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Scaling properties of a simplified bouncer model and of Chirikov’s standard map. (English) Zbl 1142.82344

Summary: Scaling properties of Chirikov’s standard map are investigated by studying the average value of \(I^{2}\), where \(I\) is the action variable, for initial conditions in (a) the stability island and (b) the chaotic component. Scaling behavior appears in three regimes, defined by the value of the control parameter \(K\): (i) the integrable to non-integrable transition (\(K \approx 0\)) and \(K < K_{c} (K_{c} \approx 0.9716)\); (ii) the transition from limited to unlimited growth of \(I^{2}\), \(K \gtrsim K_{c}\); (iii) the regime of strong nonlinearity, \(K \gg K_{c}\). Our scaling results are also applicable to Pustylnikov’s bouncer model, since it is globally equivalent to the standard map. We also describe the scaling properties of a stochastic version of the standard map, which exhibits unlimited growth of \(I^{2}\) even for small values of \(K\).

MSC:

82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
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