Scaling invariance of the diffusion coefficient in a family of two-dimensional Hamiltonian mappings

Abstract

We consider a family of two-dimensional nonlinear area-preserving mappings that generalize the Chirikov standard map and model a variety of periodically forced systems. The action variable diffuses in increments whose phase is controlled by a negative power of the action and hence effectively uncorrelated for small actions, leading to a chaotic sea in phase space. For larger values of the action the phase space is mixed and contains a family of elliptic islands centered on periodic orbits and invariant Kolmogorov-Arnold-Moser (KAM) curves. The transport of particles along the phase space is considered by starting an ensemble of particles with a very low action and letting them evolve in the phase until they reach a certain height h. For chaotic orbits below the periodic islands, the survival probability for the particles to reach h is characterized by an exponential function, well modeled by the solution of the diffusion equation. On the other hand, when h reaches the position of periodic islands, the diffusion slows markedly. We show that the diffusion coefficient is scaling invariant with respect to the control parameter of the mapping when h reaches the position of the lowest KAM island.

On the statistical and transport properties of a non-dissipative Fermi-Ulam model

“…Both analytical and numerical results in this paper, give robustness to the theory of diffusion analysis, concerning the survival probability curves, as shown also in [42]. In the future, it would be interesting to try to expand this formalism to other more complex dynamical systems, like billiards for instance.…”

“…Considering yet V hole = h and a change in the notation of the sum index from the Fourier series expansion, from l/2 to (k + 1/2), where odd and even terms are considered, one can obtain [42] …”

“…(16) furnishes an analytical approximation for the survival probability, when a hole is introduced in the chaotic sea [42]. This expression holds for any value of k, it just depends on how many terms one would consider in the Fourier series expansion.…”

“…Considering an ensemble of particles, when a particle reaches a hole, we consider it has escaped, and the time evolution of other particle from the ensemble is started. An analytical expression is obtained for the diffusion coefficient as function of the expansion in Fourier series [42]. Our theoretical findings are compared with numerical simulations obtained via the average squared velocity.…”

Scaling properties for a family of discontinuous mappings

“…As examples, we can mention some well known mappings having in common the choice of f (θ n , I n+1 ) = sin(θ n ) and h(θ n , I n+1 ) = 0: Chirikov's standard map [2,3], g(I n+1 ) = I n+1 , also known as Taylor-Chirikov's map; the bouncer model [4], g(I n+1 ) = ξ I n+1 ; the logistic twist map [5], g(I n+1 ) = I n+1 + ξ I 2 n+1 ; the Fermi-Ulam accelerator model [6,7], g(I n+1 ) = 2/I n+1 ; a generalized Fermi-Ulam accelerator (FU) model [8][9][10][11],…”