Relative Morse index and multiple homoclinic orbits for a nonperiodic Hamiltonian system

2021 ◽  
pp. 1-18
Author(s):  
Yuan Shan
Keyword(s):  
Hamiltonian System ◽  
Morse Index ◽  
Homoclinic Orbits ◽  
Relative Morse Index

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.

scholarly journals HOMOCLINIC ORBITS OF NONPERIODIC SUPERQUADRATIC HAMILTONIAN SYSTEM

2013 ◽  
Vol 17 (6) ◽  
pp. 1855-1867 ◽  
Author(s):  
Jian Zhang ◽  
Xianhua Tang ◽  
Wen Zhang
Keyword(s):  
Hamiltonian System ◽  
Homoclinic Orbits

scholarly journals Homoclinic Orbits of Nonperiodic Super Quadratic Hamiltonian System

2009 ◽  
Vol 110 (3) ◽  
pp. 1353-1371 ◽  
Author(s):  
Jian Ding ◽  
Junxiang Xu ◽  
Fubao Zhang
Keyword(s):  
Hamiltonian System ◽  
Homoclinic Orbits ◽  
Quadratic Hamiltonian

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.

Infinitely many homoclinic orbits of a Hamiltonian system with symmetry

1999 ◽  
Vol 38 (3) ◽  
pp. 391-415 ◽  
Author(s):  
Yanheng Ding ◽  
Mario Girardi
Keyword(s):  
Hamiltonian System ◽  
Homoclinic Orbits

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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