ON THE PROBABILITY OF POSITIVE FINITE-TIME LYAPUNOV EXPONENTS ON STRANGE NONCHAOTIC ATTRACTORS F. REMO G. FUHRMANN AND T. J AGER

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ON THE PROBABILITY OF POSITIVE FINITE-TIME LYAPUNOV

EXPONENTS ON STRANGE NONCHAOTIC ATTRACTORS

F. REMO, G. FUHRMANN, AND T. J ¨

AGER

Abstract. We study strange non-chaotic attractors in a class of quasiperiodically forced

monotone interval maps known as pinched skew products. We prove that the probability

of positive time-NLyapunov exponents—with respect to the unique physical measure on

the attractor—decays exponentially as N→ ∞. The motivation for this work comes from

the study of ﬁnite-time Lyapunov exponents as possible early-warning signals of critical

transitions in the context of forced dynamics.

2020 MSC numbers: 37C60, 37G35

1. Introduction

In this article, we study quasiperiodically forced interval maps of the form

Fκ:TD×[0,1] →TD×[0,1], Fκ(θ, x)=(θ+v, tanh(κx)·g(θ)) ,(1)

where κ > 0 is a real parameter, v∈TDis a totally irrational rotation vector∗and the

multiplicative forcing term g:TD→[0,1] is given by

g(θ) = 1

D·

i=1

sin(πθi).(2)

Systems of this kind are often called pinched skew products, where pinched refers to the fact

that the forcing term gvanishes for some θ∈TD(here, at θ= 0). Pinched skew-products

received considerable attention due the occurrence of so-called strange non-chaotic attractors

(SNAs) [1–7]. Due to their speciﬁc properties—in particular, the pinching in combination with

the invariance of the zero line TD×{0}—they are technically more accessible than other forced

systems that exhibit SNAs so that they have been used on various occasions for case studies

concerning the structural properties of such attractors. This led, for instance, to ﬁrst results on

the topological structure [6] and the dimensions [7] of SNAs, which have later been extended

to the more diﬃcult situation of additive quasiperiodic forcing [8–10].

In a similar spirit, the aim of this note is to establish a quantitative result on the distribution

of positive ﬁnite-time Lyapunov exponents on the SNA appearing in the system given by (1)

and (2). Given (θ, x)∈TD×[0,1] and N∈N, we deﬁne the time-N-Lyapunov exponent as

λN(θ, x) = log ∂xFN

κ(θ, x)/N .

The (asymptotic) Lyapunov exponents are then given by

λ(θ, x) = lim

N→∞ λN(θ, x).

As established in [3], for any κ > κ0:= e−RTDlog g(θ)dθ, there exists a unique physical measure

Pκof the system (1) that is ergodic and has a negative Lyapunov exponent. As a consequence,

asymptotic Lyapunov exponents are Pκ-almost surely negative. However, on the invariant

zero line TD× {0}, the pointwise Lyapunov exponents almost surely equal log κ−log κ0(see

Remark 2.1 below). Hence, for κ>κ0, positive asymptotic Lyapunov exponents are still present

in the system and lead to a positive probability of positive ﬁnite-time exponents for all times

N∈N. Our main result provides information on the scaling behaviour of these probabilities.

∗We say v∈TDis totally irrational if there is no non-zero n∈Zdwith hv, ni ∈ Z.

arXiv:2210.15292v2 [math.DS] 12 Nov 2022

2 F. REMO, G. FUHRMANN, AND T. J ¨

AGER

Theorem 1.1. Denote by Pκthe unique physical measure of (1) with forcing function (2). Let

pκ,N =Pκ({(θ, x)|λN(θ, x)>0}). Then there exists κ1> κ0such that for all κ≥κ1, there

are constants γ+≥γ−>0(depending on κ) such that

exp(−γ+N)≤pκ,N ≤exp(−γ−N)

holds for all N∈N.

Apart from its intrinsic interest, motivation for this result stems from the study of critical

transitions. One major problem in this ﬁeld is the identiﬁcation of suitable (that is, observable

and reliable) early warning signals [16–21] for such transitions. A commonly proposed and

utilized early warning signal for fold bifurcations—which are often cited as a paradigmatic

example of critical transitions—are slow recovery rates (also referred to as a critical slowing

down) [16–18,20]. Since this notion has been coined in an interdisciplinary context and is used

in a wide variety of situations, there is no comprehensive and rigorous mathematical deﬁnition

of this term and we refrain from attempting to give one here. However, in the classical case of

an autonomous fold bifurcation, recovery rates can be identiﬁed with the Lyapunov exponents

of the stable equilibria. Thus, in this situation, critical slowing down simply refers to the fact

that when the stable and unstable equilibria involved in the bifurcation approach each other

and eventually merge at the critical parameter, the resulting single ﬁxed point is neutral, that

is, it has exponent zero.

This picture changes signiﬁcantly when a fold bifurcation takes place under the inﬂuence of

external quasiperiodic forcing. First of all, the resulting non-autonomous systems generally do

not allow for ﬁxed points. Therefore, when carrying over ideas from an autonomous to a non-

autonomous setting, one needs an appropriate replacement. In the present context, this part is

played by so-called invariant graphs (see Section 2) and accordingly, non-autonomous fold bifur-

cations occur as invariant graphs approach each other upon a change of system parameters. In

stark contrast to autonomous fold bifurcations, this does not necessarily yield neutral invariant

graphs but may instead lead to a strange non-chaotic attractor-repeller-pair [14, 15] created at

the bifucation point. This alternative pattern is referred to as a non-smooth saddle-node bifur-

cation. Moreover, just as for pinched systems, under suitable conditions, there exists a unique

physical measure Pwhich is supported on the attractor and has a negative Lyapunov exponent

(see [3, 12]). However, this means that Lyapunov exponents remain P-almost surely negative

and bounded away from zero during a non-smooth saddle-node bifurcation (see Section 2 for

more details).

Figure 1. A logarithmic plot of the numerically obtained probability pN=pκ,N over Nfor

the system (1) with D= 1, κ= 3 and vthe golden mean. The graph shows the relative

frequency of non-negative ﬁnite-time Lyapunov exponents among a grid of 5 ·106initial

conditions on the SNA (see also Figure 2). Consistent with the statement of Theorem 1.1,

the plot indicates an exponential decay.

PINCHED SYSTEMS 3

While this seems to rule out the viability of slow-recovery rates as early warning signals

for non-smooth fold bifurcations, one should bear in mind that experiments never measure

the actual Lyapunov exponent but rather approximations of it. In other words, one rather

measures ﬁnite-time Lyapunov exponents instead of asymptotic ones. Since the presence of an

SNA implies that positive ﬁnite-time exponents occur with positive probability for any time

N∈N[2, 22], one may hence wonder whether the observation of non-negative ﬁnite-time Lya-

punov exponents can help to detect an SNA in practice. However, if Nis chosen too small, then

positive time-N-exponents can be observed already far from a bifurcation. Conversely, for large

N, the probability of observing positive exponents on this time-scale converges to zero since the

unique physical measure has a negative exponent. It is in this context that the scaling behaviour

of the probabilities of time-N-exponents with N→ ∞ becomes important. Numerical studies

for the quasiperiodically forced Allee model performed in [22] remained somewhat inconclusive,

which is partly explained by the fact that the simulation of continuous-time systems is consider-

ably more time-consuming than that of discrete-time systems. The exponential decay obtained

in Theorem 1.1 is an indication that very large data sets may be required to detect this kind

of early-warning signals in practice. As mentioned before, this interpretation relies on the hy-

pothesis that quasiperiodically forced systems undergoing a saddle-node bifurcation—as studied

in [22]—show a behaviour comparable to that of pinched systems treated here. We expect that

using techniques from [8, 10, 12], similar statements can be established for non-pinched sys-

tems but this would require a considerably more involved analysis due to the inherent technical

diﬃculties.

This article is organised as follows. In the next section, we introduce some technical back-

ground on forced monotone interval maps and their invariant graphs. There, we also describe

the physical measure Pfrom above in more detail. In Section 3, we specify the class of pinched

skew-products for which we prove (a more general version of) the above theorem. This proof

and the full statement—Theorem 4.4 and Theorem 4.8 (which gives the upper bound and is

the harder part)—are given in the ﬁnal section, Section 4.

Acknowledgments. This project has received funding from the European Union’s Horizon

2020 research and innovation program under the Marie Sk lodowska-Curie grant agreements No

643073 and No 750865. TJ acknowledges support by a Heisenberg grant of the German Research

Council (DFG grant OE 538/6-1).

2. Forced monotone interval maps and invariant graphs

Throughout this note, we deal with quasiperiodically forced (qpf) monotone interval maps,

that is, skew products of the form

F:TD×[0,1] →TD×[0,1],(θ, x)7→ (ρ(θ), Fθ(x)),(3)

where TD=RD/ZDis the D-dimensional torus (for some D≥1),

ρ:TD→TD, θ 7→ θ+v

is a minimal rotation with a rotation vector vand for each θ∈TD,Fθis a continuously diﬀer-

entiable non-decreasing map on [0,1] such that (θ, x)7→ F0

θ(x) is continuous. It is customary

to refer to (TD, ρ) as the forcing system (deﬁned on the base TD); the maps Fθ(θ∈TD) are

also referred to as ﬁbre maps (deﬁned on the ﬁbres {θ} × [0,1]).

An invariant graph of (3) is a measurable function φ:TD→[0,1] which satisﬁes

Fθ(φ(θ)) = φ(θ+ρ) for all θ∈TD.

From an intuitive perspective, invariant graphs are to be seen as non-autonomous ﬁxed points

of (3).†This idea is the basis for a bifurcation theory of invariant graphs, see [14,15]. Indepen-

dently of this analogy, invariant graphs of qpf monotone interval maps are key to understanding

the dynamics of (3) due to their intimate relationship with the invariant sets and ergodic mea-

sures.

Every invariant graph φcomes with an ergodic measure µφwhere µφ(A) = LebTD(φ−1A)

for each measurable A⊆TD×[0,1] and likewise, to each ergodic measure µof (3) there is an

†Observe that due to the minimality of ρ, (3) does not allow for actual ﬁxed points.

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ONTHEPROBABILITYOFPOSITIVEFINITE-TIMELYAPUNOVEXPONENTSONSTRANGENONCHAOTICATTRACTORSF.REMO,G.FUHRMANN,ANDT.JAGERAbstract.Westudystrangenon-chaoticattractorsinaclassofquasiperiodicallyforcedmonotoneintervalmapsknownaspinchedskewproducts.Weprovethattheprobabilityofpositivetime-NLyapunovexponents|withr...

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ON THE PROBABILITY OF POSITIVE FINITE-TIME LYAPUNOV EXPONENTS ON STRANGE NONCHAOTIC ATTRACTORS F. REMO G. FUHRMANN AND T. J AGER

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