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On the stability "in the large'' and unsafe disturbances in a nonlinear oscillator

Szemplińska-Stupnicka W., Tyrkiel E., Zubrzycki A.

Computer Assisted Mechanics and Engineering Sciences

2001

Vol. 8, No. 1

155-168

The problem of the stability "in the large'' and the unsafe disturbances of the equilibrium position is studied for the structures whose dynamics is governed by the equation of motion of the pendulum with parametric excitation. The system displays a variety of nonlinear and chaotic phenomena, so that the study requires the use of theoretical concepts of the mathematics of chaos. Detailed explorations are performed by the aid of the nonlinear software package Dynamics.

On the boundary crises of chaotic attractors

Tyrkiel E., Szemplińska-Stupnicka W.

Computer Assisted Mechanics and Engineering Sciences

2000

Vol. 7, No. 4

743-755

In nonlinear dissipative mechanical systems, bifurcations of chaotic attractors called boundary crises appear to be the cause of most sudden changes in chaotic dynamics. They result in a sudden loss of stability of chaotic attractor, together with destruction of its basin of attraction and its disappearance from the phase portrait. Chaotic attractor is destroyed in the collision with an unstable orbit (destroyer saddle) sitting on its basin boundary, and the structure of the saddle defines the type of the crisis - regular or chaotic one. In the paper we exemplify both types of the boundary crisis by using a mathematical model of the symmetric twin-well Duffing oscillator; we consider the regular boundary crisis of the cross-well chaotic attractor, and the chaotic boundary crisis of the single-well chaotic attractor. Our numerical analysis makes use of the underlying topological structure of the phase space, namely the geometry of relevant invariant manifolds, as well as the structure of basins of attraction of the coexisting attractors. The study allows us to establish some relevant relations between the properties of the regular and chaotic boundary crisis, and to outline the differences that result mainly in the post-crisis