Dynamics of a family of piecewise-linear area-preserving plane maps II. Invariant circles

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dc.creatorLagarias, Jeffrey C.
dc.creatorRains, Eric M.
dc.date2003-03-01
dc.date2005-07-10
dc.date.accessioned2026-07-07T06:23:22Z
dc.date.available2026-07-07T06:23:22Z
dc.descriptionThis paper studies the behavior under iteration of the maps T_{ab}(x,y)=(F_{ab}(x)-y,x) of the plane R^2, in which F_{ab}(x)=ax if x>=0 and bx if x<0. The orbits under iteration correspond to solutions of the nonlinear difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| + 1/2(a+b)x_{n+1} - x_n. This family of maps has the parameter space (a,b)\in R^2. These maps are area-preserving homeomorphisms of R^s that map rays from the origin into rays from the origin. This paper shows the existence of special parameter values where T_{ab} has every nonzero orbit an invariant circle with irrational rotation number, and these invariant circles are piecewise unions of arcs of conic sections. Numerical experiments suggest the possible existence of many other parameter values having invariant circles.
dc.descriptionv2 corresponds to second half of old part I; 27 pages latex, 9 ps figure files. Current part I is math.DS/0301294, part III is math.DS/0505103; v3 reflects prior work of Beardon, Bullett and Rippon
dc.identifierhttps://arxiv.org/abs/math/0303007
dc.identifierhttp://arxiv.org/abs/math/0303007
dc.identifierJournal of Difference Equations and Applications 11 (2005), No. 13, 1137-1163.
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/96207
dc.subjectDynamical Systems
dc.subject37E30; 52C23; 82D30
dc.titleDynamics of a family of piecewise-linear area-preserving plane maps II. Invariant circles
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