Global Bifurcations, Chaotic Saddles and the Resulting Chaotic Dynamics in Nonlinear Oscillating Systems

In the paper, the most important 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 the global bifurcations, is highlighted. Characteristic examples of the resulting multiple aspects of chaotic system behaviors (chaotic transient motions, fractal basin boundaries, unpredictability of the final state) are shown and discussed. Numerical study is carried out for two low-dimensional representative models of nonlinear, strictly dissipative oscillators (a twin-well Duffing oscillator and a plane pendulum), driven externally by periodic force. The results are presented and interpreted with the use of concepts and numerical techniques of nonlinear dynamics and chaos. The study allows to establish critical thresholds of forcing parameters which are important in safe engineering, i.e. which place limits on the domains of the safe (regular) and unsafe (chaotic, unpredictable) system motion.