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Some constructions of Lyapunov function for linear extensions of dynamical systems

Kulik V., Tkocz-Piszczek E.

Opuscula Mathematica

2011

Vol. 31, no. 3

399-409

In this note we consider some sets of linear extensions of dynamical systems and research regularity by means of the sign-changing Lyapunov function. We examine some constructions of Lyapunov functions for given systems.

Global Bifurcations, Chaotic Saddles and the Resulting Chaotic Dynamics in Nonlinear Oscillating Systems

Tyrkiel E.

Machine Dynamics Problems

2004

Vol. 28, No. 2

87-105

In the paper, the most important dynamical element underlying the build-up of chaotic responses in nonlinear vibrating systems, i.e. the formation and expansion of invariant nonattracting chaotic sets, so-called chaotic saddles, as a result of the global bifurcations, is highlighted. Characteristic examples of the resulting multiple aspects of chaotic system behaviors (chaotic transient motions, fractal basin boundaries, unpredictability of the final state) are shown and discussed. Numerical study is carried out for two low-dimensional representative models of nonlinear, strictly dissipative oscillators (a twin-well Duffing oscillator and a plane pendulum), driven externally by periodic force. The results are presented and interpreted with the use of concepts and numerical techniques of nonlinear dynamics and chaos. The study allows to establish critical thresholds of forcing parameters which are important in safe engineering, i.e. which place limits on the domains of the safe (regular) and unsafe (chaotic, unpredictable) system motion.