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

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Abstract

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.

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Notes

  1. The last invariant circle not homotopic to a point. It must be noted that due the choice of scale of the map, the rotation number is \(\omega =2\pi \gamma \), instead of the golden mean: \(\gamma =\frac{\sqrt{5}-1}{2}\).

  2. The equation still depends on the label k of each oscillator, but it can be absorbed in the angular variable: \(\zeta _k:=\zeta +\omega k\).

  3. For the perturbation analysis to work we can assume that both parameters are small. Notice that this assumption is consistent since both parameters can be traced back to the perturbation parameter of the standard map.

  4. Were both equations have been merged into 1, the lagrangian representation of a 2-dimensional map.

  5. Actually fixed points for the map iterated q times.

  6. It is important to remind that the upper index 0 is just a reference, the value of \(\gamma \) will not change from iteration to iteration even though the change of variables g and the normal form are used to evaluate iterations of the map.

  7. Where: \(\phi ^k:= \zeta ^k - \theta ^k\).

  8. Where \(\mathrm{s}(\phi )\equiv \sin (\phi )\) and \(\mathrm{c}(\phi )\equiv \cos (\phi )\).

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Acknowledgments

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.

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Authors and Affiliations

  1. IIMAS-UNAM, 04510, Mexico, D.F., Mexico

    D. Martínez-del-Río, A. Olvera & R. Calleja

  2. Oak Ridge National Laboratory, Oak Ridge, TN, 37831-8071, USA

    D. del-Castillo-Negrete

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  1. D. Martínez-del-Río
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  2. D. del-Castillo-Negrete
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  3. A. Olvera
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  4. R. Calleja
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Correspondence to A. Olvera.

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Martínez-del-Río, D., del-Castillo-Negrete, D., Olvera, A. et al. Self-Consistent Chaotic Transport in a High-Dimensional Mean-Field Hamiltonian Map Model. Qual. Theory Dyn. Syst. 14, 313–335 (2015). https://doi.org/10.1007/s12346-015-0168-6

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