A stability theorem for a linear Hamiltonian system with periodic coefficients

1975 ◽  
Vol 3 (4) ◽  
pp. 574-584
Author(s):  
V. A. Yakubovich
Keyword(s):  
Hamiltonian System ◽  
Stability Theorem ◽  
Linear Hamiltonian System ◽  
Periodic Coefficients

scholarly journals Geometry of Hamiltonian Dynamics with Conformal Eisenhart Metric

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Linyu Peng ◽  
Huafei Sun ◽  
Xiao Sun
Keyword(s):  
Hamiltonian System ◽  
Geometric Structure ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Hamiltonian Dynamics ◽  
Jacobi Field ◽  
Linear Hamiltonian System ◽  
Conformal Metric ◽  
Jacobi Equations ◽  
Special Case

We characterize the geometry of the Hamiltonian dynamics with a conformal metric. After investigating the Eisenhart metric, we study the corresponding conformal metric and obtain the geometric structure of the classical Hamiltonian dynamics. Furthermore, the equations for the conformal geodesics, for the Jacobi field along the geodesics, and the equations for a certain flow constrained in a family of conformal equivalent nondegenerate metrics are obtained. At last the conformal curvatures, the geodesic equations, the Jacobi equations, and the equations for the flow of the famous models, anNdegrees of freedom linear Hamiltonian system and the Hénon-Heiles model are given, and in a special case, numerical solutions of the conformal geodesics, the generalized momenta, and the Jacobi field along the geodesics of the Hénon-Heiles model are obtained. And the numerical results for the Hénon-Heiles model show us the instability of the associated geodesic spreads.

scholarly journals Bifurcation of Limit Cycles by Perturbing a Piecewise Linear Hamiltonian System

2013 ◽  
Vol 2013 ◽  
pp. 1-19 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han
Keyword(s):  
Hamiltonian System ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Melnikov Function ◽  
Phase Portraits ◽  
Unperturbed System ◽  
Linear Hamiltonian System ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Global Center

This paper concerns limit cycle bifurcations by perturbing a piecewise linear Hamiltonian system. We first obtain all phase portraits of the unperturbed system having at least one family of periodic orbits. By using the first-order Melnikov function of the piecewise near-Hamiltonian system, we investigate the maximal number of limit cycles that bifurcate from a global center up to first order ofε.

scholarly journals On the Effective Reducibility of a Class of Quasi-Periodic Linear Hamiltonian Systems Close to Constant Coefficients

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Nina Xue ◽  
Wencai Zhao
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Linear Hamiltonian System ◽  
Constant Matrix ◽  
Change Of Variables ◽  
Periodic Linear ◽  
Multiple Eigenvalues ◽  
Linear Hamiltonian Systems ◽  
Constant Coefficients ◽  
Exponentially Small

In this paper, we consider the effective reducibility of the quasi-periodic linear Hamiltonian system x˙=A+εQt,εx, ε∈0,ε0, where A is a constant matrix with possible multiple eigenvalues and Q(t,ε) is analytic quasi-periodic with respect to t. Under nonresonant conditions, it is proved that this system can be reduced to y˙=A⁎ε+εR⁎t,εy, ε∈0,ε⁎, where R⁎ is exponentially small in ε, and the change of variables that perform such a reduction is also quasi-periodic with the same basic frequencies as Q.

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