Parameter converting method for bifurcation analysis of nonlinear dynamical systems

Abstract

For detecting the behavior of a dynamical system, bifurcation analysis is necessary to be conducted with respect to changes in the parameters of the system. In this study, based on the solution to ordinary di erential equations from the initial value and parameters, a simple method is presented that can eciently reveal di erent bifurcations of the system. In addition to its simplicity, this method does not require a deep physical and mathematical understanding of the problem and, because of its high precision and the speed of solutions, does not need to reduce the order of models for many complex problems or problems with high degrees of freedom. This method is called Parameter Converting Method (PCM), which has two steps. In the rst step, the parameter varies as a function of time and, in the second step, time is expressed as the inverse of the assumed function. With this method at hand, bifurcation and amplitude-frequency diagrams and hidden attractors of some complex dynamics will be analyzed, and the sensitivity of the multi-potential well systems to initial conditions is studied. With this algorithm, a simple way to nd the domain of high-energy orbit in bistable systems is obtained.