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Dynamical properties of a particle in a wave packet: Scaling invariance and boundary crisis

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  • Oliveira, Diego F.M.
  • Robnik, Marko
  • Leonel, Edson D.

Abstract

Some dynamical properties present in a problem concerning the acceleration of particles in a wave packet are studied. The dynamics of the model is described in terms of a two-dimensional area preserving map. We show that the phase space is mixed in the sense that there are regular and chaotic regions coexisting. We use a connection with the standard map in order to find the position of the first invariant spanning curve which borders the chaotic sea. We find that the position of the first invariant spanning curve increases as a power of the control parameter with the exponent 2/3. The standard deviation of the kinetic energy of an ensemble of initial conditions obeys a power law as a function of time, and saturates after some crossover. Scaling formalism is used in order to characterise the chaotic region close to the transition from integrability to nonintegrability and a relationship between the power law exponents is derived. The formalism can be applied in many different systems with mixed phase space. Then, dissipation is introduced into the model and therefore the property of area preservation is broken, and consequently attractors are observed. We show that after a small change of the dissipation, the chaotic attractor as well as its basin of attraction are destroyed, thus leading the system to experience a boundary crisis. The transient after the crisis follows a power law with exponent −2.

Suggested Citation

  • Oliveira, Diego F.M. & Robnik, Marko & Leonel, Edson D., 2011. "Dynamical properties of a particle in a wave packet: Scaling invariance and boundary crisis," Chaos, Solitons & Fractals, Elsevier, vol. 44(10), pages 883-890.
    • Handle: RePEc:eee:chsofr:v:44:y:2011:i:10:p:883-890
    DOI: 10.1016/j.chaos.2011.07.001
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    References listed on IDEAS

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    1. Leonel, Edson D. & Livorati, André L.P., 2008. "Describing Fermi acceleration with a scaling approach: The Bouncer model revisited," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(5), pages 1155-1160.
    2. Leonel, Edson D. & da Silva, J.Kamphorst Leal & Kamphorst, Sylvie O., 2004. "On the dynamical properties of a Fermi accelerator model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 331(3), pages 435-447.
    3. Oliveira, Diego F.M. & Leonel, Edson D., 2010. "Suppressing Fermi acceleration in a two-dimensional non-integrable time-dependent oval-shaped billiard with inelastic collisions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(5), pages 1009-1020.
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