Construction of chaotic regions

A new method of numerical simulation is given in this chapter to describe the characteristics of the solutions of dynamical systems. The characteristics of the solutions, such as the number of solutions, the type and periodicity of the solutions and, more importantly, the existence of chaotic solutions, are of great interest in a physical parametric space. The boundaries separating qualitatively different solutions in a physical parametric space, constituting a characteristic diagram, are constructed below. The critical points in the initial bifurcation graph are used as initial approximations. Depending on the nature of the critical points, additional constraints are used to trace the neighbouring critical points by the method of parametric unfolding. As a result, the characteristics of the steady-state solutions can be read directly from the characteristic diagram. Non-linear governing differential equations arise from various types of physical and engineering problems. Owing to the presence of non-linearities, analytical solutions are rarely obtained. Apart from their deterministic nature, random-like solutions are observed for certain parameters and initial conditions. The occurrence of these chaotic motions has been studied by many authors.