Global analyses of crisis and stochastic bifurcation in the hardening Helmholtz-Duffing oscillator

Crisis and stochastic bifurcation of the hardening Helmholtz-Duffing oscillator are studied by means of the generalized cell mapping method using digraph. For the system subject to a single deterministic harmonic excitation, our study reveals that a series of crisis phenomena can occur when the system parameter passes through different critical values, including chaotic boundary crisis, regular boundary crisis and interior crisis. A chaotic boundary crisis due to the collision of regular attractor with chaotic saddle embedded in a fractal basin boundary and an interior crisis due to the collision of regular attractor with chaotic saddle of its attraction basin are discovered. A new phenomenon, namely the global properties of dynamical system show symmetric as system parameter is varied, can be also revealed according to our analysis. For the system subject to a combination of a deterministic harmonic excitation and a random excitation, it is found that stochastic bifurcation, defined as a sudden change in character of a stochastic attractor, can occur one after another when the noise intensity passes through different critical values. This kind of stochastic bifurcation corresponds to stochastic crisis essentially. Our study also reveals that the generalized cell mapping method using digraph is a powerful tool not only for the crisis behavior analysis of deterministic system, but also for the global property analysis of stochastic bifurcation.