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

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Jeffrey C. Lagarias, Eric M. Rains

Dynamical Systems · Submitted 1 Mar 2003

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv IDmath/0303007 (version 3)
Pith IDYXIE6AVL, Pith's permanent record of this paper and its review
CitationEric M. Rains, Jeffrey C. Lagarias. "Dynamics of a family of piecewise-linear area-preserving plane maps II. Invariant circles." arXiv:math/0303007 (2003). https://pith.science/paper/YXIE6AVL
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,006 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.55 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

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 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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 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. contribution 0.65 0.74 0.03 asserted 0.98
s1 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. background 0.47 0.63 0.04 asserted 1.00
s2 This family of maps has the parameter space (a,b)\in R^2. background 0.97 0.25 0.02 asserted 1.00
s3 These maps are area-preserving homeomorphisms of R^s that map rays from the origin into rays from the origin. background 0.98 0.23 0.03 asserted 1.00
s4 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. result 0.96 0.94 0.88 asserted 1.00
s5 Numerical experiments suggest the possible existence of many other parameter values having invariant circles. future work 0.35 0.82 0.04 hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
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  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

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 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.

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