On the application of KAM theory to discontinuous dynamical systems
type: Item
instance of: scholarly article
title: On the application of KAM theory to discontinuous dynamical systems
zbMATH Open document ID: 0884.34049
author: Jiangong You
author: Markus Kunze
author: Tassilo Küpper
published in: Journal of Differential Equations
publication date: 1997-11-20T00:00:00Z
review text: This paper deals with the following discontinuous systems \[ x''+ x + a \text{ sgn}(x) =p(t), \] where \(p\in C^6(S^1)\). It is proved via Moser's twist theorem that the above-mentioned equation possesses infinitely many periodic and quasiperiodic solutions and all its solutions are bounded if \(a>0\) is sufficiently large. There have been many papers concerning the application of KAM theory to continuous systems [\textit{M. Levi}, Commun. Math. Phys. 143, No. 1, 43-83 (1991; Zbl 0744.34043); \textit{R. Dieckerhoff} and \textit{E. Zehnder}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14, No. 1, 79-95 (1987; Zbl 0656.34027); \textit{R. Ortega}, J. Lond. Math. Soc., II. Ser. 53, No. 2, 325-342 (1996; Zbl 0860.34017); etc.], but this paper is the first about discontinuous systems.
reviewed by: Rong Yuan
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C27
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 37C55
zbMATH DE Number: 1067761
zbMATH Keywords: KAM theory
zbMATH Keywords: discontinuous dynamical systems
MaRDI profile type: Publication
OpenAlex ID: W1963488314
cites work: Q5558469
cites work: Unbounded solutions of semilinear equations at resonance
cites work: Q5731599
cites work: Q4692311
cites work: Q3803454
cites work: Unbounded Solutions in Non‐Smooth Dynamical Systems at Resonance
cites work: Invariant curves and time-dependent potentials
cites work: Quasiperiodic motions in superquadratic time-periodic potentials
cites work: Q4012891
cites work: Oscillatory escape in a Duffing equation with a polynomial potential
cites work: Unbounded Solutions of an Equation ÿ+g{y) =p(t) , with p(t) Periodic and Bounded, and g(y) /y → ∞ as y →±∞
cites work: Q5577283
cites work: A case of boundedness in Littlewood's problem on oscillatory differential equations
cites work: Q3294369
cites work: Quasi-periodic solutions of nonlinear elliptic partial differential equations
cites work: Boundedness in Periodically Forced Second Order Conservative Systems
cites work: Asymmetric Oscillators and Twist Mappings
DOI: 10.1006/JDEQ.1997.3286
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description: scientific article; zbMATH DE number 1067761