Turbulence Caused by Phase Singularities

This is estimated to give 1-2 for the parameter values at the accumulation point of the bifurcations. Thus, the value of the Feigenbaum constant 5, in particular, should differ from the standard value 4.669.  

The present study suggests that the probability of encountering smooth onedimensional maps with non-quadratic maxima should not be ignored as nongeneric for real physical systems. The anomalous bifurcation sequence discussed above is a consequence of a system s symmetry with respect to spatial inversion this kind, or possibly other kinds, of symmetry are commonly present in real physical systems. Experimentally, such a bifurcation sequence could easily be distinguished from the usual subharmonic bifurcations. This is because considerable elongation in period (measured in the continuous time /, and not the step number n) is expected to occur each time a closed orbit is being transformed into a homoclinic orbit. 

It is expected that rotating spiral waves obtained for the Ginzburg-Landau equation become unstable and turbulent if l + CiC2 0. This is because spiral waves in general behave asymptotically as plane waves far from the core, and under the above condition no plane waves can remain stable. However, we are not much interested in this kind of turbulence in the present section, but we are more interested in the sort of turbulence which would be caused by the very existence of the phase singularity in the core. 

Let us restrict our attention, for simplicity, to the case of vanishing Cj, by which the possibility of the first type of turbulence may be eliminated. For infinitely large systems, C2 is the only parameter involved. Remember that the spiral waves obtained numerically in Chap. 6 were also for Cj = 0, and that they persisted in being stable for some range of C2. Here we seek the possibility of their becoming unstable for still larger C2. 

By taking the complex conjugate of this equation, and changing the sign before C2 at the same time, the equation remains invariant. This says that the only relevant parameter is the absolute value of C2. As we see below, sufficiently large c21 causes turbulence. Since C2 = Im /Re, where g is the nonlinear parameter in the original form of the Ginzburg-Landau equation (2.4.10), c2 oo as Re 0 (i.e., as the system approaches the borderline between supercritical and subcritical bifurcations). A number of kinetic models can have parameter values for which Re = 0, so that such systems should in principle exhibit chemical turbulence of the type discussed below. 

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