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.

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.

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}.

scholarly journals Homoclinics for singular strong force Lagrangian systems in $${\mathbb {R}}^N$$

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Marek Izydorek ◽  
Joanna Janczewska ◽  
Nils Waterstraat
Keyword(s):  
Singular Point ◽  
Homoclinic Solution ◽  
Global Maximum ◽  
Lagrangian Systems ◽  
Type Condition ◽  
Infinite Depth ◽  
Strong Force ◽  
Action Integral ◽  
Smooth Potential ◽  
Single Well

AbstractWe will be concerned with the existence of homoclinics for Lagrangian systems in $${\mathbb {R}}^N$$ R N ($$N\ge 3 $$ N ≥ 3 ) of the form $$\frac{d}{dt}\left( \nabla \Phi (\dot{u}(t))\right) +\nabla _{u}V(t,u(t))=0$$ d dt ∇ Φ ( u ˙ ( t ) ) + ∇ u V ( t , u ( t ) ) = 0 , where $$t\in {\mathbb {R}}$$ t ∈ R , $$\Phi {:}\,{\mathbb {R}}^N\rightarrow [0,\infty )$$ Φ : R N → [ 0 , ∞ ) is a G-function in the sense of Trudinger, $$V{:}\,{\mathbb {R}}\times \left( {\mathbb {R}}^N{\setminus }\{\xi \} \right) \rightarrow {\mathbb {R}}$$ V : R × R N \ { ξ } → R is a $$C^2$$ C 2 -smooth potential with a single well of infinite depth at a point $$\xi \in {\mathbb {R}}^N{\setminus }\{0\}$$ ξ ∈ R N \ { 0 } and a unique strict global maximum 0 at the origin. Under a strong force type condition around the singular point $$\xi $$ ξ , we prove the existence of a homoclinic solution $$u{:}\,{\mathbb {R}}\rightarrow {\mathbb {R}}^N{\setminus }\{\xi \}$$ u : R → R N \ { ξ } via minimization of an action integral.

scholarly journals Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems with Some Twisted Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Qi Wang ◽  
Qingye Zhang
Keyword(s):  
Hamiltonian Systems ◽  
Index Theory ◽  
Maslov Index ◽  
Homoclinic Orbits ◽  
Asymptotically Linear ◽  
Existence And Multiplicity ◽  
Hamiltonian Functions

By the Maslov index theory, we will study the existence and multiplicity of homoclinic orbits for a class of asymptotically linear nonperiodic Hamiltonian systems with some twisted conditions on the Hamiltonian functions.

scholarly journals Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Wenping Qin ◽  
Jian Zhang ◽  
Fukun Zhao
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Point Spectrum ◽  
Homoclinic Orbits ◽  
Hamiltonian Operator ◽  
Quadratic Nonlinearity ◽  
Linking Theorem ◽  
Strongly Indefinite Functionals ◽  
Superquadratic Nonlinearity

We study the following nonperiodic Hamiltonian systemż=JHz(t,z), whereH∈C1(R×R2N,R)is the formH(t,z)=(1/2)B(t)z⋅z+R(t,z). We introduce a new assumption onB(t)and prove that the corresponding Hamiltonian operator has only point spectrum. Moreover, by applying a generalized linking theorem for strongly indefinite functionals, we establish the existence of homoclinic orbits for asymptotically quadratic nonlinearity as well as the existence of infinitely many homoclinic orbits for superquadratic nonlinearity.

MULTIPLE HOMOCLINICS IN A HAMILTONIAN SYSTEM WITH ASYMPTOTICALLY OR SUPER LINEAR TERMS

2006 ◽  
Vol 08 (04) ◽  
pp. 453-480 ◽  
Author(s):  
YANHENG DING
Keyword(s):  
Hamiltonian System ◽  
Critical Point ◽  
Critical Point Theory ◽  
Variational Formulation ◽  
Homoclinic Orbits ◽  
Point Theory ◽  
Asymptotically Linear ◽  
Recent Information ◽  
Existence And Multiplicity ◽  
The Matrix

This paper is concerned with homoclinic orbits in the Hamiltonian system [Formula: see text] where H is periodic in t with Hz(t, z) = L(t)z + Rz(t, z), Rz(t, z) = o(|z|) as z → 0. We find a condition on the matrix valued function L to describe the spectrum of operator [Formula: see text] so that a proper variational formulation is presented. Supposing Rz is asymptotically linear as |z| → ∞ and symmetric in z, we obtain infinitely many homoclinic orbits. We also treat the case where Rz is super linear as |z| → ∞ with assumptions different from those studied previously in relative work, and prove existence and multiplicity of homoclinic orbits. Our arguments are based on some recent information on strongly indefinite functionals in critical point theory.

scholarly journals Existence and Multiplicity of Homoclinic Orbits for Second-Order Hamiltonian Systems with Superquadratic Potential

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Ying Lv ◽  
Chun-Lei Tang
Keyword(s):  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Homoclinic Orbits ◽  
Second Order ◽  
Mountain Pass ◽  
Fountain Theorem ◽  
Existence And Multiplicity ◽  
Second Order Hamiltonian Systems ◽  
Superquadratic Potential

We investigate the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems with local superquadratic potential by using the Mountain Pass Theorem and the Fountain Theorem, respectively.

scholarly journals Nonexistence of Homoclinic Orbits for a Class of Hamiltonian Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Xiaoyan Lin ◽  
Qi-Ming Zhang ◽  
X. H. Tang
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Satisfying Condition ◽  
Sufficient Conditions ◽  
Homoclinic Orbits ◽  
First Order ◽  
Nonlinear Hamiltonian System

We give several sufficient conditions under which the first-order nonlinear Hamiltonian systemx'(t)=α(t)x(t)+f(t,y(t)),  y'(t)=-g(t,x(t))-α(t)y(t)has no solution(x(t),y(t))satisfying condition0<∫-∞+∞[|x(t)|ν+(1+β(t))|y(t)|μ]dt<+∞‍,whereμ,ν>1and(1/μ)+(1/ν)=1,0≤xf(t,x)≤β(t)|x|μ,xg(t,x)≤γ0(t)|x|ν,β(t),γ0(t)≥0, andα(t)are locally Lebesgue integrable real-valued functions defined onℝ.

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