On the role of chaotic saddles in generating chaotic dynamics in nonlinear driven oscillators

摘要

In the paper, the most important common dynamical element underlying the build-up of chaotic responses in nonlinear vibrating systems, i.e. the formation and expansion of invariant nonattracting chaotic sets, so-called chaotic saddles, as a result of transverse intersections of stable and unstable invariant manifolds of particular unstable orbits, is highlighted. Characteristic examples of the resulting multiple aspects of chaotic system behaviors, such as chaotic transient motions, fractal basin boundaries and unpredictability of the final state, are shown and discussed with the use of geometrical interpretation of the results completed by color computer graphics. Numerical study is carried out for two low-dimensional but representative models of nonlinear, strictly dissipative oscillators driven externally by periodic force, i.e. the twin-well Duffing oscillator and the plane pendulum. In particular, it is demonstrated that the formation of chaotic saddles (equivalent to the creation of horseshoes in the system dynamics) is the primary mechanism triggering chaotic transient motions independently of either single or multiple attractors exist. The aspect of formation of chaotic saddles as a result of a sequence of global (homoclinic and heteroclinic) bifurcations, which is useful in establishing criteria for the occurrence of chaotic system behaviors as the control parameter changes, is presented.