1 Hopf - Turing mixed mode and pattern selection in reaction diffusion systems

The amplitude equation of Gierer-Mainhardt model has been actually derived near the boundary abuot which Turing and Hopf modes exist. In a parameter region where Hopf-Turing mixed mode solution is stable, a chaotic state that generally results from interaction between mixed modes, is observed. This chaotic region follows a strong selection of a spatially periodic order followed by a local, resonant, very large frequency temporal oscillation. A spatio-temporal forcing, responsible for what obseved, has been identified. 1 Turbulence due to interaction of a spatial (Turing) and temporal (Hopf) modes has caught attention of physicists for more than a decade [1, 2]. A Hopf-Turing coupled system shows turbulence of mainly two types, 1) phase turbulene and 2) amplitude turbulence [3, 4, 5]. Phase turbulence or Phase chaos, is generally observed in a parameter region where adiabatic elimination of amplitude modes leads to a nonlinear phase equation. This phase turbulence is weak compared to the second type, the amplitude turbulene. The parameter region where amplitude turbulence occurs is that region where adiabatic elimination of amplitude is not possible because phase and amplitude are strongly coupled. A region of defect turbulence [3, 6]is believed to interpolate between these two regions. Turbulence resulting from interation of mixed modes, or on the other hand stability of different modes against this turbulence in the region where Turing and Hopf modes interact, has been studied on by numerical integration of prototypical amplitude equations. An actual derivation of amplitude equation in the region of interest, as has been done here, not only puts constraints on the variation of relavant coefficients of the equation but also gives the proper selection for the wave number and frequencies over which variations are investigated. Gierer-Meinhardt (GM) model which was originally put forward to explain some observed features of regeneration of hydra [7] has a codimension-2 point where two bifurcation boundaries intersect [8]. One of those boundaries is between a stable spatially periodic (Turing) region and homogeneous osillatory (Hopf) region and is of present interest. Near this boundary a coupled equation for slowly variing Turing and Hopf mode amplitudes has been derived by the use of multiple scale perturbation technique. A numerical investigation of this equation reveals strong selection of a wave number in a region where a spatiotemporal chaos results due to interation of a Turing mode and a travelling wave solution of the coupled amplitude equation. This mode loses stability to …