Periodic Solutions of Hamiltonian Systems

By Yanheng Ding, Cheng Lee. SIAM Journal on Mathematical Analysis, 2000

316 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/29584d7d5813b40fc7597225995bb408500c3374

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In Applied Mathematics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Multiplicity results for periodic solutions to delay differential equations via critical point theory (2005, background): “However, there is another effective approach, calculus of variation, which proved to be very useful in determining the existence and multiplicity of periodic solutions, subharmonic solutions, homoclinic orbits and heteroclinic orbits for ordinary differential equation provided that the equation with”
Maslov index for homoclinic orbits of hamiltonian systems (2007, background): “Started with the pioneering work [33] of P.”
On periodic solutions of superquadratic Hamiltonian systems (2002, methodology): “We shall use the linking theorem [13, Theorem 5.29] to prove our results. The idea comes from [11, 12 , 13].”
Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbb{R}^{2n}$ (2001, unknown): “Rabinowitz in [22] (for star-shaped hypersurfaces) and A.”
Quantitative symplectic geometry (2005, unknown): “Indeed, this inequality and the Hofer-Zehnder Theorem imply existence of periodic orbits on almost every energy surface of any Hamil tonian with support in U provided only that U is displaceable in M .”
A FLOER HOMOLOGY FOR EXACT CONTACT EMBEDDINGS (2007, methodology): “09 72 In this paper we construct the Floer homology for an action functional which was introduced by Rabinowitz and prove a vanishing theorem.”
Bounds for Minkowski Billiard Trajectories in Convex Bodies (2011, background): “nd-Hofer and Hofer-Zehnder capacities for the class of smooth convex bodies, is a combination of results from [13] and [25], and is based on the dual action principle introduced by Clarke in [8] (cf. [35, 44]). Theorem 2.2. Let Σ ⊆ R2n be a convex bounded domain with smooth boundary ∂Σ. Then there e”
LECTURES ON FLOER THEORY AND SPECTRAL INVARIANTS OF HAMILTONIAN FLOWS (2006, methodology): “In the classical mini-max theory for the indefinite functionals as in [ Ra ], [BnR], there was implicitly used the notion of ‘semi-infinite cycles’ to carry out the minimax procedure.”
Closed characteristics on compact convex hypersurfaces in $\R^{2n}$ (2001, unknown): “Rabinowitz in [37] (for star-shaped hypersurfaces) and A.”
Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds (2004, methodology): “spectrum bundle Spec(M,ω) is deﬁned by Spec(M,ω) := [ eφ∈H]am(M,ω) Spec(φe) ⊂ H]am(M,ω) ×R. We will investigate this question elsewhere. In the classical mini-max theory for the indeﬁnite functionals [Ra], [BnR], there was implicitly used the notion of ‘semi-inﬁnite cycles’ to carry out the mini-max”
Periodic solutions of weakly coupled superlinear systems (2016, unknown): “[1, 2, 3, 9, 16, 19, 20].”
Lectures on Holomorphic Curves in Symplectic and Contact Geometry (2010, background): “llowing result (and the related result of A. Weinstein [Wei78] for convex energy surfaces) singles out a special class of hypersurfaces for which the answer is always yes: Theorem 1.29 (P. Rabinowitz [Rab78]). Every star-shaped hypersurface in the standard symplectic R2n admits a closed orbit. Recal”
Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces (2007, methodology): “n that these concepts are independent of the choice of α > 1. For the existence and multiplicity of geometrically distinct closed characteristics on convex compact hypersurfaces in R2n we refer to [Rab1], [Wei1], [EkL1], [EkH1], [Szu1], [HWZ1], [LoZ1], 2 [LLZ1], and references therein. Note that ”
Variational methods in relativistic quantum mechanics (2007, background): “Rabinowitz concerning the study of nonlinear waves [134] and also the study of periodic solutions of Hamiltonian systems [135].”
Dissipation-induced instabilities in finite dimensions (2007, methodology): “The variational methods in proving this result were introduced later by Rabinowitz in his seminal paper (Rabinowitz, 1978), but this field still has many open problems (Ekeland, 1990).”
The Restricted Three-Body Problem and Holomorphic Curves (2018, background): “Although Moser explicitly wrote in ([87]) that A is useless in finding periodic orbits, it was used shortly later by Rabinowitz in his celebrated paper [93] to prove existence of periodic orbits on star-shaped hypersurfaces in R and so opened the way for”
Spectral Flow and Bifurcation of Critical Points of Strongly Indefinite Functionals (2000, background): “Complete calculation, based on the normal form of a Hamiltonian matrix, can be obtained assuming semisimplicity of the eigenvalues (see [1]).”