The analytical predictive criteria for chaos and escape in nonlinear oscillators: A survey

Abstract

The purpose of this paper is to provide a brief summary of the various analytical predictive criteria in order for "strange phenomena" to occur in a class of softening nonlinear oscillators, oscillators which may exhibit escape from a potential well. Implications of Melnikov's criteria are discussed first and transient chaos in the twin-well potential oscillator is illustrated. Three different heuristic criteria for steady state chaos or escape solution, proposes by F. Moon, G. Schmidt and W. Szemplifiskia-Stupnicka, are then presented and compared to computer simulation results.

Transition to chaos in the self-excited system with a cubic double well potential and parametric forcing

“…Such a bifurcation correspond to transit chaotic behaviour of the system and with some further increasing of the excitation amplitude can lead to the permanent chaos [7,8,14,26].…”

“…It has found numerous applications in mechanical engineering [1,2,3,4,5,6,7,8] and control theory [9,10,11,12]. It was also one of intriguing examples of simple nonlinear systems which showed complex behaviour including chaotic oscillations [13].…”

Suppression of chaos by weak resonant excitations in a non-linear oscillator with a non-symmetric potential

“…have been a subject of studies for many years [1,2,3,4,5,6,7,8,9,10]. These investigations were motivated by possible applications in description of physical systems mostly mechanical [1,3,7,8] and electrical systems [4].…”

Vibration of generalized double well oscillators

“…Such a treatment has been already performed successfully to selected problems with various potentials [8,9,14]. Vibrations of a single Duffing oscillator have got a large bibliography [1,2,3,4,5,6,8,9,10,11,12,13,14,15,16,17,18]. In the last decade coupled Duffing oscillators [19,20,21,22,23] with numerous modifications to potential and forcing parts have been studied.…”