Symmetry-breaking and first period-doubling following a Hopf bifurcation in a three dimensional system

A new perturbation procedure for limit cycle analysis in three-dimensional nonlinear autonomous dynamical systems

“…3, the limit cycle of two three-dimensional systems are analyzed by the present perturbation procedure. Our results and those of previous researchers [13][14][15][16][17] are compared. Numerical solutions are also obtained and our results are noted to be highly accurate.…”

“…System (21) can indeed be considered as a feedback control system [13]. It has been studied by Rand [13] using the normal form method and by Nayfeh [14], Belhaq [15][16][17] using the multiple scales method. We chose this system in order to demonstrate the effectiveness and accurateness of our new perturbation procedure.…”

“…Although their methods are different, their first-order approximate formulations for the limit cycle are identical. Belhaq et al [15][16][17] also applied the method of multiple scales to obtain the analytical approximation up to fifth order.…”

Bifurcation and route-to-chaos analyses for Mathieu–Duffing oscillator by the incremental harmonic balance method

“…Among the aforementioned routes to chaos, cascaded period-doubling bifurcations to chaos constitute a familiar scenario which has been studied extensively. Many methods, including the method of multiple scales [2,3], numerical time-integration [4][5][6][7][8][9], shooting method [10][11][12][13], the incremental harmonic balance (IHB) method [14][15][16][17][18][19], and the experimental method [20], etc., have been developed and applied to predict period-doubling bifurcations.…”

“…The method of multiple scales is an analytical method which had been used to predict the first and second period-doubling bifurcations in a threedimensional system [2,3]. In general, however, it is not able to predict the subsequent period-doubling bifurcations and the onset of chaos.…”