We show that stable localized waves can be generated in the vicinity of an inverted Hopf bifurcation. We compute the size of the localized wave envelope perturbatively in the case of slightly dissipative systems. The size selection traces back to the broken scale invariance by the dissipative terms. This mechanism is a possible explanation for the localized structures, widely observed in various hydrodynamic flows in dissipative systems driven far from equilibrium.
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Fauve et al. (1990) studied this question.
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