Qualitative Theory of Dynamical Systems

, Volume 14, Issue 2, pp 313–335 | Cite as

Self-Consistent Chaotic Transport in a High-Dimensional Mean-Field Hamiltonian Map Model

  • D. Martínez-del-Río
  • D. del-Castillo-Negrete
  • A. OlveraEmail author
  • R. Calleja


Self-consistent chaotic transport is studied in a Hamiltonian mean-field model. The model provides a simplified description of transport in marginally stable systems including vorticity mixing in strong shear flows and electron dynamics in plasmas. Self-consistency is incorporated through a mean-field that couples all the degrees-of-freedom. The model is formulated as a large set of N coupled standard-like area-preserving twist maps in which the amplitude and phase of the perturbation, rather than being constant like in the standard map, are dynamical variables. Of particular interest is the study of the impact of periodic orbits on the chaotic transport and coherent structures. Numerical simulations show that self-consistency leads to the formation of a coherent macro-particle trapped around the elliptic fixed point of the system that appears together with an asymptotic periodic behavior of the mean field. To model this asymptotic state, we introduced a non-autonomous map that allows a detailed study of the onset of global transport. A turnstile-type transport mechanism that allows transport across instantaneous KAM invariant circles in non-autonomous systems is discussed. As a first step to understand transport, we study a special type of orbits referred to as sequential periodic orbits. Using symmetry properties we show that, through replication, high-dimensional sequential periodic orbits can be generated starting from low-dimensional periodic orbits. We show that sequential periodic orbits in the self-consistent map can be continued from trivial (uncoupled) periodic orbits of standard-like maps using numerical and asymptotic methods. Normal forms are used to describe these orbits and to find the values of the map parameters that guarantee their existence. Numerical simulations are used to verify the prediction from the asymptotic methods.


Self-consistent transport Normal forms Sequential periodic orbits Single-wave model 



This work was founded by PAPIIT IN104514, FENOMEC-UNAM and by the Office of Fusion Energy Sciences of the US Department of Energy at Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the US Department of Energy under Contract DE-AC05-00OR22725. We also express our gratitude to the graduate program in Mathematics of UNAM for making the GPU servers available to perform our computations and especially to Ana Perez for her invaluable help. Finally, we would also like to thank the anonymous referee whose valuable comments have improved the presentation of the paper.


  1. 1.
    Bofetta, G., del Castillo, D., López, C., Pucacco, G., Vulpiani, A.: Diffusive transport and self-consistent dynamics in coupled maps. Phys. Rev. E 67, 026224 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Calleja, R., de la Llave, R.: Fast numerical computation of quasi-periodic equilibrium states in 1D statistical mechanics, including twist maps. Nonlinearity 22(6), 1311–1336 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Carbajal, L., del-Castillo-Negrete, D., Martinell, J.J.: Dynamics and transport in mean-field coupled, many degrees-of-freedom, area-preserving nontwist maps. Chaos 22, 1,013137 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    del-Castillo-Negrete, D.: Weekly nonlinear dynamics of electrostatic perturbations in marginally stable plasmas. Phys. Plasmas 5(11), 3886–3900 (1998)CrossRefGoogle Scholar
  5. 5.
    del-Castillo-Negrete, D.: Self-consistent chaotic transport in fluids and plasmas. Chaos 10, 75 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    del-Castillo-Negrete, D.: Dynamics and self-consistent chaos in a mean field Hamiltonian model. In: Dauxois, T., Ruffo, S., Arimondo, E., Wilkens, M. (eds.) Dynamics and Thermodynamics of Systems with Long Range Interactions, Lecture Notes in Physics, vol. 602, Springer, Berlin (2002)Google Scholar
  7. 7.
    Delshams, A., de la Llave, R.: KAM theory and a partial justification of Greene’s criterion for nontwist maps. SIAM J. Math. Anal. 31(6), 1235–1269 (2000)Google Scholar
  8. 8.
    Doedel, E.J.: Lectures Notes on Numerical Analysis of Nonlinear Equations. (2010)
  9. 9.
    Greene, J.M.: A method for computing the stochastic transition. J. Math. Phys. 20, 1183–1201 (1979)CrossRefGoogle Scholar
  10. 10.
    Haragus, M., Iooss, G.: Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems. Springer, New York (2011)zbMATHCrossRefGoogle Scholar
  11. 11.
    Kook, H.T., Meiss, J.D.: Periodic orbits for reversible, symplectic mappings. Phys. D 35, 65–86 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Meiss, J.D.: Symplectic maps, variational principles and transport. Rev. Mod. Phys. 64(3), 795–848 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Meyer, K.R., Hall, G.R.: Introduction to Hamiltonian Dynamical Systems and N-Body Problem. Springer, New York (1992)zbMATHCrossRefGoogle Scholar
  14. 14.
    Moser, J.: On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math. Phys. Kl 1(1), 1–20 (1962)Google Scholar
  15. 15.
    O’Neil, T.M., Winfrey, J.H., Malmberg, J.H.: Nonlinear interaction of a small cold beam and a plasma. Phys. Fluids 14, 1204 (1971)CrossRefGoogle Scholar
  16. 16.
    Olvera, A.: Estimation of the amplitude of resonance in the general standard map. Exp. Math. 10(3), 401–418 (2001)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.IIMAS-UNAMMexicoMexico
  2. 2.Oak Ridge National LaboratoryOak RidgeUSA

Personalised recommendations