# Arnold Diffusion in Weakly Coupled Standard Maps

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

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/ERL M90/42

1990

The standard map has a divided phase space in which two dimensional regions of stochasticity are isolated by one dimensional KAM curves which form a barrier to diffusion in action. When two standard maps are coupled together, the two dimensional KAM surfaces no longer divide the four dimensional phase space, and particles diffuse slowly along stochastic layers by the process of Arnold diffusion. We compare an analytic calculation of the rate of localized Arnold diffusion with numerically determined rates in regions having rotational and liberational KAM curves for a single map, in the weak coupling limit for which the three resonance model holds. We then determine the rate of global Arnold diffusion across many cells of the 2pi periodic mapping. The global diffusion rate depends both on the local diffusion rate and on the relative volume occupied by the various stochastically accessible regions in the four dimensional phase space.

BibTeX citation:

@techreport{Wood:M90/42, Author = {Wood, B.P. and Lichtenberg, Allan J. and Lieberman, Michael A.}, Title = {Arnold Diffusion in Weakly Coupled Standard Maps}, Institution = {EECS Department, University of California, Berkeley}, Year = {1990}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1990/1488.html}, Number = {UCB/ERL M90/42}, Abstract = {The standard map has a divided phase space in which two dimensional regions of stochasticity are isolated by one dimensional KAM curves which form a barrier to diffusion in action. When two standard maps are coupled together, the two dimensional KAM surfaces no longer divide the four dimensional phase space, and particles diffuse slowly along stochastic layers by the process of Arnold diffusion. We compare an analytic calculation of the rate of localized Arnold diffusion with numerically determined rates in regions having rotational and liberational KAM curves for a single map, in the weak coupling limit for which the three resonance model holds. We then determine the rate of global Arnold diffusion across many cells of the 2pi periodic mapping. The global diffusion rate depends both on the local diffusion rate and on the relative volume occupied by the various stochastically accessible regions in the four dimensional phase space.} }

EndNote citation:

%0 Report %A Wood, B.P. %A Lichtenberg, Allan J. %A Lieberman, Michael A. %T Arnold Diffusion in Weakly Coupled Standard Maps %I EECS Department, University of California, Berkeley %D 1990 %@ UCB/ERL M90/42 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1990/1488.html %F Wood:M90/42