Bifurcation of periodic solution

We show in Fig. 7 a bifurcation diagram around the point marked by in Fig. 4 (the point of intersection of D-type of branching set ₀d₂ and Hopf bifurcation set ₀h₁). Figure 8 shows a bifurcation diagram when the parameters change along the curve l in Fig. 7. ``Almost in-phase'' solutions generated by D-type of branching D₃ (3) of in-phase solution meet Neimark-Sacker bifurcation N (4) and disappear by the Hopf bifurcation ₂h₁ (5) of $\sigma_2$-invariant equilibrium points.

  

Figure7: Bifurcation diagram around the intersection point of Hopf bifurcation and D-type of branching. N represents Neimark-Sacker bifurcation.

  

Figure8: Bifurcation diagram corresponding to the curve l in Fig. 7. Heavy curves (D) and light curves (O) indicate periodic solutions and equilibrium points, respectively. Subscripts of D and O represent a dimension of unstable subspace.

Figure 9 shows a bifurcation diagram around the point marked by in Fig. 4 (the intersection point of ₂d₁ and ₂h₂). In small parameter region there exist many bifurcation sets and the bifurcation diagram becomes complicated therefore we use a schematic diagram. Figure 10 shows a bifurcation diagram when the parameters change along the curve l in Fig. 9. Bifurcation structure is similar to that of Fig. 7, but in Fig. 9 there exists tangent bifurcation set G₄ and asymmetrical solution is folded (the point marked by 8 in Fig. 10). Four unstable ICCs (Invariant Closed Curves) which correspond to quasi-periodic solutions are generated by N₂ (6). Those results permit us to predict that similar bifurcation structure would exist around the points marked by �� in Fig. 4.

  

Figure9: Schematic bifurcation diagram around the intersection point of Hopf bifurcation and D-type of branching.

  

Figure10: Bifurcation diagram corresponding to the curve l in Fig. 9.

We show in Fig. 11 a bifurcation diagram around the point marked by 5 in Fig. 4 (the intersection point of ₀h₁ and ₀h₂). In the shaded region , one stable equilibrium point with full symmetry exists. From this region increasing the parameter $\epsilon$ and crossing ₀h₁, we obtain a stable in-phase solution. On the other hand decreasing the parameter $\delta$ and crossing ₀h₂, a stable anti-phase solution appears. Figure 12 shows a bifurcation diagram when the parameters change along the curve l in Fig. 11. The in-phase and the anti-phase solution appeared at 2, 7 and 1, 6 respectively, generate ``$\bar{I_4}$-invariant'' periodic solutions by D-type of branchings D₁ (3) and D₂ (5). On the axis of $\delta=0$, the dimension of unstable subspace is changed through a cusp point.

  

Figure11: Bifurcation diagram around the intersection point of double Hopf bifurcations. D represents D-type of branching of periodic solution.

  

Figure12: Bifurcation diagram corresponding to the curve l in Fig. 11.

We show in Fig. 13 a bifurcation diagram around the point marked by 5 in Fig. 4 (the intersection point of ₂h₁ and ₂h₂ and also of ₁h₁ and ₁h₂). Bifurcation structure is the same as that of Fig. 11, but symmetrical properties and stability are changed, see Fig. 14.

  

Figure13: Bifurcation diagram around the intersection point of double Hopf bifurcations.

  

Figure14: Bifurcation diagram corresponding to the curve l in Fig. 13.

Figure 15 shows a bifurcation diagram in the large value of parameter $\epsilon$. The in-phase solution and the anti-phase solution generated by the Hopf bifurcations ₀h₁ and ₀h₂ disappear by tangent bifurcations G₁ and G₂, respectively [12]. ``$\bar{I_4}$-invariant'' periodic solutions caused by D₁ of the in-phase solution and D₂ of the anti-phase solution meet D₄ and generate four asymmetrical solutions. Figure 16 shows a bifurcation diagram of the asymmetrical solutions. Inside period-doubling bifurcation set I₁ there is a cascade of period-doubling bifurcations and asymmetrical chaotic state appears, see Fig. 17. From Fig. 17 (b), this chaotic attractor has any symmetry operations therefore in total four chaotic attractors exist. In a system of coupled two oscillators where the single oscillator does not have any chaotic oscillations, the existence of stable asymmetry periodic solution is one of the most important condition for existing a chaotic attractor.

  

Figure15: Bifurcation diagram of in-phase and anti-phase solution.

  

Figure16: Enlarged diagram of Fig. 15. In the shaded region the asymmetrical solutions stably exist.

  

Figure17: Chaotic attractor. $\alpha = 1.546$, $\delta = 0.05$.

Figure 18 shows a detailed bifurcation diagram of Fig. 13. In the shaded region , there exist ``almost in-phase'' 2-periodic solutions generated by period-doubling bifurcation set I<sub>2</sub> of ``shifted anti-phase'' solution. By crossing the L, two ``shifted anti-phase'' solutions bifurcate to one anti-phase solution after separatrix loops.

  

Figure18: Bifurcation diagram of ``shifted anti-phase'' solution created by Hopf bifurcation set ₂h₂. The curve L denotes a global bifurcation set.

In Fig. 19 we summarize the results obtained from calculating the bifurcation sets of equilibrium points and periodic solutions in Eqs. (7).

  

Figure: Possible symmetry-breaking bifurcations observed in Eqs. (7). Small squares represent (r₁, r₂) phase plane. Heavy and light solid lines indicate Hopf bifurcation and D-type of branching, respectively. The dotted lines I and the dashed lines L indicates period-doubling bifurcations and global bifurcations, respectively.