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

Peter Ashwin,Julian Newman

doi:10.1140/epjs/s11734-021-00114-z

Peter Ashwin, Julian Newman

Open Access

https://doi.org/10.1140/epjs/s11734-021-00114-z

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Abstract

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.

Highlights

Physical measures are invariant measures that characterise “typical” behaviour of trajectories started in the basin of chaotic attractors for autonomous dynamical systems
We propose a notion of physical measures appropriate for pullback attractors of nonautonomous systems, in the case that there are autonomous past and future limits with chaotic attractors
To [17], these probabilities can be understood as the probability that the climate state x(t) will tip to a given attractor of the future limit system conditional on both the specific time-dependent dynamical system governing the evolution of x(t) and on knowing which of the attractors of the past-limit system the climate state was close to in the distant past, but without the knowledge of the exact state x(t) at any time t

Summary

Physical measures and pullback attractors

For the deterministic nonautonomous model (1), a “physical measure” aims to describe, at each time t, the probability distribution for the location of a solution x(t) when the exact location is not known, but it is known that in the distant past, the solution was close to a particular attractor A− of the past-limit system. Definition 4 Given an attractor A− for the past-limit system (6) and a pullback attractor A of (1) starting at A−, a physical measure on A is a probability measure μ on A, such that supp(μt) = A(t) at each t ∈ R and for Lebesgue-almost every x0 ∈ B, the following holds: for each t ∈ R, as T → ∞, the empirical measure μt−T,t,x0 converges weakly to μt. Definition 5 Given an attractor A− for the past-limit system (6) and a pullback attractor A of (1) starting at A−, a pullback-PF-attracting physical measure (or just “pullback-attracting measure” for short) on A is a probability measure μ on A, such that supp(μt) = A(t) at each t ∈ R and for every probability measure ν0 of smooth density h supported within B, for each t ∈ R, as T → ∞, the measure νTt := Φ(r)(t, t − T, ν0) on Rd computed as νTt (S) = (LΦ(r)(t,t−T )h)(y) dy. In [24], the question of when an (attracting) physical measure of the past-limit system can be extended to a (pullback-attracting) physical measure of the nonautonomous system is explored in detail

Tipping between past and future attractors

Example 1: double scroll with parameter shift

Example 2

Discussion

A Proof of Theorem 6