Weak tracking in nonautonomous chaotic systems

Alkhayuon, Hassan; Ashwin, Peter

doi:10.1103/physreve.102.052210

Cited by 5 publications

(21 citation statements)

References 26 publications

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“…Furthermore, the concept of P-tipping, for base states that are attracting limit cycles with regular basin boundaries, naturally extends to more complicated base states, such as quasiperiodic tori and chaotic attractors, and to irregular (e.g. fractal) basin boundaries [28,31,32,92]. Defining phase for more complicated cycles in higher dimensions, and for non-periodic oscillations, will usually require a different approach.…”

Section: Discussionmentioning

confidence: 99%

Phase tipping: how cyclic ecosystems respond to contemporary climate

2021

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We identify the phase of a cycle as a new critical factor for tipping points (critical transitions) in cyclic systems subject to time-varying external conditions. As an example, we consider how contemporary climate variability induces tipping from a predator–prey cycle to extinction in two paradigmatic predator–prey models with an Allee effect. Our analysis of these examples uncovers a counterintuitive behaviour, which we call phase tipping or P-tipping , where tipping to extinction occurs only from certain phases of the cycle. To explain this behaviour, we combine global dynamics with set theory and introduce the concept of partial basin instability for attracting limit cycles. This concept provides a general framework to analyse and identify easily testable criteria for the occurrence of phase tipping in externally forced systems, and can be extended to more complicated attractors.

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Section: Discussionmentioning

confidence: 99%

Phase tipping: how cyclic ecosystems respond to contemporary climate

2021

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“…This corresponds to a parameter shift between two asymptotic values, one in the distant past and one in the distant future. Such systems, and their dependence on the rate of the parameter shift, have previously been investigated for shifts between equilibrium attractors [4,5] and periodic or more general attractors [1,2,16]. Rigorous results on the existence of such nonautonomous physical measures are presented in a companion paper [24].…”

Section: Tipping Points and Chaotic Multistabilitymentioning

confidence: 99%

“…For each x 0 ∈ R d , τ ∈ R, and T > 0, define the empirical measure μ τ −T,τ,x0 as follows: for each Borel set S ∈ B(R d ), μ τ −T,τ,x0 (S) is the proportion of time t within the interval [τ − T, τ ] for which x(t) ∈ S, where x(•) is a solution of (1) with x(τ − T ) = x 0 . In other words 2 To be more precise, provided that the sets A(t) are Borel subsets of R d , we can naturally equip A with the σ-algebra {{x ∈ A : x(t) ∈ S} : S ∈ B(R d )} generated by the natural identification of A with A(t) by x → x(t); this σ-algebra does not depend on the time t. A probability measure μ can then be defined on this σ-algebra.…”

Section: Is the Perron-frobenius (Transfer) Operator Associated With The Map ϕ(T •)mentioning

confidence: 99%

Physical invariant measures and tipping probabilities for chaotic attractors of asymptotically autonomous systems

Ashwin

Newman

2021

Eur. Phys. J. Spec. Top.

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Physical measures are invariant measures that characterise “typical” behaviour of trajectories started in the basin of chaotic attractors for autonomous dynamical systems. In this paper, we make some steps towards extending this notion to more general nonautonomous (time-dependent) dynamical systems. There are barriers to doing this in general in a physically meaningful way, but for systems that have autonomous limits, one can define a physical measure in relation to the physical measure in the past limit. We use this to understand cases where rate-dependent tipping between chaotic attractors can be quantified in terms of “tipping probabilities”. We demonstrate this for two examples of perturbed systems with multiple attractors undergoing a parameter shift. The first is a double-scroll system of Chua et al., and the second is a Stommel model forced by Lorenz chaos.

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“…For arbitrary time-dependent inputs, the theory of nonautonomous systems [56] summarises work in this area and gives general results on attraction and stability. Here, we focus on a case that is more specific, relevant to applications, and allows us to make further progress on the nonautonomous problem (3). In particular, it allows us to extend results from [9] to arbitrary dimension and to different cases of R-tipping.…”

Section: Asymptotically Constant Inputs: Future and Past Limit Systemsmentioning

confidence: 99%

“…to denote a solution 3 to system (3) at time τ started from x 0 at initial time τ 0 for a fixed rate r. We also write trj [r]…”

Section: Solutions and Trajectories Of The Parametrised Nonautonomous...mentioning

confidence: 99%

Rate-Induced Tipping: Thresholds, Edge States and Connecting Orbits

Wieczorek¹,

Xie²,

Ashwin³

2021

Preprint

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Rate-induced tipping, or simply R-tipping, occurs when time-variation of input parameters of a dynamical system interacts with system timescales to give genuine nonautonomous instabilities that cannot, in general, be understood in terms of autonomous bifurcations in the frozen system with a fixed-in-time input. Such instabilities appear as the input varies at some critical rates. Finding these critical rates and characterising what happens when they are exceeded is of great interest in natural science. The challenge is that it requires development of mathematical concepts and techniques beyond classical autonomous bifurcation theory.This paper develops an accessible mathematical framework and gives testable criteria for R-tipping in multidimensional nonautonomous dynamical systems with an autonomous future limit. Our focus is on R-tipping via loss of tracking of base attractors that are equilibria in the frozen system, due to crossing what we call regular thresholds. These thresholds are associated with regular edge states: compact hyperbolic invariant sets with one unstable direction and orientable stable manifold, that lie on a basin boundary in the frozen system. We define Rtipping and critical rates for the nonautonomous system in terms of special solutions that limit to a compact invariant set of the future limit system that is not an attractor. We then focus on the case when the limit set is a regular edge state of the future limit system, which we call the regular R-tipping edge state that anchors the associated regular R-tipping threshold at infinity. We then introduce the concept of edge tails to rigorously classify R-tipping into reversible, irreversible and degenerate cases. The main idea is to use autonomous dynamics and regular edge states of the future limit system to analyse R-tipping in the nonautonomous system. To that end, we compactify the original nonautonomous system to include the limiting autonomous dynamics. This allows us to give easily verifiable conditions in terms of simple properties of the frozen system and input variation that are sufficient for the occurrence of Rtipping. Additionally, we give necessary and sufficient conditions for the occurrence of reversible and irreversible R-tipping in terms of computationally verifiable (heteroclinic) connections to regular R-tipping edge states in the compactified system. Thus, our work extends existing results for R-tipping in one dimension to arbitrary dimension and to different cases of R-tipping, some of which can occur only in higher dimensions.

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Weak tracking in nonautonomous chaotic systems

Cited by 5 publications

References 26 publications

Phase tipping: how cyclic ecosystems respond to contemporary climate

Phase tipping: how cyclic ecosystems respond to contemporary climate

Physical invariant measures and tipping probabilities for chaotic attractors of asymptotically autonomous systems

Rate-Induced Tipping: Thresholds, Edge States and Connecting Orbits

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