Multistability and Rare attractors in van der Pol-Duffing oscillator

Nonlinear systems exhibit a rich variety of different long-term behaviors such as: fixed points, limit cycles, quasiperiodic and chaotic behavior. In a complex system several attractors may coexist for a given set of system parameters. This coexistence is termed multistability and has been found in almost all research areas of natural science, such as: mechanics, electronics, biology, environmental science and neuroscience [Atteneave, 1971; Knorre et al., 1975; Arecchi et al. 1982; Feudel, 2008; Shrimali et al, 2008; Lysyansky et al., 2008; Perlikowski et al., 2010]. Multistable systems are characterized by a high degree of complexity in dynamical behavior due to the interaction among the co-existing attractors. Generally, in the multistable systems one observes the interaction among the attractors. First, the dynamics of a multistable system is extremely sensitive to initial conditions. Due to the coexistence of different attractors and complex fractal basin boundary structures very small perturbations of the initial state may influence the final attractor. Second, the qualitative behavior of the system often changes under the variation of the system parameters as some attractors exist only in the small intervals of the system parameters. A slight change in a control parameter may cause a rapid change in the number and type of coexisting attractors. Third, multistable systems are extremely sensitive to noise. Noise may cause a popping process between various attractors.