Complex behaviours in two bi-directionally coupled Lorenz systems

The aim of this paper is to study complex behaviours generated by loss of stability of chaotic attractors in two bidirectionally and nonlinearly coupled Lorenz systems. Owing to the characteristics of the coupling elements, the dynamic system turns out to be invariant with respect to linear transformations. In particular, the dynamics of the whole system is described by the same set of equations that defines a single autonomous Lorenz system. This allows one to exclude the presence of identical synchronization and anti-synchronization. On the other hand, the analysis of Lyapunov exponents as function of the coupling strength points out the occurrence of complex behaviours, e.g., generalized synchronization, on-off intermittency and hyperchaotic oscillations. A blow-out bifurcation is numerically located, confirming a transition route from lower dimensional chaos (generalized synchronization) to higher dimensional chaos (hyperchaos). Such complex behaviors indicate possible applications to the design of chaotic systems with proper statistical properties.