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Stability and Bifurcation Theory for Non-Autonomous Differential Equations pp 265–300Cite as

Twist Mappings with Non-Periodic Angles

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Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2065))

Abstract

Consider an annulus A with coordinates \((\theta,r)\), \(\theta + 2\pi \equiv \theta \), \(r \in [a,b]\). A map \((\theta,r)\mapsto ({\theta }_{1},{r}_{1})\) is twist if it satisfies \(\frac{\partial {\theta }_{1}} {\partial r} > 0\). Twist maps have been extensively studied and they are useful to understand the dynamics of autonomous or periodic Hamiltonian systems. In this course we study twist maps without assuming periodicity in θ. In other words, the annulus A is replaced by a strip \(S = \mathbb{R} \times [a,b]\). This new class of twist maps can be applied to the study of generalized standard maps or ping-pong models with a general non-autonomous time dependence.

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Notes

  1. 1.

    Notice that no smoothness in t has been assumed.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Duisburg-Essen, 45117, Essen, Germany

    Markus Kunze

  2. Departamento de Matemática Aplicada, Universidad de Granada, 18071, Granada, Spain

    Rafael Ortega

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  1. Markus Kunze
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  2. Rafael Ortega
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Correspondence to Rafael Ortega .

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Kunze, M., Ortega, R. (2013). Twist Mappings with Non-Periodic Angles. In: Stability and Bifurcation Theory for Non-Autonomous Differential Equations. Lecture Notes in Mathematics(), vol 2065. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32906-7_5

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