Electrical Engineering
      and Computer Sciences

Electrical Engineering and Computer Sciences

COLLEGE OF ENGINEERING

UC Berkeley

Arnold Diffusion in Many Dimensions

B.P. Wood, Allan J. Lichtenberg and Michael A. Lieberman

EECS Department
University of California, Berkeley
Technical Report No. UCB/ERL M93/13
1993

When several standard maps are coupled together, KAM surfaces cannot isolate stochastic regions, and particles diffuse along stochastic layers by the process of Arnold diffusion. For the case of two coupled standard maps the rate of Arnold diffusion has previously been calculated both locally around a particular KAM curve and globally across many cells of the 2 pi periodic mapping. When more than two standard maps are coupled, the Arnold diffusion rate increases, depending on the total number of maps, N, and the number of phases in each coupling term, m, where 2 is less than or equal to m is less than or equal to N. As N is increased, the diffusion rate increases as N to the 1/2, the length of the diagonal in the action space. As m is increased, the diffusion rate increases because the phase of the coupling term for a particular map becomes less correlated with the phase of the map itself. In the limit of large m, the coupling term is randomized with each iteration. When the effect of N is removed by dividing the diffusion distance delta I rms by N to the 1/2, a global diffusion delta I rms versus m is found which is determined by the effect of M on the local rate of Arnold diffusion and thE relative volume occupied by the various stochastically accessible regions in the coupled phase space. For local Arnold diffusion, the increase in delta Irms for a particular m depends strongly on the stocasticity parameter K. An analytic calculation of this K dependence for the cases of two and three coupled maps and an analytic calculation for the m dependence at fixed K are presented, which are in good agreement with numerical results.


BibTeX citation:

@techreport{Wood:M93/13,
    Author = {Wood, B.P. and Lichtenberg, Allan J. and Lieberman, Michael A.},
    Title = {Arnold Diffusion in Many Dimensions},
    Institution = {EECS Department, University of California, Berkeley},
    Year = {1993},
    URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/2284.html},
    Number = {UCB/ERL M93/13},
    Abstract = {When several standard maps are coupled together, KAM surfaces cannot isolate stochastic regions, and particles diffuse along stochastic layers by the process of Arnold diffusion. For the case of two coupled standard maps the rate of Arnold diffusion has previously been calculated both locally around a particular KAM curve and globally across many cells of the 2 pi periodic mapping. When more than two standard maps are coupled, the Arnold diffusion rate increases, depending on the total number of maps, N, and the number of phases in each coupling term, m, where 2 is less than or equal to m is less than or equal to N. As N is increased, the diffusion rate increases as N to the 1/2, the length of the diagonal in the action space. As m is increased, the diffusion rate increases because the phase of the coupling term for a particular map becomes less correlated with the phase of the map itself. In the limit of large m, the coupling term is randomized with each iteration. When the effect of N is removed by dividing the diffusion distance delta I rms by N to the 1/2, a global diffusion  delta I rms versus m is found which is determined by the effect of M on the local rate of Arnold diffusion and thE relative volume occupied by the various stochastically accessible regions in the coupled phase space. For local Arnold diffusion, the increase in delta Irms for a particular m depends strongly on the stocasticity parameter K. An analytic calculation of this K dependence for the cases of two and three coupled maps and an analytic calculation for the m dependence at fixed K are presented, which are in good agreement with numerical results.}
}

EndNote citation:

%0 Report
%A Wood, B.P.
%A Lichtenberg, Allan J.
%A Lieberman, Michael A.
%T Arnold Diffusion in Many Dimensions
%I EECS Department, University of California, Berkeley
%D 1993
%@ UCB/ERL M93/13
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/2284.html
%F Wood:M93/13