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

Lagarias, Jeffrey C. and Rains, Eric (2005) Dynamics of a family of piecewise-linear area-preserving plane maps II. Invariant circles. Journal of Difference Equations and Applications, 11 (13). pp. 1137-1163. ISSN 1023-6198. doi:10.1080/10236190500273127.

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This 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 difference equation x_(n+2) = ½(a-b)|x_(n+1)| + ½(a+b)x_(n+1) – x_n. This family of piecewise-linear maps of the plane has the parameter space (a,b) ϵ ℝ^2. These maps are area-preserving homeomorphisms of ℝ^2 that map rays from the origin into rays from the origin. We show the existence of special parameter values where T_(ab) has every nonzero orbit contained in an invariant circle with an irrational rotation number, with invariant circles that are piecewise unions of arcs of conic sections. Numerical experiments indicate the possible existence of invariant circles for many other parameter values.

Item Type:Article
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URLURL TypeDescription Paper
Additional Information:© 2005 Taylor & Francis. Received 25 May 2005, Accepted 05 Jul 2005, Published online: 21 Aug 2006.
Subject Keywords:Area preserving map, Iterated map, Symbolic dynamics, Plane maps
Issue or Number:13
Classification Code:AMS Subject Classification: Primary: 37E30 Secondary: 52C23, 82D30
Record Number:CaltechAUTHORS:20171002-100946430
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:81941
Deposited By: Tony Diaz
Deposited On:02 Oct 2017 17:20
Last Modified:15 Nov 2021 19:47

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