Violation of hyperbolicity in a diffusive medium with local hyperbolic attractor

TL;DR

This paper investigates how hyperbolic chaos in a spatially extended system of coupled local hyperbolic attractors is lost as the system size increases, leading to violations of hyperbolicity and extensive spatiotemporal chaos.

Contribution

It demonstrates the transition from hyperbolic to non-hyperbolic chaos in a diffusive medium with local hyperbolic attractors as the system size grows.

Findings

Hyperbolicity persists in small systems with synchronized oscillations.

Loss of hyperbolicity occurs at a critical system size where tangent subspaces intersect.

Beyond this point, the system exhibits extensive spatiotemporal chaos with linear growth of complexity measures.

Abstract

Departing from a system of two non-autonomous amplitude equations, demonstrating hyperbolic chaotic dynamics, we construct a 1D medium as ensemble of such local elements introducing spatial coupling via diffusion. When the length of the medium is small, all spatial cells oscillate synchronously, reproducing the local hyperbolic dynamics. This regime is characterized by a single positive Lyapunov exponent. The hyperbolicity survives when the system gets larger in length so that the second Lyapunov exponent passes zero, and the oscillations become inhomogeneous in space. However, at a point where the third Lyapunov exponent becomes positive, some bifurcation occurs that results in violation of the hyperbolicity due to the emergence of one-dimensional intersections of contracting and expanding tangent subspaces along trajectories on the attractor. Further growth of the length results in two-dimensional intersections of expanding and contracting subspaces that we classify as a stronger type of the violation. Beyond of the point of the hyperbolicity loss, the system demonstrates an extensive spatiotemporal chaos typical for extended chaotic systems: when the length of the system increases the Kaplan-Yorke dimension, the number of positive Lyapunov exponents, and the upper estimate for Kolmogorov-Sinai entropy grow linearly, while the Lyapunov spectrum tends to a limiting curve.