Tetsushi Ueta, Masafumi Tsueike, Hiroshi Kawakami, Tetsuya Yoshinaga, Yuji Katsuta, A computation of bifurcation parameter values for limit cycles, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, Vol.EA80, No.9, 1725-1728, Sep. 1997.
Abstract: This letter describes a new computational method to obtain the bifurcation parameter value of a limit cycle in nonlinear autonomous systems. The method can calculate a parameter value at which local bifurcations; tangent, period-doubling and Neimark-Sacker bifurcations are occurred by using properties of 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 needs only whether the orbit reaches a cross-section or not. In our method, the period is treated as an independent variable for Newton's method, so an accurate location of the fixed point, its period and the bifurcation parameter value can be calculated simultaneously. Although the number of variables increases, the Jacobian matrix becomes simple and the recurrence procedure converges rapidly compared with conventional methods.