Abstract
The chaotic diffusion for a family of Hamiltonian mappings whose angles diverge in the limit of vanishingly action is investigated by using the solution of the diffusion equation. The system is described by a two-dimensional mapping for the variables action, I, and angle, \(\theta \) and controlled by two control parameters: (i) \(\epsilon \), controlling the nonlinearity of the system, particularly a transition from integrable for \(\epsilon =0\) to non-integrable for \(\epsilon \ne 0\) and; (ii) \(\gamma \) denoting the power of the action in the equation defining the angle. For \(\epsilon \ne 0\) the phase space is mixed and chaos is present in the system leading to a finite diffusion in the action characterized by the solution of the diffusion equation. The analytical solution is then compared to the numerical simulations showing a remarkable agreement between the two procedures.
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Notes
Similar in the sense of mixed with coexistence of chaos and regularity including periodic islands and invariant spanning curves.
We assume them as reflecting boundaries, however a short discussion must be made here. When a particle passes near enough of a regular region it might suffers a dynamical trapping called stickiness. The particle stays confined in such a region for a while, that may be eventually very long, until escape such region and visit other regions of the phase space. During a temporary trapping the diffusion is no longer normal but rather anomalous.
We have also compared the dynamics with \(k=10\), \(k=10^3\) and \(k=10^4\) and no difference was noticed.
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Acknowledgements
EDL acknowledges support from CNPq (303707/2015-1), FAPESP (2012/23688-5), (2017/14414-2) and FUNDUNESP. CMK thanks to CAPES for support.
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Leonel, E.D., Kuwana, C.M. An Investigation of Chaotic Diffusion in a Family of Hamiltonian Mappings Whose Angles Diverge in the Limit of Vanishingly Action. J Stat Phys 170, 69–78 (2018). https://doi.org/10.1007/s10955-017-1920-x
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DOI: https://doi.org/10.1007/s10955-017-1920-x