The Approximate Analytical Methods in the Study of Bifurcations and Chaos in Nonlinear Oscillators

Abstract

We consider the third order dynamical systems which are reduced to the form:

$$\begin{array}{*{20}{c}} {\ddot x + \Omega _0^2x + f(x,\dot x,\omega {\text{t}}) = 0,\quad T = \frac{{2\pi }}{\omega },} \\ {f(x,\dot x,\omega {\text{t}}) = h\dot x + {\alpha _2}{x^2} + {\alpha _3}{x^3} - F\;\cos \;\omega {\text{t}};\quad h > 0,} \end{array}$$

(1)

and we focus on the question of prediction of the bifurcations which are related to,and which precede the escape from a potential well. This class of nonlinear oscillators models a wide spectrum of physical problems and has an extensive literature. Complex bifurcations phenomena in systems characterized in Fig. 1 (a-c) have been reported since 1979, the results being based mostly on computer based or experimental investigations [e.g. 1–5].