Riddled basins of chaotic synchronization and unstable dimension variability in coupled Lorenz-like systems

Unstable dimension variability is an extreme form of non-hyperbolic behavior that causes a severe shadowing breakdown of chaotic trajectories. This phenomenon can occur in coupled chaotic systems possessing symmetries, leading to an invariant attractor with riddled basins of attraction. We consider the coupling of two Lorenz-like systems, which exhibits chaotic synchronized and anti-synchronized states, with their respective basins of attraction. We demonstrate that these basins are riddled, in the sense that they verify both the mathematical conditions for their existence, as well as the characteristic scaling laws indicating power-law dependence of parameters. Our simulations have shown that a biased random-walk model for the log-distances to the synchronized manifold can accurately predict the scaling exponents near blowout bifurcations in this high-dimensional coupled system. The behavior of the finite-time Lyapunov exponents in directions transversal to the invariant subspace has been used as numerical evidence of unstable dimension variability.