Bifurcations primary equations

The simplest new phenomenon induced by this mechanism is a secondary bifurcation from the first primary branch, arising from the interaction between the latter and another nearby primary branch. It leads to the loss of stability of the first primary branch or to the stabilization of one of the subsequent primary branches, as illustrated in Fig. 1. The analysis of this branching follows similar lines as in Section I. A, except that one has now two control parameters X and p., which are both expanded [as in equation (5)] about the degeneracy point (X, p.) corresponding to a double eigenvalue of the linearized operator L. Because of this double degeneracy, the first equation (7) is replaced by... 

Figure 5. Bifurcation diagram on the plane of the two control parameters p and a. The solid lines 1 and 2 mark the primary instability, where the homogeneous homeotropic orientation becomes unstable. At 1, the bifurcation is a stationary (pitchfork) bifurcation, and a Hopf one at 2. The two lines connect in the Takens-Bogdanov (TB) point. The solid lines 3 and 4 mark the first gluing bifurcation and the second gluing bifurcation respectively. The dashed lines 2b and 3b mark the lines of the primary Hopf bifurcation and the first gluing bifurcation when calculated without the inclusion of flow in the equations.

The analysis of the bifurcation equations in this case indicates that in addition to the primary states given by (2.11) and (2.12), there exists a secondary bifurcation to quasi-periodic solutions which satisfies (2.16) with P e a2 P+ bj o). The condition for its existence and its location depends on higher order terms in the bifurcation equations. This analysis is carried out by Erneux and Matkowsky in [6]. Figure 3 exhibits a typical bifurcation diagram of the amplitude as a function of X. 

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