av_{n}(1-v_{n})+\omega_{n}-\dfrac{a}{4}\exp(2u_{n}).\end{array}$

The benefit of this change of coordinates is two-fold. First, it allows for
accurate computation of fluctuations of the distance between two trajectories
close to synchronization. Secondly, exponential growth or decay of the
distance between two orbits manifests itself as linear growth or decay of
$u_{n}$.

Figure 2: Time series evolution of $u_{n}$ for an initial condition
$(x_{0},y_{0})$ of the two-point motion for phase II
($\varepsilon_{D}<\varepsilon=0.00125<\varepsilon_{L},\Lambda<0$), at the
transition point ($\varepsilon=0.00146\simeq\varepsilon_{L},\Lambda=0$), and
phase III ($\varepsilon_{L}<\varepsilon=0.00175,\Lambda>0$).

In FIG. 2 we illustrate the typical evolution of $u_{n}$ for a pair of close
initial conditions and for three representative values of $\varepsilon$: one
in phase II ($\varepsilon_{D}<\varepsilon<\varepsilon_{L},\Lambda<0)$, one at
the transition point $(\varepsilon=\varepsilon_{L},\Lambda=0)$, and one in
phase III ($\varepsilon>\varepsilon_{L},\Lambda>0)$. On each time series,
characteristic bursts stand out as finite-time trajectories where $u_{n}$
grows or decays linearly for relatively long time intervals, representing
desynchronising and synchronizing excursions, respectively. Moreover, we
observe in each time-series that desynchronising bursts have similar growth
rates and, likewise, desynchronising ones have similar decay rates.

##  III A conditioned random dynamics perspective

We view the noise-induced transition to chaos as characterized by the change
of sign of the Lyapunov exponent due to a competition between stable and
unstable regimes. This leads us to propose a compartmental model, partitioning
the attractor $A$ into two regions, $M_{\text{exp}}$ and $M_{\text{cont}}$,
with predominantly expanding and contracting dynamics, respectively. Each
compartment is characterized by its own expected escape time,
$\tau_{\text{exp}}$ and $\tau_{\text{cont}}$, as well as its effective
exponential rate of expansion and contraction, $\Lambda_{\text{exp}}$ and
$\Lambda_{\text{cont}}$. Thus, the overall exponential growth rate of the
model can be expressed as the weighted average

$\overline{\Lambda}\coloneqq\dfrac{{\tau}_{\text{exp}}\Lambda_{\text{exp}}+{\tau}_{\text{cont}}\Lambda_{{\text{cont}}}}{{\tau}_{\text{exp}}+{\tau}_{\text{cont}}}.$
(2)

If the two-state model correctly reflects the dynamics of the random logistic
map in Eq. (1), then of course $\overline{\Lambda}\simeq\Lambda$.

We note that a compartmental point of view towards noise-induced chaos, and
Eq. (2), were also proposed in Ref. 19, cf. also Ref. 13. The crucial
difference between their treatment and ours is that the former relies on
properties of transient dynamics in the deterministic limit. In particular,
growth rates are identified with local Lyapunov exponents of attractors and
non-attracting chaotic sets, which are established in a heuristic way,
supported by numerical observations. Expected escape times are related to
phenomenological quasipotentials, in a perturbative - small noise - setting.
In contrast, in this paper we identify the escape times and exponential growth
rates in Eq. (2) with expected escape times[31] and conditioned Lyapunov
exponents[14, 15] from the theory of conditioned random dynamical systems.[32,
33] These are mathematically precise and do not rely on small noise
assumptions, nor refer to deterministic limits.

The theory of conditioned random dynamics concerns the behavior of
trajectories that stay within a domain $M$ of the state space for
asymptotically long times. The corresponding statistics are described by the
so-called quasi-stationary density $m$, which is the leading eigenmode of the
conditioned transfer operator $\mathcal{L}_{M}$:[33]

$\mathcal{L}_{M}m\coloneqq\mathbb{E}(m\circ f_{\omega}\cdot\mathbf{1}_{M}\circ
f_{\omega})=\lambda_{M}m,$ (3)

where the expectation $\mathbb{E}$ is taken with respect to the noise
realizations $\omega$, $g\circ f_{\omega}$ is the evolution of the density
under the dynamics, $\mathbf{1}_{M}\circ f_{\omega}$ controls survival of
trajectories within $M$, and $\lambda_{M}\in(0,1)$ is the largest eigenvalue
of $\mathcal{L}_{M}$. Indeed, the conditioned time evolution of any initial
density converges - up to scalar multiple - to the quasi-stationary density
$m$.[33]

The eigenvalue $\lambda_{M}$ is the so-called (asymptotic) escape rate from
$M$ since it follows from Eq. (3) that

$\lambda_{M}=\int_{M}\mathbb{P}(f_{\omega}(x)\in M)m(x)\mathrm{d}x,$ (4)

i.e. $\lambda_{M}$ expresses the probability that trajectories remain in $M$
given that their initial conditions are distributed according to $m$. The
associated expected escape time from $M$ is given by[31]

${\tau}_{M}=\sum_{n=1}^{\infty}n(1-\lambda_{M})\lambda_{M}^{n-1}=\frac{1}{1-\lambda_{M}},$
(5)

where $(1-\lambda_{M})\lambda_{M}^{n-1}$ is the average over $m$ of the
probability that a trajectory leaves $M$ for the first time at the $n$-th
iterate.

The theory of conditioned random dynamics admits a version of Birkhoff’s
ergodic theorem involving averages over so-called quasi-ergodic, rather than
quasi-stationary, densities.[34, 32, 21] A probability density $q$ is quasi-
ergodic if in the limit $n\to\infty$, the expectation of the time average for
any observable $g$ conditioned on survival on $M$ equals the expectation of
$g$ with respect to $q$. In particular, the conditioned Lyapunov exponent
$\Lambda_{M}$ for the random logistic map in Eq. (1) is given by

$\Lambda_{M}\coloneqq\int_{M}\ln|a(1-2y)|q(y)\mathrm{d}y.$

This conditioned Lyapunov exponent represents the infinitesimal exponential
growth rate of orbits that remain asymptotically long within $M$. Like
Lyapunov exponents, conditioned Lyapunov exponents provide useful dynamical
insights.[14]

The formal relationship between the quasi-stationary and quasi-ergodic
densities is given by (see e.g. Ref. 21, Theorem C)

$q(x)=\dfrac{v(x)m(x)}{\int_{M}v(y)m(y)\mathrm{d}y},$ (6)

where $v$ is the unique non-negative continuous eigenfunction of the adjoint
operator of $\mathcal{L}_{M}$ with eigenvalue $\lambda_{M}$:

$\lambda_{M}v(x)=\int_{M}v(y)\mathcal{\mathbb{P}}(f_{\omega}(x)\in\mathrm{d}y).$

In practice, quasi-stationary and quasi-ergodic distributions rarely admit
analytic expressions, but numerical approximations of these densities can be
obtained using Ulam’s method.[35, 36, 37] This entails a coarse-grained
approximation of the conditioned transfer operator on a $k$-component
partition of $M$, represented by a $k\times k$ sub-stochastic transition
matrix $\mathbf{P}$, where the matrix elements $\mathbf{P}_{ij}$ denote the
probability to evolve from the $i$-th to the $j$-th component. The maximal
eigenvalue ${\lambda}$ of $\mathbf{P}$ approximates the escape rate
$\lambda_{M}$ in Eq. (4) as $k\to\infty$. The left probability eigenvector
$\mathbf{m}=(m_{1},\dots,m_{k})$ of $\mathbf{P}$ associated with ${\lambda}$
approximates the quasi-stationary density $m$, and the right probability
eigenvector $\mathbf{v}=(v_{1},\dots,v_{k})$ of $\mathbf{P}$ associated with
${\lambda}$ approximates the function $v$.[34] Therefore, the quasi-ergodic
density $q$ of the conditioned process in Eq. (6) is approximated by the
probability vector $\mathbf{q}=(q_{1},\dots,q_{k})$ with

$q_{i}=\dfrac{v_{i}m_{i}}{\sum_{j=1}^{k}v_{j}m_{j}},\quad i=1,\dots,k.$

Returning to our two-state model, with expanding and contracting compartments
$M_{\text{exp}}$ and $M_{\text{cont}}$, we propose to approximate the escape
times $\tau_{\text{exp}}$ and $\tau_{\text{cont}}$ with the expected
asymptotic escape times $\tau_{M_{\text{exp}}}$ and $\tau_{M_{\text{cont}}}$
obtained from the respective quasi-stationary densities, cf. Eq. (5).
Similarly, we approximate the rates of expansion and contraction
$\Lambda_{\text{exp}}$ and $\Lambda_{\text{cont}}$ with the respective
conditioned Lyapunov exponents. In other words,

$\begin{array}[]{cccccc}\tau_{\text{exp}}&\coloneqq&\tau_{M_{\text{exp}}},&\quad\tau_{\text{cont}}&\coloneqq&\tau_{M_{\text{cont}}},\\\
\Lambda_{\text{exp}}&\coloneqq&\Lambda_{M_{\text{exp}}},&\quad\Lambda_{\text{cont}}&\coloneqq&\Lambda_{M_{\text{cont}}}.\end{array}$

This choice implies the assumption that conditioned long-term asymptotic
behavior dominates the dynamics in each compartment, neglecting short-term
transients to this asymptotic behavior.

Inspired by the three-periodic attractor of the underlying deterministic
logistic map, we consider the partition of the attractor $A$ into the
following compartments:

$\begin{array}[]{lcl}M_{\text{exp}}&\coloneqq&\\{x\in
A\,|\,|(f^{3}_{\omega})^{\prime}(x)|\geq 1,\,\forall\,\omega\\},\\\
M_{\text{cont}}&\coloneqq&\\{x\in A\,|\,\exists\,\omega\text{ s.t.
}|(f^{3}_{\omega})^{\prime}(x)|<1\\}.\end{array}$

For random dynamical systems like the logistic map in Eq. (1), and with this
type of compartments (that do not contain the support of any stationary
density), it is known[21] that unique quasi-stationary and quasi-ergodic
densities exist, so all proposed expected escape times and conditioned
Lyapunov exponents are well-defined.

Importantly, the theory of conditioned random dynamics applies equally to the
bounded and unbounded noise settings. In particular, this theory can in
principle also be used to give the numerical observations on transient
dynamics for the random logistic map with unbounded Gaussian noise in Ref. 23
a rigorous mathematical footing.

##  IV Noise-induced chaos

The numerically obtained Lyapunov exponent $\overline{\Lambda}$ of the
corresponding two-state model is presented in FIG. 3 for a range of noise
amplitudes $\varepsilon$ increasing from phase II to phase III. For
comparison, the Lyapunov exponent $\Lambda$ of the random logistic map is also
depicted. We obtain excellent qualitative and good quantitative agreement. We
discuss the reason for the slight overestimation of $\Lambda$ by
$\overline{\Lambda}$ later in this section, but first focus on the insights we
obtain about the noise-induced transition to chaos from the compartmental
model.

Figure 3:  Lyapunov exponent $\Lambda$ of Eq. (1) as a function of noise
amplitude obtained from the stationary density, and $\overline{\Lambda}$
obtained from the quasi-stationary and quasi-ergodic densities.

$\overline{\Lambda}$ is calculated from the expected escape times and
conditioned Lyapunov exponents in the compartments $M_{\text{exp}}$ and
$M_{\text{cont}}$, which in turn are obtained through the quasi-stationary and
quasi-ergodic densities. For illustration and comparison, these densities are
depicted in FIG. 4 for $M_{\text{exp}}$ and noise amplitude
$\varepsilon=0.00125$ (in phase II). The function $v$ that provides the
connection between these densities, cf. Eq. (6), is also given.

(a) quasi-stationary density, $m$ (b) right eigenfunction, $v$ (c) quasi-
ergodic density, $q$

Figure 4:  From top to bottom, the quasi-stationary density $m$, the function
$v$ and the quasi-ergodic density $q$, on the expanding region
$M_{\text{exp}}$ for $\varepsilon=0.00125$. These have been obtained using
Ulam’s method with $2^{14}$ components. The compartment $M_{\text{exp}}$ has
been highlighted in bold on the bottom horizontal axis.

The behavior of the Lyapunov exponent as a function of $\varepsilon$ can be
understood from the corresponding behavior of the constituents of
$\overline{\Lambda}$ in Eq. (2). As the noise strength $\varepsilon$ increases
from phase II to phase III we track the conditioned Lyapunov exponents
$\Lambda_{\text{exp}}$ and $\Lambda_{\text{cont}}$ in FIG. 5, as well as the
expected escape times ${\tau}_{\text{exp}}$ and ${\tau}_{\text{cont}}$ in FIG.
6. We observe in FIG. 5 that the growth and decay rates of the compartmental
model are almost unchanged as a function of the noise amplitude, compared to
the Lyapunov exponent $\overline{\Lambda}$. In FIG 6 we observe minor
variability of ${\tau}_{\text{exp}}$ and fast decay of $\tau_{\text{cont}}$.
Thus, with $\tau_{\text{exp}}$, $\Lambda_{\text{exp}}$ and
$\Lambda_{\text{cont}}$ effectively constant, it is the significant decay of
the expected escape time from the contracting region $\tau_{\text{cont}}$ that
drives the noise-induced transition to chaos.

Figure 5: Conditioned Lyapunov exponent on the expanding and contracting
components, $\Lambda_{\text{exp}}$ and $\Lambda_{\text{cont}}$, as a function
of $\varepsilon$. The Lyapunov exponent $\overline{\Lambda}$ is presented to
illustrate the role of the escape times in Eq. (2) as $\Lambda_{\text{exp}}$
and $\Lambda_{\text{cont}}$ remain almost constant. Figure 6: Expected escape
times ${\tau}_{\text{exp}}$ and ${\tau}_{\text{cont}}$ from the expanding and
contracting regions, respectively, as a function of the noise strength
$\varepsilon$. Note that the intersection of $\tau_{\text{exp}}$ and
$\tau_{\text{cont}}$ near the transition point $\varepsilon_{L}$ is not an
indicator of phase change.

We conclude this section with a discussion of the slight overestimation of the
Lyapunov exponent $\Lambda$ by $\overline{\Lambda}$. This is ultimately a
consequence of the fact that the dynamics in each component are not entirely
dominated by asymptotic conditioned behavior. It turns out that the expected
escape time from $M_{\text{cont}}$ and effective contraction rate in this
compartment are well approximated by $\tau_{\text{cont}}$ and
$\Lambda_{\text{cont}}$, respectively. The accuracy of the corresponding
approximations by $\tau_{\text{exp}}$ and $\Lambda_{\text{exp}}$ in
$M_{\text{exp}}$ are somewhat less reliable when the noise amplitude is not
large. First of all, $\tau_{\text{exp}}$ is theoretically correct when the
empirical distribution in $M_{\text{exp}}$ equals the quasi-stationary
distribution. However, the true distribution is influenced by the way
trajectories enter this region from $M_{\text{cont}}$, leading to an
overestimation of escape times by $\tau_{\text{exp}}$. The appropriateness of
the conditioned Lyapunov exponent $\Lambda_{\text{exp}}$ similarly relies on
the closeness of the empirical distribution to the quasi-stationary
distribution. In addition, it requires a sufficiently long escape time so that
finite-time conditioned Lyapunov exponents at the time-scale of the expected
escape time are close to the asymptotic conditioned Lyapunov exponent. We have
observed in numerical experiments that, at the time scale of the expected
escape time, the finite-time Lyapunov distribution is essentially bi-modal
rather than entirely concentrated around $\Lambda_{\text{exp}}$. As the right-
most peak of the bi-modal distribution lies around $\Lambda_{\text{exp}}$,
$\Lambda_{\text{exp}}$ overestimates the real effective growth rate. All in
all, inaccuracies in both $\tau_{\text{exp}}$ and $\Lambda_{\text{exp}}$ lead
to a slight overestimation of $\overline{\Lambda}$, as observed in FIG. 3. The
quantitative aspects of the compartmental model could be improved by taking
account more accurately of the inhomogeneity of escape times and the actual
distribution of finite-time conditioned Lyapunov exponents, rather than
relying on asymptotic values. We envisage this to be achievable considering
higher order corrections beyond the leading asymptotics.

##  V Conclusion and outlook

In this paper, we have shown how the noise-induced transition to chaos in a
random logistic map can be understood from a conditioned random dynamics point
of view. In a two-compartment model, representing the competition between
expanding and contracting behavior, we have found that the bifurcation is
characterized by a fast decay of the escape time from the contracting
compartment. We conjecture this feature to be universal in noise-induced
transitions of this type, and have indeed observed this also in other
examples, such as the two-dimensional Hénon map with bounded additive noise in
the period 7 window.[38] A full treatment of such higher dimensional examples,
analogous to the random logistic map, is in principle possible, but requires
substantially more numerical effort. Firstly, bifurcations from phase I to
phase II in such examples require more analysis from the set-valued point of
view.[39] Secondly, the computation of conditioned Lyapunov exponents relies
on estimating the quasi-ergodic density in the tangent bundle of
compartments.[15] Examples of noise-induced chaos in SDEs (with unbounded
noise) involve similar challenges for the determination of conditioned
Lyapunov exponents, as chaotic dynamics in such systems can arise only in
dimension two or higher.[40] We intend to report on higher dimensional
examples in forthcoming publications.

Long transients near deterministic repellers have been previously proposed as
an essential ingredient for noise-induced chaos in the small noise regime.[20,
13] However, from our case study it transpires that small noise is not
essential, nor sufficient, to explain noise-induced chaos. Indeed, while
“stickiness” of the repelling region may well be related to its deterministic
limit, crucially, noise needs to be sufficiently large in order to facilitate
expedient escape from the attracting region, enabling the transition from
negative to positive Lyapunov exponent. Moreover, we have observed noise-
induced chaos to arise also in the random logistic map Eq. (1) with noise
amplitude $\varepsilon=0.074$ and parameter value $a=3.4$ (between the first
and second period-doubling bifurcation in the deterministic logistic map,
where there are no chaotic transients).

Conditioned random dynamics is also useful to elucidate the understanding of
chaotic transients in deterministic systems by considering the zero noise
limit.[41] Indeed, the quasi-ergodic measures that underlie conditioned
Lyapunov exponents are reminiscent of the “natural measures” on repellers
introduced in Ref. 17 (see Fig. 4 therein), which have been effectively used
to define and compute local Lyapunov exponents[19, 13]. While quasi-ergodic
measures in random systems have a solid mathematical foundation[34, 32, 21],
in the context of long chaotic transients, natural measures on repellers are
defined with reference to numerical explorations.[17, 13] It turns out[41]
that quasi-ergodic measures near random hyperbolic repellers converge to such
natural invariant measures in the zero noise limit, and closely relate to
invariant measures for deterministic dynamical systems with holes.[42]

###### Acknowledgements.

The authors gratefully acknowledge support from the EPSRC Centre for Doctoral
Training in Mathematics of Random Systems: Analysis, Modelling and Simulation
(EP/S023925/1). JSWL thanks IRCN (Tokyo) and GUST (Kuwait) for their research
support. We would like to thank M. M. Castro, H. Chu, E. Gibson, V. P. H.
Goverse, T. Pereira, M. Rasmussen, Y. Sato, M. Tabaro, G. Tenaglia and D.
Turaev for useful discussions.

## AUTHOR DECLARATION

### Conflict of Interest

The authors have no conflicts to disclose.

### Data Availability

The data and code that support the findings of this paper are openly available
in https://github.com/Bernat-BC/Ni-Chaos.

### Author Contributions

Bernat Bassols-Cornudella: Conceptualization (equal); software and
visualizations (lead); writing (equal). Jeroen S.W. Lamb: Conceptualization
(equal); writing (equal).

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