Bifurcation of equilibrium point

The Jacobian matrix of the system at the equilibrium point (r₁₀, s₁₀, r₂₀, s₂₀) [11] is

Parameters in Eqs. (7) are fixed as

$$\sigma=0.8, \: \beta = 1.0, \: \omega = 1.0.$$ (12)

We investigate bifurcation problems in the $(\delta, \epsilon)$parameter plane. By calculating the eigenvalues $\mu$ of Eq. (16), we obtain Hopf bifurcation () and D-type of branching ($\mu = 0$) sets. The results are shown in Fig. 4 as a bifurcation diagram.

  

Figure4: Bifurcation diagram of equilibrium points. Solid lines (h) and dashed lines (d) indicate Hopf bifurcation set and D-type of branching set, respectively.

In Fig. 4 the notations _mh_k and _md_k indicate, respectively, Hopf bifurcation set and D-type of branching set of type m equilibrium point, see Table 2; k denotes the bifurcating direction (I or II) in Fig. 2). The symbols ��, �� and �� represent codimension two, three and four bifurcations which are the points of intersection of double Hopf bifurcations, D-type of branching and Hopf bifurcation, and double D-type of branchings, respectively. In the regions , and there exist stable equilibrium points whose type is, respectively, full symmetry, $\sigma_1$ -invariant and $\sigma_2$ -invariant.

At first we explain bifurcation structure and stability of equilibrium points around the point marked by . In Sect. 4.3 we will show detailed bifurcation diagrams including bifurcation sets of periodic solutions around the points marked by , , 4 and 5.

Figure 5 shows only D-type of branching sets in Fig. 4. A schematic bifurcation diagram is shown in Fig. 6 when the parameters $\epsilon$ and $\delta$ change along the curve l in Fig. 5. In Fig. 6 from the points marked by 1, 2, 5 and 6 ``two'' equilibrium points are generated by D-type of branching, but we omit one of them because two branches have same bifurcation structure. From Fig. 6, we see that there exist one equilibrium point with full symmetry in whole parameter plane, two $\sigma_2$-invariant equilibrium points between 1 and 5 two $\sigma_1$ -invariant equilibrium points between 2 and 6, and four equilibrium points without symmetry between 3 and 4. Since the equilibrium point with full symmetry is already completely unstable (₄O) by two Hopf bifurcations ₀h₁ and ₀h₂, equilibrium points generated by D-type of branchings (₀d₁, ₀d₂, ₁d₂ and ₂d₁) are all unstable.

  

Figure5: Bifurcation diagram around the intersection point of double D-type of branchings.

  

Figure6: Bifurcation diagram corresponding to the curve l in Fig. 5. The symbol O indicates equilibrium point and its subscript represents a dimension of unstable subspace.

In Table 4 we show that Hopf bifurcations in Fig. 4 generate what kind of periodic solutions. From this table we see that six kinds of periodic solutions are generated by Hopf bifurcations of three kinds of equilibrium points and ``$\bar{I_4}$-invariant'' periodic solution never appear by Hopf bifurcation.

  Table4: Hopf bifurcations of equilibrium points shown in Fig. 4.

notation bifurcation

₀h₁ full symmetry -- in-phase

₀h₂ full symmetry -- anti-phase ₁h_{1 1}h_{2 2}h_{1 2}h₂