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- W2061569886 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 1δ = 18.721097200…, 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." @default.
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- W2061569886 date "1981-08-01" @default.
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- W2061569886 title "Universal behaviour in families of area-preserving maps" @default.
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- W2061569886 doi "https://doi.org/10.1016/0167-2789(81)90034-8" @default.
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