Homoclinic solutions for singular Hamiltonian systems without the strong force condition

2019 ◽  
Vol 472 (1) ◽  
pp. 352-371 ◽  
Author(s):  
Mohamed Antabli ◽  
Morched Boughariou
Keyword(s):  
Hamiltonian Systems ◽  
Homoclinic Solutions ◽  
Force Condition ◽  
Strong Force ◽  
Singular Hamiltonian Systems

scholarly journals Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Ziheng Zhang ◽  
Fang-Fang Liao ◽  
Patricia J. Y. Wong
Keyword(s):  
Hamiltonian Systems ◽  
Bounded Function ◽  
Homoclinic Solution ◽  
Second Order ◽  
Homoclinic Solutions ◽  
Global Maximum ◽  
Continuous Bounded Function ◽  
Almost Periodic ◽  
Strong Force ◽  
Singular Hamiltonian Systems

We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systemsu¨+atWuu=0, (HS) where-∞<t<+∞,u=u1,u2, …,uN∈ℝNN≥3,a:ℝ→ℝis a continuous bounded function, and the potentialW:ℝN∖{ξ}→ℝhas a singularity at0≠ξ∈ℝN, andWuuis the gradient ofWatu. The novelty of this paper is that, for the case thatN≥3and (HS) is nonautonomous (neither periodic nor almost periodic), we show that (HS) possesses at least one nontrivial homoclinic solution. Our main hypotheses are the strong force condition of Gordon and the uniqueness of a global maximum ofW. Different from the cases that (HS) is autonomousat≡1or (HS) is periodic or almost periodic, as far as we know, this is the first result concerning the case that (HS) is nonautonomous andN≥3. Besides the usual conditions onW, we need the assumption thata′t<0for allt∈ℝto guarantee the existence of homoclinic solution. Recent results in the literature are generalized and significantly improved.

scholarly journals Homoclinics for singular strong force Lagrangian systems

2019 ◽  
Vol 9 (1) ◽  
pp. 644-653 ◽  
Author(s):  
Marek Izydorek ◽  
Joanna Janczewska ◽  
Jean Mawhin
Keyword(s):  
Singular Point ◽  
Homoclinic Solutions ◽  
Global Maximum ◽  
Lagrangian Systems ◽  
Infinite Depth ◽  
Force Condition ◽  
Strong Force ◽  
Action Integral ◽  
Smooth Potential ◽  
Single Well

Abstract We study the existence of homoclinic solutions for a class of Lagrangian systems $\begin{array}{} \frac{d}{dt} \end{array} $(∇Φ(u̇(t))) + ∇uV(t, u(t)) = 0, where t ∈ ℝ, Φ : ℝ2 → [0, ∞) is a G-function in the sense of Trudinger, V : ℝ × (ℝ2 ∖ {ξ}) → ℝ is a C1-smooth potential with a single well of infinite depth at a point ξ ∈ ℝ2 ∖ {0} and a unique strict global maximum 0 at the origin. Under a strong force condition around the singular point ξ, via minimization of an action integral, we will prove the existence of at least two geometrically distinct homoclinic solutions u± : ℝ → ℝ2 ∖ {ξ}.

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.

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