Violation of hyperbolicity in a diffusive medium with local hyperbolic attractor.

Departing from a system of two nonautonomous amplitude equations, demonstrating hyperbolic chaotic dynamics, we construct a one-dimensional medium as an ensemble of such local elements introducing spatial coupling via diffusion. When 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 occur 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 the two-dimensional intersections of expanding and contracting subspaces that we classify as a stronger type of the violation. Beyond 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.