Stability of equilibrium solutions of Hamiltonian systems with n-degrees of freedom and single resonance in the critical case

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:264

Article

dos Santos, Fabio¹  Vidal, Claudio² 

[1] Univ Fed Sergipe, Dept Matemat, Av Marechal Rondon S-N, Sao Cristovao, SE, Brazil

[2] Univ Bio Bio, GISDA, Dept Matemat, Fac Ciencias, Casilla 5-C,8 Reg, Concepcion, Chile

关键词: Hamiltonian system;    Equilibrium solution;    Type of stability;    Normal form;    Resonances;    Critical cases;    Lyapunov's Theorem;    Chetaev's Theorem;    Cosmological problems;   

DOI  :  10.1016/j.jde.2017.12.033

来源: Elsevier

【 摘 要 】

In this paper we give new results for the stability of one equilibrium solution of an autonomous analytic Hamiltonian system in a neighborhood of the equilibrium point with n-degrees of freedom. Our Main Theorem generalizes several results existing in the literature and mainly we give information in the critical cases (i. e., the condition of stability and instability is not fulfilled). In particular, our Main Theorem provides necessary and sufficient conditions for stability of the equilibrium solutions under the existence of a single resonance. Using analogous tools used in the Main Theorem for the critical case, we study the stability or instability of degenerate equilibrium points in Hamiltonian systems with one degree of freedom. We apply our results to the stability of Hamiltonians of the type of cosmological models as in planar as in the spatial case. (c) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free