Hyperchaotic Synchronization under Square Symmetry

We study two identical hyperchaotic oscillators symmetrically coupled. Each oscillator represents a codimension-2 Takens–Bogdanov bifurcation under square symmetry, and was used to model a convection experiment in a time-dependent state. In the coupled system, the Lyapunov exponents behavior against the coupling parameter is used to detect changes in the dynamics, and the synchronization state is controlled by checking the phase planes. Complete synchronization is achieved without chaos suppression in a coupling parameter interval. Outside this window, complete synchronization cannot be generally achieved. As a consequence of a bubbling transition, synchronization is obtained only for some particular values of the initial conditions.