Violation of hyperbolicity via unstable dimension variability in a chain with local hyperbolic chaotic attractors

TL;DR

This paper investigates how decreasing coupling strength in a chain of hyperbolic chaotic oscillators leads to unstable dimension variability, causing a transition from hyperbolic to non-hyperbolic chaos, analyzed through Lyapunov exponents.

Contribution

It provides a detailed Lyapunov analysis of the transition from hyperbolic to non-hyperbolic chaos due to unstable dimension variability in coupled oscillator chains.

Findings

Strong coupling yields hyperbolic chaos with one positive Lyapunov exponent.

We observe the second Lyapunov exponent approaching zero and fluctuating due to unstable dimension variability.

The transition to non-hyperbolic chaos is characterized by sign changes in finite-time Lyapunov exponents.

Abstract

We consider a chain of oscillators with hyperbolic chaos coupled via diffusion. When the coupling is strong the chain is synchronized and demonstrates hyperbolic chaos so that there is one positive Lyapunov exponent. With the decay of the coupling the second and the third Lyapunov exponents approach zero simultaneously. The second one becomes positive, while the third one remains close to zero. Its finite-time numerical approximation fluctuates changing the sign within a wide range of the coupling parameter. These fluctuations arise due to the unstable dimension variability which is known to be the source for non-hyperbolicity. We provide a detailed study of this transition using the methods of Lyapunov analysis.