An analytical method for analyzing symmetry-breaking bifurcation and period-doubling bifurcation

Abstract

A new modification of homotopy analysis method (HAM) is proposed in this paper. The auxiliary differential operator is specifically chosen so that more than one secular term must be eliminated. The proposed method can capture asymmetric and period-2 solutions with satisfactory accuracy and hence can be used to predict symmetry-breaking and period-doubling bifurcation points. The variation of accuracy is investigated when different number of frequencies are considered.

Periodic solutions of delay nonlinear system by multi-frequency homotopy analysis method

“…x 00 1 ðtÞ þ cx 0 1 ðtÞ À x 2 ðt À sÞ þ x 3 1 ðtÞ ¼ fcosXt (24a) First, we fix the parameters c ¼ 0:4; f ¼ 2:85; X ¼ 1:15, and s ¼ 1:25, and further take h ¼ 0:0001; h ¼ 1; h ¼ 1000; h ¼ 10;000, and h ¼ 100; 000, respectively, Figures 6 to 10 compare the approximate and exact solutions of equation (24). We find that the three images obtained by h ¼ 0:0001 ( Figure 6), h ¼ 1 (Figure 7) and h ¼ 1000 ( Figure 8) are basically the same, and the approximate solutions of x 1 ðtÞ and x 2 ðtÞ are in good agreement with the exact solution.…”

“…Results showed that the method was effective in studying the dynamic response of a MDOF nonlinear coupling system. Zou and Nagarajaiah 24 proposed the multi-frequency HAM (MFHAM) to predict symmetry-breaking bifurcation and period-doubling bifurcation points accurately. Fu and Qian 25 used the modified HAM to study the single periodic and period-doubling solutions of a two-DOF (TDOF) coupled Duffing system.…”

Asymmetric Solutions of SDOF System with Wire Rope Vibration Isolator Subjected to Harmonic Excitation

“…[10][11][12] HAM still works accurately when the parameters in nonlinear ODEs and PDEs are large and has been successfully applied to a wide range of nonlinear problems. [13][14][15][16][17][18][19][20][21][22][23] However, as will be shown in the present paper, the standard HAM cannot capture asymmetric periodic solutions of nonlinear ODEs or PDEs. In the present work, a technique is put forward in the frame of HAM to capture asymmetric periodic solutions of dynamic systems with wire rope isolators which exhibit asymmetric restoring force (described in detail in the next section).…”

On the homotopy analysis method for backward/forward-backward stochastic differential equations

“…Especially, there is a convergence-control parameter c 0 in the series solutions, which provides us a convenient way to guarantee the convergence of series solutions gained by the HAM. It is these merits that distinguish the HAM from other analytic approaches, and thus enable the HAM to be successfully applied to many complicated problems with high nonlinearity [39][40][41][42][43][44][45][46][47]. Note that the HAM was successfully applied to give an analytic approximation with much longer expiry for the optimal exercise boundary of an American put option than perturbation approximations [48].…”