Universal behaviour in families of area-preserving maps |
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Authors: | J.M. Greene R.S. MacKay F. Vivaldi M.J. Feigenbaum |
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Affiliation: | Plasma Physics Laboratory, P.O. Box 451, Princeton, NJ 08544, USA;School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA;Los Alamos Scientific Laboratory, P.O. Box 1663, Los Alamos, NM 87545, USA |
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Abstract: | We have investigated numerically the behaviour, as a perturbation parameter is varied, of periodic orbits of some reversible area-preserving maps of the plane. Typically, an initially stable periodic orbit loses its stability at some parameter value and gives birth to a stable orbit of twice the period. An infinite sequence of such bifurcations is accomplished in a finite parameter range. This period-doubling sequence has a universal limiting behaviour: the intervals in parameter between successive bifurcations tend to a geometric progression with a ratio of …, and when examined in the proper coordinates, the pattern of periodic points reproduces itself, asymptotically, from one bifurcation to the next when the scale is expanded by α = ?4.018076704… in one direction, and by β = 16.363896879… in another. Indeed, the whole map, including its dependence on the parameter, reproduces itself on squaring and rescaling by the three factors α, β and δ above. In the limit we obtain a universal one-parameter, area-preserving map of the plane. The period-doubling sequence is found to be connected with the destruction of closed invariant curves, leading to irregular motion almost everywhere in a neighbourhood. |
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