Loss of synchronization in coupled Rِssler systems

This paper considers the loss of synchronization for a system of two coupled Rِssler oscillators. Bifurcation curves for the transverse destabilization of low-periodic orbits embedded in the synchronized chaotic state are obtained, and we show that desynchronization for a pair of symmetrically coupled, identical Rِssler systems is associated with different orbits undergoing transverse pitchfork or period-doubling bifurcations. The transverse destabilization of the period-1 orbit is examined in detail, and we follow the sequence of bifurcations that the asynchronous periodic cycles undergo. In the presence of an asymmetry in the coupling, the transverse period-doubling bifurcation remains essentially the same. The transverse pitchfork bifurcation, on the other hand, is transformed into a transcritical riddling bifurcation. If the interacting Rِssler oscillators have different parameter values, the non-generic character of the pitchfork bifurcation leads it to be replaced by a saddle-node bifurcation occurring off the symmetric sub-space. Finally, we show how the transverse stability properties of the equilibrium point can be used to obtain approximative analytical results for the transverse stability of the coupled chaotic oscillators.