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

Juliano A. de Oliveira*, Carl P. Dettmann, Diogo R. da Costa, Edson D. Leonel

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

14 Citations (Scopus)


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.

Original languageEnglish
Article number062904
Number of pages5
JournalPhysical Review E: Statistical, Nonlinear, and Soft Matter Physics
Issue number6
Publication statusPublished - 10 Jun 2013


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