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 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 ≈ 0) and K < K c (K c ≈ 0.9716); (ii) the transition from limited to unlimited growth of I 2 , K K c ; (iii) the regime of strong nonlinearity, K 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.

Scaling properties of discontinuous maps

“…We found that the scaling of I 2 for map (2) when K K c and K K c obey the same scaling laws as CSM in the regimes of weak and strong nonlinearity [13], respectively. Except for that in the slow diffusion regime, due to the absence of KAM tori to bound the motion, I 2 ∝ n for large enough n. Also, we conclude that the scaling I 2 ∝ n α K β applies to discontinuous maps with (i) α ≈ 2 and β ≈ 2 for K K c and small n; (ii) α ≈ 1 and β ≈ 5/2 for K K c and large n; and (iii) α ≈ 1 and β ≈ 2 for K K c and large n. Our results are summarized in Table I. …”

“…056212-3 This behavior for I 2 is completely equivalent to that for CSM in the strong nonlinearity regime [13]. That is, the scaling given in Eq.…”

Shrimp-shape domains in a dissipative kicked rotator

“…On the other hand, when weak dissipation is taken into account, a drastic change occurs in the behavior of the average energy. The unlimited energy growth present in the Hamiltonian case 38 for the case K ) K c is no longer observed. The average action exhibits a characteristic saturation value which can be described using scaling arguments.…”

A dynamical phase transition for a family of Hamiltonian mappings: A phenomenological investigation to obtain the critical exponents

“…Since the idea was used with success in the Fermi-Ulam model [12], different authors extended the formalism and hence applied with great success the scaling approach in other mappings [25][26][27][28][29][30][31].…”