COMPUTATIONAL BIFURCATION THEORY

Abstract

The present study mainly focuses on computational bifurcation theory. The continuation method is used for the numerical calculation of bifurcation points. An algorithm for the detection of Hopf bifurcation from dynamical solutions is proposed and examined with several examples. A nonlinear system which describes a shallow arch subjected to horizontal and vertical harmonic dynamic loads or a two-bar mechanism containing a nonlinear spring and a dashpot is thoroughly studied. In chapter 2, a review of the basic principles of the continuation method is given and the method of numerical detection of bifurcation is described. In chapter 3, a new method of calculation Hopf bifurcation points is proposed and applied to some examples. The method can be applied for both autonomous and nonautonomous systems. In Chapter 4 and Chapter 5, a nonlinear parametric system is investigated with the new algorithm. The primary bifurcation of the system is shown to exist in a large range of excitation frequency. Periodic doubling, multiple solutions are found at both fundamental and principal parametric resonance. Finally in chapter 6, the Lyapunov exponent will be used to study the effects of other parameters in the system on the stability and bifurcation of the trivial.