IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v44y2011i10p883-890.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077911001202
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2011.07.001?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Oliveira, Diego F.M. & Leonel, Edson D., 2014. "Statistical and dynamical properties of a dissipative kicked rotator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 413(C), pages 498-514.
    2. Hermes, Joelson D.V. & dos Reis, Marcelo A. & Caldas, Iberê L. & Leonel, Edson D., 2022. "Break-up of invariant curves in the Fermi-Ulam model," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    3. Jousseph, C.A. & Kruger, T.S. & Manchein, C. & Lopes, S.R. & Beims, M.W., 2016. "Weak dissipative effects on trajectories from the edge of basins of attraction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 456(C), pages 68-74.
    4. Oliveira, Diego F.M. & Roberto Silva, Mario & Leonel, Edson D., 2015. "A symmetry break in energy distribution and a biased random walk behavior causing unlimited diffusion in a two dimensional mapping," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 436(C), pages 909-915.
    5. Cetin, Kivanc & Tirnakli, Ugur & Oliveira, Diego F.M. & Leonel, Edson D., 2024. "Statistical mechanical characterization of billiard systems," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:44:y:2011:i:10:p:883-890. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.