Homoclinic orbits for a class of singular second order Hamiltonian systems in ℝ3

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Joanna Janczewska ◽  
Jakub Maksymiuk
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Homoclinic Orbits ◽  
Second Order ◽  
Singular Points ◽  
Global Maximum ◽  
Strong Force ◽  
Second Order Hamiltonian Systems ◽  
Order Hamiltonian System ◽  
Second Order Hamiltonian System

AbstractWe consider a conservative second order Hamiltonian system $$\ddot q + \nabla V(q) = 0$$ in ℝ3 with a potential V having a global maximum at the origin and a line l ∩ {0} = ϑ as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.

Odd Homoclinic Orbits for a Second Order Hamiltonian System

2012 ◽  
Vol 12 (1) ◽  
Author(s):  
Liliane A. Maia ◽  
Olimpio H. Miyagaki ◽  
Sergio H. M. Soares
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Second Order ◽  
Concentration Compactness ◽  
Concentration Compactness Principle ◽  
Order Hamiltonian System ◽  
Compactness Principle ◽  
Second Order Hamiltonian System

AbstractThe aim of this paper is to find an odd homoclinic orbit for a class of reversible Hamiltonian systems. The proof is variational and it employs a version of the concentration compactness principle of P. L. Lions in a lemma due to Struwe.

Subharmonics near an equilibrium for some second-order Hamiltonian systems

1993 ◽  
Vol 123 (5) ◽  
pp. 819-834 ◽  
Author(s):  
Patricio L. Felmer ◽  
Elves A. de B. e Silva
Keyword(s):  
Hamiltonian System ◽  
Periodic Solutions ◽  
Second Order ◽  
Point Of View ◽  
Subharmonic Solutions ◽  
Second Order Hamiltonian Systems ◽  
Order Hamiltonian System ◽  
Existence Of Periodic Solutions ◽  
Image Position ◽  
Second Order Hamiltonian System

SynopsisThis work is devoted to the study of subharmonic solutions of a second-order Hamiltonian systemnear an equilibrium point, say q = 0. The problem of existence of periodic solutions from the global point of view is also considered.This problem has been studied for the case where the potential is positive and superquadratic. In this work a potential V that has change in sign is considered. The potential is decomposed aswhere P is homogeneous, superquadratic and nondegenerate, and is of higher order near 0. In this paper the existence is shown of a sequence of subharmonic solutions of the equation above that converges to the equilibrium, such that their minimal periods converge to infinity.This problem is approached from a variational point of view. Existence of subharmonic and periodic solutions is obtained via minimax techniques.

The shadowing chain lemma for singular Hamiltonian systems involving strong forces

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Marek Izydorek ◽  
Joanna Janczewska
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Homoclinic Orbits ◽  
Global Maximum ◽  
Infinite Depth ◽  
Strong Force ◽  
Autonomous Hamiltonian System ◽  
Existence And Multiplicity ◽  
Single Well ◽  
Singular Hamiltonian Systems

AbstractWe consider a planar autonomous Hamiltonian system :q+∇V(q) = 0, where the potential V: ℝ2 \{ζ}→ ℝ has a single well of infinite depth at some point ζ and a strict global maximum 0at two distinct points a and b. Under a strong force condition around the singularity ζ we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits — the shadowing chain lemma — via minimization of action integrals and using simple geometrical arguments.

Orbits with minimal period for a class of autonomous second order one-dimensional Hamiltonian systems

2018 ◽  
Vol 25 (1) ◽  
pp. 117-122 ◽  
Author(s):  
Chouhaïd Souissi
Keyword(s):  
Periodic Solution ◽  
Hamiltonian System ◽  
Variational Method ◽  
Hamiltonian Systems ◽  
Second Order ◽  
Minimal Period ◽  
One Dimensional ◽  
Order Hamiltonian System ◽  
Second Order Hamiltonian System

AbstractWe show, under an iterative condition which is similar to but stronger than that of Ambrosetti and Rabinowitz and by using a variational method, the existence of aT-periodic solution of the autonomous superquadratic second order Hamiltonian system with even potential\ddot{z}+V^{\prime}(z)=0,\quad z\in\mathbb{R},for any{T>0}. Moreover, such a solution has{T/k}as a minimal period for some integer{1\leq k\leq 3}.

scholarly journals Periodic Solution of Second-Order Hamiltonian Systems with a Change Sign Potential on Time Scales

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
You-Hui Su ◽  
Wan-Tong Li
Keyword(s):  
Periodic Solution ◽  
Hamiltonian System ◽  
Critical Theory ◽  
Time Scales ◽  
Second Order ◽  
Second Order Hamiltonian Systems ◽  
Minimax Methods ◽  
Order Hamiltonian System ◽  
Second Order Hamiltonian System ◽  
First Time

This paper is concerned with the second-order Hamiltonian system on time scales𝕋of the formuΔΔ(ρ(t))+μb(t)|u(t)|μ−2u(t)+∇¯H(t,u(t))=0, Δ-a.e.t∈[0,T]𝕋 ,u(0)−u(T)=uΔ(ρ(0))−uΔ(ρ(T))=0,where0,T∈𝕋. By using the minimax methods in critical theory, an existence theorem of periodic solution for the above system is established. As an application, an example is given to illustrate the result. This is probably the first time the existence of periodic solutions for second-order Hamiltonian system on time scales has been studied by critical theory.

scholarly journals New Periodic Solutions for the Singular Hamiltonian System

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Yi Liao
Keyword(s):  
Hamiltonian System ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Second Order ◽  
Singular Case ◽  
Strong Force ◽  
Second Order Hamiltonian Systems ◽  
Palais Smale Condition ◽  
Singular Hamiltonian System ◽  
Force Potential

By use of the Cerami-Palais-Smale condition, we generalize the classical Weierstrass minimizing theorem to the singular case by allowing functions which attain infinity at some values. As an application, we study certain singular second-order Hamiltonian systems with strong force potential at the origin and show the existence of new periodic solutions with fixed periods.

Homoclinic orbits for a class of Hamiltonian systems

1990 ◽  
Vol 114 (1-2) ◽  
pp. 33-38 ◽  
Author(s):  
Paul H. Rabinowitz
Keyword(s):  
Hamiltonian System ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Homoclinic Orbit ◽  
Mountain Pass Theorem ◽  
Local Maximum ◽  
Homoclinic Orbits ◽  
Order Hamiltonian System ◽  
Image Position ◽  
Second Order Hamiltonian System

SynopsisConsider the second order Hamiltonian system:where q ∊ ℝn and V ∊ C1 (ℝ ×ℝn ℝ) is T periodic in t. Suppose Vq (t, 0) = 0, 0 is a local maximum for V(t,.) and V(t, x) | x| → ∞ Under these and some additional technical assumptions we prove that (HS) has a homoclinic orbit q emanating from 0. The orbit q is obtained as the limit as k → ∞ of 2kT periodic solutions (i.e. subharmonics) qk of (HS). The subharmonics qk are obtained in turn via the Mountain Pass Theorem.

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