0 Fe b 20 05 1 D spirals : is multi stability essential ?

A complex Ginzburg-Landau type amplitude equation which is valid in the Hopf region of phase space near an instability threshold admits solutions like antisymmet-ric pulses traveling in alternate direction from a core. Such pulse solutions come out as a steady state solution of the system on moving frame. The pulses have spatial profile like Hermite polynomial of order unity with the negative(positive) part of it less well defined on the right(left) wards moving frames due to nonlinear effects. One dimensional spiral pattern, experimentally observed in CIMA reaction is a fascinating phenomenon of nonlinear waves. As described in the experimental part of [1] that at first a transition between a stationary periodic and a traveling wave like Hopf mode is observed by lowering the starch concentration in the chemical reactor. Now, keeping a low starch concentration and by increasing the malonic acid concentration a similar transition takes place which makes an almost 1D Turing state loose its stability to a Hopf state when malonic acid concentration is doubled. It has been observed in this experiment that when the Hopf state takes up from the Turing state very often there remain a few spots, reminiscent of the previous Turing state, acting as the source of one dimensional anti synchronous wave trains. Bands of maximum intensity spreads alternatively toward right and left directions with a time delay. As experiment suggests, this phenomenon is essentially one dimensional and variation of two parameters like concentrations of starch and malonic acid indicates proximity to a co-dimension 2 point. On the basis of this observation a theory has been developed being based on multisability and non variational effects. Such localized structures near a Hopf-Turing instability boundary have been analyzed in many subsequent papers [2-9]. A common belief is that in the range of parameters far away from depinning transitions a pinned Turing state in a global traveling wave phase helps sustain 1D spirals by accepting one on one side of it while leaving the other on the *