scholarly journals On the Reducibility for a Class of Quasi-Periodic Hamiltonian Systems with Small Perturbation Parameter

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Jia Li ◽  
Junxiang Xu
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Small Perturbation ◽  
Perturbation Parameter ◽  
Nondegeneracy Condition ◽  
Two Dimensional ◽  
Symplectic Transformation ◽  
Periodic Hamiltonian System

We consider the following real two-dimensional nonlinear analytic quasi-periodic Hamiltonian systemx˙=J∇xH, whereH(x,t,ε)=(1/2)β(x12+x22)+F(x,t,ε)withβ≠0,∂xF(0,t,ε)=O(ε)and∂xxF(0,t,ε)=O(ε)asε→0. Without any nondegeneracy condition with respect to ε, we prove that for most of the sufficiently small ε, by a quasi-periodic symplectic transformation, it can be reduced to a quasi-periodic Hamiltonian system with an equilibrium.

scholarly journals A DIFFERENTIAL INTEGRABILITY CONDITION FOR TWO-DIMENSIONAL HAMILTONIAN SYSTEMS

2014 ◽  
Vol 54 (2) ◽  
pp. 139-141
Author(s):  
Ali Mostafazadeh
Keyword(s):  
Partial Differential Equations ◽  
Hamiltonian System ◽  
Differential Equations ◽  
Hamiltonian Systems ◽  
Harmonic Functions ◽  
Integrability Condition ◽  
Complete Integrability ◽  
Two Dimensional ◽  
Partial Differential ◽  
Generalized Coordinates

We review, restate, and prove a result due to Kaushal and Korsch [Phys. Lett. A 276, 47 (2000)] on the complete integrability of two-dimensional Hamiltonian systems whose Hamiltonian satisfies a set of four linear second order partial differential equations. In particular, we show that a two-dimensional Hamiltonian system is completely integrable, if the Hamiltonian has the form <em>H = T + V</em> where <em>V</em> and <em>T</em> are respectively harmonic functions of the generalized coordinates and the associated momenta.

Coupled systems of two-dimensional turbulence

2013 ◽  
Vol 732 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Hamiltonian System ◽  
Energy Conservation ◽  
Strong Correlation ◽  
Conservation Law ◽  
Hamiltonian Dynamics ◽  
Coupled Systems ◽  
The Other ◽  
Two Dimensional ◽  
Coherent Vortices ◽  
Fluid Particles

AbstractOrdinary two-dimensional turbulence corresponds to a Hamiltonian dynamics that conserves energy and the vorticity on fluid particles. This paper considers coupled systems of two-dimensional turbulence with three distinct governing dynamics. One is a Hamiltonian dynamics that conserves the vorticity on fluid particles and a quantity analogous to the energy that causes the system members to develop a strong correlation in velocity. The other two dynamics considered are non-Hamiltonian. One conserves the vorticity on particles but has no conservation law analogous to energy conservation; the other conserves energy and enstrophy but it does not conserve the vorticity on fluid particles. The coupled Hamiltonian system behaves like two-dimensional turbulence, even to the extent of forming isolated coherent vortices. The other two dynamics behave very differently, but the behaviours of all four dynamics are accurately predicted by the methods of equilibrium statistical mechanics.

scholarly journals Gap opening in two-dimensional periodic systems

2019 ◽  
Vol 23 (01) ◽  
pp. 1950080
Author(s):  
D. I. Borisov ◽  
P. Exner
Keyword(s):  
Differential Operator ◽  
Finite Number ◽  
Distinctive Feature ◽  
Perturbation Parameter ◽  
Periodic Systems ◽  
Order Differential Operator ◽  
Two Dimensional ◽  
Gap Opening ◽  
Gap Control ◽  
Waveguide System

We present a new method of gap control in two-dimensional periodic systems with the perturbation consisting of a second-order differential operator and a family of narrow potential “walls” separating the period cells in one direction. We show that under appropriate assumptions one can open gaps around points determined by dispersion curves of the associated “waveguide” system, in general any finite number of them, and to control their widths in terms of the perturbation parameter. Moreover, a distinctive feature of those gaps is that their edge values are attained by the corresponding band functions at internal points of the Brillouin zone.

On the Dynamics Near a Homoclinic Network to a Bifocus: Switching and Horseshoes

2015 ◽  
Vol 25 (11) ◽  
pp. 1530030 ◽  
Author(s):  
Santiago Ibáñez ◽  
Alexandre Rodrigues
Keyword(s):  
Invariant Manifolds ◽  
Switching Behavior ◽  
Nondegeneracy Condition ◽  
Two Dimensional

We study a homoclinic network associated to a nonresonant hyperbolic bifocus. It is proved that on combining rotation with a nondegeneracy condition concerning the intersection of the two-dimensional invariant manifolds of the equilibrium, switching behavior is created: close to the network, there are trajectories that visit the neighborhood of the bifocus following connections in any prescribed order. We discuss the existence of suspended horseshoes which accumulate on the network and the relation between these horseshoes and the switching behavior.

Hamiltonian systems of Schrödinger equations with vanishing potentials

2020 ◽  
pp. 2050074
Author(s):  
E. Toon ◽  
P. Ubilla
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Positive Solutions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Cerami Sequence ◽  
Cerami Sequences ◽  
Vanishing Potentials

In this paper, by means of minimax techniques involving Cerami sequences, we prove the existence of at least one pair of positive solutions for a Hamiltonian system of Schrödinger equations in [Formula: see text] with potentials vanishing at infinity and subcritical nonlinearities which are superlinear at the origin and at infinity. We establish new estimates to prove the boundedness of a Cerami sequence.

scholarly journals The number of limit cycles for a class of quintic Hamiltonian systems under quintic perturbations

2002 ◽  
Vol 73 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Guowei Chen ◽  
Yongbin Wu ◽  
Xinan Yang
Keyword(s):  
Hopf Bifurcation ◽  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Homoclinic Bifurcation

AbstractThe Hopf bifurcation and homoclinic bifurcation of the quintic Hamiltonian system is analyzed under quintic perturbations by using unfolding theory in this paper. We show that a quintic system can have at least 29 limit cycles.

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