Families Of Area-preserving Maps Research Articles

Reversing Symmetries in a Two-Parameter Family of Area-Preserving Maps

The integrable twist map is used to derive an invertible two-parameter family of areapreserving maps specified by two arbitrary functions of a real variable. It has been shown that the integrable case has infinitely many reversing symmetries (not necessarily all involutory). Almost all of these symmetries are destroyed under addition of non-integrable terms. However some involutory reversing symmetries survive if certain restrictions on the functions are imposed. An involutory reversing symmetry exists in three different cases. In the first two cases, the symmetry lines are continuous curves leading to infinitely many symmetric periodic orbits. The third case is an interesting example of an involutory reversing symmetry in which the corresponding symmetry lines are isolated points leading to none or few symmetric periodic orbits. The combined restrictions of the first two cases lead to a family of maps with double reversing symmetries. Furthermore there exists a subset of this family (identified by an additional restriction) with quadruple reversing symmetries.

Locally most robust circles and boundary circles for area-preserving maps

What is the rotation number of the last rotational invariant circle to break in a family of area-preserving maps? And what are the rotation numbers of the outermost invariant circles around elliptic islands? Under the hypothesis that the breakup of invariant circles of arbitrary rotation number for a significant class of area-preserving maps is governed by a critical set of a renormalization operator with certain properties, the authors show that for maps in this universality class the rotation number of locally most robust circles is always noble, and that the rotation number of every boundary circle lies on the stable manifold of a certain Cantor set of numbers of constant type under the Gauss map.

Universal behaviour in families of area-preserving maps

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 1 δ = 1 8.721097200 …, 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.