Chaotic Behavior in Shell Models and Shell Maps

We study the chaotic behavior of the GOY'' shell model by measuring the variation of the maximal Lyapunov exponent with the parameter $\epsilon$ which determines the nature of the second invariant (the generalized helicity’’ invariant). After a Hopf bifurcation, we observe a critical point at ${ϵ}_c\sim 0.38704$ 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.