Global dynamics for a class of new bistable nonlinear oscillators with bilateral elastic collisions

Abstract

How to model non-smooth dynamical systems and study its non-smooth global dynamics by developing analytical methods is now an important topic. In this paper, a new bistable nonlinear oscillator with bilateral elastic constraints and a three-piecewise nonlinear restoring force is established to study the perturbation of viscous damping and an external harmonic excitation. A five-piecewise linear approach to the restoring force can transform the original dimensionless dynamical equations into a two-dimensional piecewise-defined Hamiltonian systems, separated by four switching manifolds with symmetrical characteristic and subjected to damping dissipation and a periodic excitation. The geometrical structure of the unperturbed system has a pair of symmetrical piecewise-defined homoclinic orbits and each of them transversally and continuously crosses two switching manifolds. The analytical Melnikov method is extended here to fit the theoretical framework for analyzing homoclinic bifurcations and chaotic dynamics for non-smooth oscillators with multiple switching manifolds. A global perturbation technique is presented to calculate the distance between the stable and unstable manifolds and derive the Melnikov function in the form of piecewise-integration. A major innovation here is no need to extend the vector filed near the switching manifolds and carry out rigorous perturbation analysis with geometrical intuition to derive the Hamilton energy differences of trajectories involving infinite time. The developed Melnikov function can be directly used to detect the parameter threshold of homoclinic chaos in the bistable nonlinear oscillator under bilateral elastic collision. Finally, the effectiveness of the Melnikov analysis for homoclinic chaos is verified by simulations.