Bifurcation imperfect

Golubitsky, M. and Schaeffer, D. (1979). A theory for imperfect bifurcation via singularity theory. Comm Pure Appl. Math., 32, 21-98. 

In many applications involving nonequilibrium instabilities and dissipative structures, the sharp transitions corresponding to bifurcation rarely occur. Small impurities, imperfections, or external fields tend to distort these transitions. Many experiments, particularly in fluid dynamics, illustrate this fact and demonstrate, in addition, that small imperfections may have large or even qualitative effects. This very general phenomenon is at the basis of the enhanced sensitivity of systems operating near a bifurcation point discussed in the chapter by I. Prigogine. 

The first of these equations admits a solution of the type given in equation (7). In order to specify the amplitude of this solution, which remains undetermined in this stage, one has to solve equations (14) for j 2. As the operator °( ) admits a nontrivial null solution, one has to verify that these latter equations satisfy suitable solvability conditions. If this is the case for all j, then the imperfection constitutes a smooth perturbation whose effects can be handled by the expansion (13). In particular, bifurcation will subsist and the bifurcation points K will be identical to those determined in the absence of the imperfection. 

Substituting into equation (12) one then obtains a system of linear equations for xj which incorporate the effect of the imperfection already at the dominant order. By working out these relations one finally arrives at the bifurcation equations for the amplitude of the dominant part of the solution, exi. 

Note the near-horizontal and slightly disconnected black line at y yc = 1.13 in Figure 3.21. For the chosen value of a = 1.23706 106 = io6 092389299 the bifurcation diagram becomes an imperfect pitchfork. A perfect pitchfork bifurcation diagram occurs when a is slightly decreased, so that the corresponding black horizontal line in Figure 3.19... 

The bifurcation diagram represents an imperfect pitchfork diagram, where the middle steady state persists over the entire range of Kc, even for negative values of Kc, i.e., even for positive feed back control, which would destabilize the system. 

Imperfect Bifurcation. In the context of spatial selforganization, the bifurcation diagram of a system is a plot of the amplitude of a pattern as a function of the various control parameters. For the cell in a plane parallel applied field of strength E and with some system parameter X (as a bath concentration or as in (22)), the pattern amplitude can be described by the g s in (23-30) and especially (29). 

In the present problem the imperfect bifurcation theory results in clarifying the close analogy of the development of polarity in Fucus with a ferromagnetic transition. To demonstrate this we multiply (kk) by e3 and let ge s m, aX e2 s t and bE e3 = h. Then (kk) becomes 2 3... 

Figure 4a. Imperfect bifurcation showing the breaking of the symmetry of the diagram of Figure 3a due to the imposition of an external electric field of strength...

Imposed Field Effects. In this section we have set forth a set of equations to describe pattern formation in a multicellular electrophysiological system. A central goal of the theory is to study the effects of applied electric fields. This is done by imposing appropriate boundary conditions on the equations developed here. For example, assume we subject a one dimensional tissue to fixed ionic currents 1. Then if the tissue is in the interval 0 x along the x axis, the boundary conditions for the electro-diffusion model of the small gradient theory, i.e. (6k), are replaced by J = I at x = 0, L. One expects the richness of effects to include hyperpolarizability, induction of new phenomena and imperfect bifurcations to be found in these systems... 

Ortoleva, P. "inherent Asymmetry and Imperfect Bifurcation in the Developmental Scenario" (in preparation). 

As we mentioned earlier, pitchfork bifurcations are common in problems that have a symmetry. For example, in the problem of the bead on a rotating hoop (Section 3.5), there was a perfect symmetry between the left and right sides of the hoop. But in many real-world circumstances, the symmetry is only approximate—an imperfection leads to a slight difference between left and right. We now want to see what happens when such imperfections are present. 

If /i = 0, we have the normal form for a supercritical pitchfork bifurcation, and there s a perfect symmetry between x and -x. But this symmetry is broken when h 0 for this reason we refer to h as an imperfection parameter. 

As a simple example of imperfect bifurcation and catastrophe, consider the following mechanical system (Figure 3.6.7). 

Imperfect transcritical bifurcation) Consider the system x — h + rx — x. When h-0, this system undergoes a transcritical bifurcation at r-0. Our goal is to see how the bifurcation diagram of x vs. r is affected by the imperfection parameter h. 

Imperfect saddle-node) What happens if you add a small imperfection to a system that has a saddle-node bifurcation ... 

Research project on asymmetric waterwheel) Our derivation of the waterwheel equations assumed that the water is pumped in symmetrically at the top. Investigate the asymmetric case. Modify Q(6) in (9.1.5) appropriately. Show that a closed set of three equations is still obtained, but that (9.1.9) includes a new term. Redo as much of the analysis in this chapter as possible. You should be able to solve for the fixed points and show that the pitchfork bifurcation is replaced by an imperfect bifurcation (Section 3.6). After that, you re on your own This problem has not yet been addressed in the literature. 

An interesting situation also came to light in the limit of normal incidence. This case was impossible to analyze in the framework of the approximate model, as the modes become large quickly and violate the initial assumptions. It turned out that for a = 0 (which is a peculiar case, since the external symmetry breaking in the x direction vanishes), another stationary instability precedes the secondary Hopf bifurcation that spontaneously breaks the reflection symmetry with respect to x. It is shown by point A in Fig. 18. It is also seen from this figure, that the secondary pitchfork bifurcation is destroyed in tbe case of oblique incidence, which can be interpreted as an imperfect bifurcation with respect to the angle a [43]. 

To precipitate stable deformation beyond the point of bifurcation, a slight imperfection in the system was introduced in the form of an anisotropic mismatch strain. Typically, the mismatch strain in the x-direction y-direction) was taken to be 0.01% larger (smaller) than the nominal value Cm- With this level of imperfection, the deformation prior to bifurcation... 

Figure 2.29 (shown by curves in a lighter shade here) was redone with a mismatch strain that is 1% larger (smaller) than the nominal value Cm in the a —direction (y—direction). The result is shown in Figure 2.30, where it can be seen that a 1% imperfection in mismatch strain obliterates the sharp bifurcation transition. Instead, the system undergoes a long, gradual transition from axially symmetric deformation to asymmetric deformation as Cm increases. 

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