Scaling properties of a simplified bouncer model and of Chirikov's standard map

Abstract

Scaling properties of Chirikov's standard map are investigated by studying the average value of I2, 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 ≈ 0) and K < Kc (Kc ≈ 0.9716); (ii) the transition from limited to unlimited growth of I2, K ≳ Kc; (iii) the regime of strong nonlinearity, K ≫ Kc. 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 I2 even for small values of K.