Lyapunov Exponents and Correlation Decay for Random Perturbations of Some Prototypical 2D Maps

Alex Blumenthal, Jinxin Xue, Lai-Sang Young

Research output: Contribution to journalArticle

Abstract

To illustrate the more tractable properties of random dynamical systems, we consider a class of 2D maps with strong expansion on large—but non-invariant—subsets of their phase spaces. In the deterministic case, such maps are not precluded from having sinks, as derivative growth on disjoint time intervals can be cancelled when stable and unstable directions are reversed. Our main result is that when randomly perturbed, these maps possess positive Lyapunov exponents commensurate with the amount of expansion in the system. We show also that initial conditions converge exponentially fast to the stationary state, equivalently time correlations decay exponentially fast. These properties depend only on finite-time dynamics, and do not involve parameter selections, which are necessary for deterministic maps with nonuniform derivative growth.

Original languageEnglish (US)
Pages (from-to)1-27
Number of pages27
JournalCommunications in Mathematical Physics
DOIs
StateAccepted/In press - Oct 12 2017

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Random Perturbation
Lyapunov Exponent
exponents
Decay
perturbation
decay
Random Dynamical Systems
Derivative
expansion
Parameter Selection
Stationary States
sinks
dynamical systems
Phase Space
Disjoint
Initial conditions
Unstable
intervals
Converge
Interval

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Lyapunov Exponents and Correlation Decay for Random Perturbations of Some Prototypical 2D Maps. / Blumenthal, Alex; Xue, Jinxin; Young, Lai-Sang.

In: Communications in Mathematical Physics, 12.10.2017, p. 1-27.

Research output: Contribution to journalArticle

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