Singular Perturbation Methods For Limit Cycles’ Approximations In Strongly Nonlinear Systems

This thesis is mainly concerned with singular perturbation methods for limit cycles’approxi-mations in strongly nonlinear systems with applications. The thesis is divided into four chapters.Chapter1is the introduction, in which, singular perturbation problems, singular perturbation methods and their developments are reviewed. The main results of this thesis are also given in this chapter.Chapter2gives a two-dimensional modified Lindstedt-Poincare method for general planar autonomous systems. This method is composed of two essential steps. The first one is to change the general planar autonomous systems to their canonical form through a series of linear transfor-mations including translation, linear change of coordinates and time rescalings, and the second one is to define the suitable parametric transformation, nonlinear frequency expansion and solutions’ expansions. As an application, this method is applied to a Rosenzweig-MacArthur predator-prey model. Comparisons of the analytical results with those of direct numerical integrations show that this method is effective and accurate.In Chapter3, by utilizing an essential idea for limit cycle’s approximation in van der Pol equa-tion, the homotopy analysis method is applied to deduce analytical approximations of limit cycles and their frequencies in general planar self-excited systems with strong nonlinearity. Like Chapter2, the general planar autonomous systems are first changed to their canonical form, in which, the auxiliary linear operators and the initial guess solutions are obtained, and the zeroth-order equa-tions are also set up. More importantly, in solving the higher-order deformation equations, the above-mentioned idea is utilized.By applying the procedure in the present chapter to a Rosenzweig-MacArthur predator-prey model, it is shown by comparing with the numerical integration solutions that the accuracy of the analytical results obtained by the present procedure is very high, even when the control parameter deviates from the Hopf bifurcation value greatly.In Chapter4, the idea and method in Chapter3is extended to a three-dimensional nonlinear autonomous feedback control system. To obtain the suitable auxiliary linear operators and the initial guess solutions, this three-dimensional system is first transformed into a coupled system consisting of a second-order nonlinear equation and a first-order nonlinear equation. Then, the zeroth-order nonlinear equations and the higher-order deformation equations are obtained. Like the procedures in Chapter3, the analytical approximations of the limit cycle and its frequency are obtained. The comparisons with the numerical integration solutions show the high accuracy of the analytical results obtained in this chapter.