Adiabatic invariance in the standard map

I. Dana, W. P. Reinhardt

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18 Scopus citations


Adiabatic invariance of the action is investigated in the standard map, under slow changes of the stochasticity parameter K. A fixed action representation of the rotational tori is developed perturbatively in K, and is connected with the usual (KAM) representation at fixed winding number. The notion of adiabatic invariance to a given order in K, K≪1, is then introduced. It involves an approximation to exact dynamics by essentially a power-series expansion in both K and a slowness parameter. Adiabatic invariance is explicitly verified to second order, and the dependence of the nonadiabaticity on the form of the switching function and the slowness of the change is investigated. The case of adiabatic switching to larger values of K, K<1, is approached phenomenologically, taking into account the problem of separatrix crossing. It is shown that this crossing leads in general to nonadiabatic effects in the limit of infinitely slow change. These reflect the finite widths, or intrinsic action uncertainties, associated with the main island chains crossed. The latter are determined from the Farey tree in the range of variation of the winding number. An estimate of the critical slowness parameter, corresponding to the onset of the intrinsic nonadiabatic effects, is derived. It involves typical time-scales associated with the main island chains crossed.

Original languageEnglish
Pages (from-to)115-142
Number of pages28
JournalPhysica D: Nonlinear Phenomena
Issue number1-2
StatePublished - Sep 1987
Externally publishedYes

Bibliographical note

Funding Information:
The authors have benefited from useful discussions with Drs. I. Benjamin, J.R. Cary, E. Chandler, G. Ezra, Bruce R. Johnson, P. Pechukas, and I.C. Percival and with Mr. R. Gillilan and Mr. C. Martens. In particular, we wish to thank Dr. Bruce R. Johnson and Mr. C. Martens for bringing ref. 31 and ref. 11, respectively, to our attention. We thank also Dr. I.C. Percival for sending us a copy of the manuscript in ref. 33, and the referees for several suggestions. The support of the National Science Foundation through grants DMR85-19059 and CHE84-16459 is gratefully acknowledged.


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