Complicated Dynamical Behaviors of a Geometrical Oscillator with a Mass Parameter

Abstract

In this paper, we consider a special kind of geometrical nonlinear oscillator with a mass parameter admitting two different dynamical states leading to a double-valued potential energy. A cylindrical manifold is introduced to formulate the equation of motion to describe the distinguished dynamical behaviors. With the help of Hamiltonian, complex bifurcations are demonstrated with varying parameters including periodic solutions, the steady states and the blowing up phenomenon near [Formula: see text] to infinity. A toroidal manifold is introduced to map the infinities into [Formula: see text] on the torus exhibiting saddle-node-like behavior, where the uniqueness of solution is lost, for which a special “collision” parameter is introduced to define the possible motion leaving from infinities. Numerical calculation is carried out to generate bifurcation diagrams using Poincaré sections for the perturbed system to exhibit complex dynamics including the coexistence of periodic solutions, chaos from the coexisting periodic doubling and also instant chaos from the coexisting periodic solutions. The results demonstrated herein this paper provide a brand new insight into the understanding of enriched nonlinear dynamics and an essential explanation about “collision” of mechanical system with both geometrical and mass parameters.