Bifurcation secondary

Expanding the sample size to 2Xc admits the other shape families shown on Fig. 6 into the analysis and leads to additional codimension-two interactions between the shapes is the (1A<.)- family and shapes with other numbers of cells in the sample. The bifurcation diagram computed for this sample size with System I and k = 0.865 is shown as Fig. 11. The (lAc)- and (Ac/2)-families are exactly as computed in the smaller sample size, but the stability of the cell shapes is altered by perturbations that are admissible is the larger sample. The secondary bifurcation between the (lAc)- and (2Ae/3)-families is also a result of a codimension two interaction of these families at a slightly different wavelength. Two other secondary bifurcation points are located along the (lAc)-family and may be intersections with the (4Ac and (4A<./7) families, as is expected because of the nearly multiple eigenvalues for these families. 

The exchange-of-stabilities associated with each of the secondary bifurcations... 

There are two more important advantages of these models. One is that it is possible under some conditions to carry out an exact stability analysis of the nonequilibrium steady-state solutions and to determine points of exchange of stability corresponding to secondary bifurcations on these branches. The other is that branches of solutions can be calculated that are not accessible by the usual approximate methods. We have already seen a case here in which the values of parameters correspond to domain 2. This also happens when the fixed boundary conditions imposed on the system are arbitrary and do not correspond to some homogeneous steady-state value of X and Y. In that case Fig. 20 may, for example,... 

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... 

All transitions associated with secondary bifurcations have now been completely classified. Moreover, several examples of tertiary or even quaternary branchings are known both for steady-state and for time-periodic solutions (see, e.g., Iooss6 and Erneux and Reiss7 for some recent results). 

Although stability may in principle be computed, the calculation is extremely complicated. Numerical calculations suggest the asymptotic stability of the limit cycle, but the stability has not been rigorously established. Assuming that the solution is asymptotically stable, a secondary bifurcation can be shown to occur. The argument is quite technical and requires a form of a Poincare map in the appropriate function space it is analogous to the bifurcation theorem used in Chapter 3 for bifurcation from a simple eigenvalue. The principal theorem takes the form of a bifurcation statement. 

Many features become more transparent when formulated in real (position) space in terms of ampbtude (envelope) or Ginzburg-Landau equations (GLE). Then one sees that the important information is really condensed in a few parameters and the universal aspects of the systems become apparent. By model calculations, which can often be performed analytically, stability boundaries and secondary bifurcation scenarios are traced out. The real space formulation is essential when it comes to the description of more complex spatio-temporal patterns with disorder and defects, which have been studied extensively in EHC slightly above threshold (Figs. 13.1b, 13.3b). One introduces a modulation ampbtude y4(x) defined as... 

The terms occurring are those allowed by symmetry. Due to the anisotropy more terms appear than in isotropic systems [94]. The clue for the characteristic appearance of the zigzag instabihty as a secondary bifurcation is that 4 is typically negative in nematics [23, 24] leading, in contrast to isotropic fluids, to amplification of transverse modulations of roll patterns. Model calculations that include this feature [25, 26] were quite successful in describing quahtatively the secondary instability and the behaviour beyond. 

From the foregoing discussion we know that the weakly non-linear behaviour of EHC in the range of validity of the GLE is imderstood fairly well. The situation is not quite as satisfactory when it comes to secondary bifurcations, which confine the q- range and which are captured in the theory only if corrections to GLE are included (Section 13.3) or a frill evaluation is done to the hydrodynamic 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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