Omega-limit sets and bounded solutions

2010/03/16 by Dăng Vũ Giang, Dang Vu Giang, Giang, Dang Vu
Computer Science · Mathematics · #37-XX #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Equations Stability Results #General Mathematics (math.GM) #Optimization and Variational Analysis #math.GM #msc:37-XX

paper · pdf · doi:10.48550/arxiv.1003.3069

openalex publication_date 2010/03/16 · arxiv created 2016/05/30 · arxiv updated 2016/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove among other things that the omega-limit set of a bounded solution of a Hamilton system \\beginaligned \mathbfp=(∂ H)/(∂ q) \mathbfq=-(∂ H)/(∂ p)
\endaligned . is containing a full-time solution so there are the limits of \frac 1t∫₀^t \mathbf p(s)ds and \frac 1t∫₀^t \mathbf q(s)ds as t→∞ for any bounded solution (\mathbf p,q) of the Hamilton system. These limits are stationary points of the Hamilton system so if a Hamilton system has no stationary point then every solution of this system is unbounded.

Related

• Bounded solutions for a class of Hamiltonian systems
• Two Results for the Omega Limit Sets of Dynamical Systems
• A remark on omega limit sets for non-expansive dynamics
• A topological characterization of omega-limit sets on dynamical systems
• The Omega limit set of a family of chords