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A turbulent state in a distributed chemical reaction system is studied theoretically. The chaotic behavior here is basically due to the unstable growth of a spatial inhomogeneity taking place in an oscillating medium. It is argued that phase turbulence and amplitude turbulence have to be discriminated from each other according to their distinct origins. Two prototype equations describing respective types of turbulence are derived by means of some asymptotic methods, and their solutions turn out toe exhibit successive bifurcations. It is found that the phase turbulence arises essentially from the interaction among a few unstable phase modes, while the amplitude turbulence may well appear in the presence of only one unstable mode with is a mixed mode of the phase and amplitude. In particular, some similarity of the amplitude turbulence to the Lorenz chaos is pointed out. Throughout the present paper, the chaotic behavior is discucssed in connection with spatial pattern, so that no discrete approximations, such as a box-model or mode-truncation, are employed.
Yoshiki Kuramoto (1978) studied this question.