Chaotic Behavior in Shell Models and Shell Maps

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Abstract

We study the chaotic behavior of the GOY shell model by measuring the variation of the maximal Lyapunov exponent with the parameter ε which determines the nature of the second invariant (the generalized “helicity” invariant). After a Hopf bifurcation, we observe a critical point at εc∼0.38704 above which the maximal Lyapunov exponent grows nearly linearly. For high values of ε the evolution becomes regular again, which can be explained by a simple analytic argument. A model with few shells shows two transitions. To simplify the model substantially we introduce a shell map which exhibits similar properties as the GOY model.

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