Scaling laws for symmetry breaking by blowout bifurcation in chaotic systems

Recent works have demonstrated that a blowout bifurcation can lead to symmetry breaking in chaotic systems with a simple kind of symmetry. That is, as a system parameter changes, when a chaotic attractor lying in some invariant subspace becomes unstable with respect to perturbations transverse to the invariant subspace, a symmetry-broken attractor can be born. As the parameter varies further, a symmetry-increasing bifurcation can occur, after which the attractor possesses the system symmetry. The purpose of this paper is to present numerical experiments and heuristic arguments for the scaling laws associated with this type of symmetry breaking and symmetry-increasing bifurcations. Specifically, we investigate (1) the scaling of the average transient time preceding the blowout bifurcation and (2) the scaling of the average switching time after the symmetry-increasing bifurcation. We also study the effect of noise. It is found that small-amplitude noise can restore the symmetry in the attractor after the blowout bifurcation and that the average time for trajectories to switch between the symmetry-broken components of the attractor scales algebraically with the noise amplitude.