Primary bifurcation parameters

Fig. 4. Bifurcation diagram for variable X of the Brusselator (A = 4.5, Dy/Dx = 8) as a function of the parameter B. It exhibits the standard hex-stripe competition (hysteresis loop) near the primary bifurcation (Be = 6.71). Reentrant hexagons become stable for higher values of the bifurcation parameter B. [Xmax — A] is represented.

This idea was proposed by Andronov and Leontovich in their first work [9] which deals with primary bifurcations of limit cycles on the plane. Further developments of the theory of bifurcations, internal to the Morse-Smale class, has also confirmed the sufficiency of using finite-parameter families for a rather large number of problems. 

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... 

In the following, we first discuss the situations where EC occurs as a primary forward bifurcation and where the standard model is directly apphcable (cases A and B). Then we discuss configurations where EC sets in as a secondary instability upon an already distorted Freedericksz ground state and compare it with experiments (cases C and D). Note that in this case the linear analysis based on the standard model already becomes numerically demanding. Finally, we address those combinations of parameters where a direct transition to EC is not very robust, since it is confined to a narrow Ca range around zero. For cases E and H this range may be accessible experimentally while for cases F and G it is rather a theoretical curiosity only. 

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.

There is no doubt that some subtle aspects of the behavior of homoclinic and heteroclinic trajectories might not be important for nonlinear dynamics since they refiect only fine nuances of the transient process. On the other hand, when we deal with non-wandering trajectories, such as near a homoclinic loop to a saddle-focus with i/ < 1, the associated fi-moduli (i.e. the topological invariants on the non-wandering set) will be of primary importance because they may be employed as parameters governing the bifurcations see [62, 63]. 

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