Determination of the Chaos Onset in Mechanical Systems with Several Equilibrium Positions

Abstract

Determination of the chaos onset in some mechanical systems with several equilibrium positions are analyzed. Namely, the snap-through truss and the oscillator with a nonlinear dissipation force, under the external periodical excitation, are considered. Two approaches are used for the chaos onset determination. First, Pade and quasi-Pade approximants are used to construct closed homoclinic trajectories for a case of small dissipation. Convergence condition used earlier in the theory of nonlinear normal vibration modes as well conditions at infinity make possible to evaluate initial amplitude values for the trajectories with admissible precision. Mutual instability of phase trajectories is used as criterion of chaotic behavior in nonlinear systems for a case of not small dissipation. The numerical realization of the Lyapunov stability definition gives us a possibility to observe a process of appearance and fast enlargement of the chaotic behavior regions if some selected parameters of the dynamical systems under consideration are changing.