Symmetry breaking crisis of chaotic attractors

Most nonlinear dynamic systems may exhibit a certain symmetrical form. Symmetry is a kind of invariance, maybe appearing in different topological forms under different situations, but always keeping the characteristic of symmetry. The transition between different symmetric forms often leads to symmetry breaking bifurcation or crisis. Many studies oil symmetry breaking bifurcations of phase trajectories of a nonlinear dynamical system have been reported, most of which are related to periodic or quasi-periodic orbits. Only a few of them ever mentioned that "duality" might also exist for a couple of chaotic attractors in symmetrical nonlinear dynamical systems. Recently, in the study of the saddle-node bifurcation resulting from symmetry breaking of periodic phase orbits in a Duffing oscillator driven by a sinusoidal excitation, an interesting phase portrait of the flow pattern of discrete Poincare mapping points has been obtained after symmetry breaking bifurcation. Along with the flow pattern, two stable periodic nodes and one periodic saddle, together with its stable and unstable manifolds, are shown, which are all in a regular form yet. In this study, as an extension of the above results, a complicated portrait for attractive basins of coexisting periodic and chaotic attractors in a parametrically driven double-well Duffing system is obtained, which is fractal, interwoven, yet symmetrical. In addition, the neighboring symmetry breaking crises are studied qualitatively.