THE HOPF-BIFURCATION AND LIMIT-CYCLE BY THE INCREMENTAL HARMONIC-BALANCE METHOD

It is well recognized that periodic solutions and their stabilities under small perturbations play a very important role in the study of nonlinear dynamic systems. These problems have been investigated by various perturbation methods which are, however, subjected to the limitation of weak nonlinearity. Moreover, a perturbation method cannot define the parameter range within which a correct solution or a satisfactory approximation can be obtained by the method. This situation may lead to totally incorrect results when the parameter is out of a certain range. By introducing a variable parameter the incremental harmonic balance method, which can treat strongly nonlinear dynamic problems, can be used to perform parametric studies on the Hopf bifurcation and limit cycle problems. This method is ideally suited for parametric studies, giving solution diagrams. In this paper it is applied to study the Van der Pol oscillator and coupled oscillators. Examples show that beyond a certain limit, the standard perturbation methods will give incorrect results.