Analysis of symmetry breaking bifurcation in Duffing system with random parameter

The symmetry breaking bifurcation (SBB) phenomenon in a deterministic parameter Duffing system (DP-DS) is well known, yet the problem how would SBB phenomenon happen in a Duffing system with random parameter (RP-DS) is still open. For comparison study, the results for DP-DS are summarized at first: in short, SBB in DP-DS is just a transition of response phase trajectories from a single self-symmetric one about the origin into two mutual symmetric once, or vice versa. However, in DP-DS case, the two mutual symmetric phase trajectories are never commutable. In view of every sample of RP-DS is a DP-DS, we think that SBB phenomenon might also happen in a RP-DS as an ensemble mean behavior. Since the orthogonal polynomial approach is a practical method to study the dynamical behavior of nonlinear system with random parameters, so we apply the Chebyshev polynomials approach to reduce the RP-DS to an equivalent deterministic system (EDS) to study its dynamical behavior in ensemble average mean. Numerical simulations on both DP-DS and EDS show that though SBB may happen in similar apparent forms, but for EDS the two coexisting symmetric phase trajectories are occasionally commutable. We cannot but resort to study the different features of attractive basins for these two kind of mutual symmetric phase trajectories. We found that the boundary of attractive basins in EDS case is fractal-like, while that in DP-DS case is smooth.