The refined approximate criterion for chaos in a two-state mechanical oscillator

Using Lyapunov exponents to predict the onset of chaos in nonlinear oscillators

An analytic technique for predicting the emergence of chaotic instability in nonlinear nonautonomous dissipative oscillators is proposed. The method is based on the Lyapunov-type stability analysis of an arbitrary phase trajectory and the standard procedure of calculating the Lyapunov characteristic exponents. The concept of temporally local Lyapunov exponents is then utilized for specifying the area in the phase space where any trajectory is asymptotically stable, and, therefore, the existence of chaotic attractors is impossible. The procedure of linear coordinate transform optimizing the linear part of the vector field is developed for the purpose of maximizing the stability area in the vicinity of a stable fixed point. By considering the inverse conditions of asymptotic stability, this approach allows formulating a necessary condition for chaotic motion in a broad class of nonlinear oscillatory systems, including many cases of practical interest. The examples of externally excited one- and two-well Duffing oscillators and a planar pendulum demonstrate efficiency of the proposed method, as well as a good agreement of the theoretical predictions with the results of numerical experiments. The comparison of the proposed method with Melnikov's criterion shows a potential advantage of using the former one at high values of dissipation parameter and/or multifrequency type of excitation in dynamical systems.

Indeterminate jumps to resonance from a tangled saddle-node bifurcation

We identify in this paper a new type of bifurcation which yields an unpredictable outcome and therefore seems to be an important new ingredient of nonlinear dissipative dynamics. It arises when the unstable manifold of the saddle of a saddle-node bifurcation is heteroclinically tangled with the inset of a distant saddle which is itself homoclinically tangled so that it forms a fractal basin boundary between two remote attractors. At the saddle-node fold, a slowly evolving system will thus find itself sitting precisely on a fractal basin boundary, and in the presence of even infinitesimal noise we cannot predict to which of the two remote attractors the system will jump. We show here that such an indeterminate tangled saddle-node bifurcation is a common ingredient in the resonance of softening systems.

Steady states in the twin-well potential oscillator: Computer simulations and approximate analytical studies

The paper is focused on the phenomena of various steady-state oscillations exhibited by the twin-well potential system. Regions of existence of different attractors in the system parameter domain are examined and a picture book of different steady states for fixed damping and forcing is presented: 20 different combinations of single or coexisting, small orbit or large orbit, periodic and chaotic attractors are displayed. Computer simulations are followed by an approximate analytical analysis: A study of various forms of instability of periodic solutions gives close form approximate criteria for occurrence of T-periodic small orbit and large orbit oscillations, and for cross-well chaos.