A computation of bifurcation parameter values for limit cycles

This paper describes a new computational method to obtain the bifurcation parameter value of limit cycles in nonlinear autonomous systems. Local bifurcations, e.g., tangent, period-doubling and Neimark-Sacker bifurcations, can be calculated by using the characteristic equation for a fixed point of the Poincare mapping. Conventionally a period of the limit cycle is not used explicitly since the Poincare mapping is needed whether the orbit reaches a cross-section or not. In our method, we regard the period as an independent variable for Newton's method and obtain location of a fixed point, the bifurcation parameter and the period simultaneously. Although the number of variables increases, the Jacobi matrix becomes simple and the method converges rapidly against the conventional method.