This chapter investigates direct methods for evaluating the bifurcation parameter values of periodic solutions in nonlinear systems. The method can numerically calculate parameter values with high convergence rate and accuracy, at which local bifurcations; tangent, period-doubling, Neimark-Sacker and pitchfork bifurcations are occurred. The method uses some basic properties of the characteristic equation for a fixed point of the corresponding Poincar map. The location and parameter value of the fixed point are used as independent variables in the computation that employs quadratically convergency in Newton method. In addition, in the case of Neimark-Sacker bifurcation, the argument of the complex conjugate multiplier or the other parameter is used as the independent variable. This direct method utilize accurate solution of variational equations with numerical integration, thus implementation of algorithm becomes very simple. Some numerical examples are given.

28.1 Introduction

Nonlinear circuits, neural networks, and biological cooperative behaviors are frequently described by ordinary differential equations. This kind of dynamical systems usually exhibits complex phenomena, such as oscillatory synchronization, chaos, chaotic synchronization, spatiotemporal chaos, and so on ^[1]. Most of these phenomena are caused by various bifurcations which, in turn, depend on some variable system parameters. To capture certain feature behaviors of such dynamical systems via monitoring and controlling their variable parameters is an important task in engineering analysis and applications.

Typically, there...