HYSTERESIS BIFURCATION THEORY

The values of X (within the domain of interest) at which the number of solutions of Eq. (1) changes are called bifurcation points. At these points F 9F/3x = 0. Using bifurcation theory it can be shown that the nature of a bifurcation diagram can change only if the parameter values cross one of three hypersurfaces [3]. The first called the Hysteresis variety (H) is the set of all points in the parameter space satisfying... 

Further highlights were the identification of thermal noise shghtly below threshold by Rehberg et al. [38-40] and finally the clear identification of a Hopf bifurcation leading to travelling rolls or waves in sufficiently thin layers, below about 50 pm and clean material (low conductivity) by Refs. [18, 41-43]. It is not possible to explain the Hopf bifurcation within the conventional theoretical framework the standard model (SM), see Section 13.2, where the LC is treated as an anisotropic ohmic conductor. Indeed some of the theoretical effort, in particular the inclusion of the rather complicated flexoelectric terms [14, 44 46], was aimed primarily at resolving this problem. The situation is fiirther complicated by the fact that the bifurcation is often observed to be slightly subcritical, i.e. with a very small hysteresis [38-40, 47], whereas the theory predicts a supercritical bifurcation. 

This discrepancy is in line with other observed quantitative discrepancies such as in the threshold behaviour when a stochastic component is present in the applied voltage [96]. As already mentioned, the most drastic non-standard behaviour, which is not understood from the SM, is the observed Hopf bifurcation in sufficiently thin and clean specimens [18, 41-43, 49-51] and the very small hysteresis sometimes observed at threshold [38-40, 47], In fact a little further above threshold (but still near it) the amplitude of the pattern does appear to coincide reasonably with the results of the weakly non-linear theory [59],... 

In the frame of this weakly nonlinear theory the hexagons are the first to appear, subcritically on increasing the value of the bifurcation parameter /x, the hexagons become unstable with respect to stripes. Reversing the variation of /X allows one to recover the hexagonal structure but by undergoing an hysteresis loop. This is the universal hex-stripes competition scenario that comes up in many different fields of study. It is also that which is observed in the quasi-2D Turing experiments [20, 34] and in the theoretical analysis [35-39] and numerical simulations of most nonlinear chemical models [40-44]. 

For go < gj D and in the weakly nonlinear theory (equivalent to the 2D hex-stripes competition and thus negligeable v renormalization) the stability study leads to sc and fee structures unstable with respect to the bcc, hpc and lam patterns. Thus, on increasing g the bcc structure is the first to appear sub-critically it is followed, also subcritically by the hpc pattern that finally yields to the lamellae. On reversing the variation of the bifurcation parameter one backtracks through these structures with the corresponding hysteresis loops. These structures have been observed, in that order, in numerical simulations of the Brusselator [52]. There is also experimental evidence for the bcc and hpc patterns [51]. 

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