Periodic-orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system

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.

Chaotic, informational and synchronous behaviour of multiplex networks

The understanding of the relationship between topology and behaviour in interconnected networks would allow to charac- terise and predict behaviour in many real complex networks since both are usually not simultaneously known. Most previous studies have focused on the relationship between topology and synchronisation. In this work, we provide analytical formulas that shows how topology drives complex behaviour: chaos, information, and weak or strong synchronisation; in multiplex net- works with constant Jacobian. We also study this relationship numerically in multiplex networks of Hindmarsh-Rose neurons. Whereas behaviour in the analytically tractable network is a direct but not trivial consequence of the spectra of eigenvalues of the Laplacian matrix, where behaviour may strongly depend on the break of symmetry in the topology of interconnections, in Hindmarsh-Rose neural networks the nonlinear nature of the chemical synapses breaks the elegant mathematical connec- tion between the spectra of eigenvalues of the Laplacian matrix and the behaviour of the network, creating networks whose behaviour strongly depends on the nature (chemical or electrical) of the inter synapses.

Intermingled basins in coupled Lorenz systems

We consider a system of two identical linearly coupled Lorenz oscillators presenting synchronization of chaotic motion for a specified range of the coupling strength. We verify the existence of global synchronization and antisynchronization attractors with intermingled basins of attraction such that the basin of one attractor is riddled with holes belonging to the basin of the other attractor and vice versa. We investigated this phenomenon by verifying the fulfillment of the mathematical requirements for intermingled basins and obtained scaling laws that characterize quantitatively the riddling of both basins in this system.

Parametric evolution of unstable dimension variability in coupled piecewise-linear chaotic maps

In the presence of unstable dimension variability numerical solutions of chaotic systems are valid only for short periods of observation. For this reason, analytical results for systems that exhibit this phenomenon are needed. Aiming to go one step further in obtaining such results, we study the parametric evolution of unstable dimension variability in two coupled bungalow maps. Each of these maps presents intervals of linearity that define Markov partitions, which are recovered for the coupled system in the case of synchronization. Using such partitions we find exact results for the onset of unstable dimension variability and for contrast measure, which quantifies the intensity of the phenomenon in terms of the stability of the periodic orbits embedded in the synchronization subspace.