scholarly journals A nonzero solution for bounded selfadjoint operator equations and homoclinic orbits of Hamiltonian systems

2021 ◽  
pp. 1-13
Author(s):  
Mingliang Song ◽  
Runzhen Li
Keyword(s):  
Existence Theorem ◽  
Hamiltonian Systems ◽  
Homoclinic Orbit ◽  
Selfadjoint Operator ◽  
Existence Result ◽  
Homoclinic Orbits ◽  
Nonzero Solution ◽  
Operator Equations ◽  
Second Order Systems ◽  
Special Case

We obtain an existence theorem of nonzero solution for a class of bounded selfadjoint operator equations. The main result contains as a special case the existence result of a nontrivial homoclinic orbit of a class of Hamiltonian systems by Coti Zelati, Ekeland and Séré. We also investigate the existence of nontrivial homoclinic orbit of indefinite second order systems as another application of the theorem.

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.

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.

On the Maslov index of multi-pulse homoclinic orbits

2009 ◽  
Vol 465 (2109) ◽  
pp. 2897-2910 ◽  
Author(s):  
Frédéric Chardard ◽  
Frédéric Dias ◽  
Thomas J. Bridges
Keyword(s):  
Numerical Method ◽  
Hamiltonian Systems ◽  
Homoclinic Orbit ◽  
Maslov Index ◽  
Homoclinic Orbits ◽  
Surprising Result

Multi-pulse homoclinic orbits of Hamiltonian systems on can be classified by a sequence of integers. In this paper, we find the surprising result that this string of integers encodes the value of the Maslov index of the homoclinic orbit. Our results include a computable formulation of the Maslov index for homoclinic orbits and a robust numerical method for the evaluation of the Maslov index.

On the existence of homoclinic orbits on Riemannian manifolds

1994 ◽  
Vol 14 (1) ◽  
pp. 103-127 ◽  
Author(s):  
Fabio Giannoni
Keyword(s):  
Riemannian Manifold ◽  
Hamiltonian Systems ◽  
Riemannian Manifolds ◽  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Second Order ◽  
Time Variable ◽  
Image Position

AbstractWe prove the existence of a non-trivial homoclinic orbit on a Riemannian manifold (possibly non-compact), for Hamiltonian systems of the second order of the form:where the potential V is T-periodic in the time variable.

FAMILIES OF TRANSVERSE POINCARÉ HOMOCLINIC ORBITS IN 2N-DIMENSIONAL HAMILTONIAN SYSTEMS CLOSE TO THE SYSTEM WITH A LOOP TO A SADDLE-CENTER

1996 ◽  
Vol 06 (06) ◽  
pp. 991-1006 ◽  
Author(s):  
O. YU. KOLTSOVA ◽  
L. M. LERMAN
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Homoclinic Orbit ◽  
Degrees Of Freedom ◽  
Homoclinic Orbits ◽  
New Criterion

We prove the theorem: if an n-degrees-of-freedom Hamiltonian system has an equilibrium of the saddle-center type (there is a pair of simple eigenvalues ±iω; the rest of the spectrum consists of eigenvalues with nonzero real parts) with a homoclinic orbit to it then this system, and all those close to it, have transversal Poincaré homoclinic orbits to Lyapunov periodic orbits if some genericity conditions are satisfied. These conditions are pointed out explicitly. Thus a new criterion of nonintegrability has been obtained.

scholarly journals Homoclinic orbits and Lie rotated vector fields

2000 ◽  
Vol 24 (3) ◽  
pp. 187-192
Author(s):  
Jie Wang ◽  
Chen Chen
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Homoclinic Orbits ◽  
Singular Points ◽  
Definition Of

Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well.

scholarly journals Existence Theorems ofε-Cone Saddle Points for Vector-Valued Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Tao Chen
Keyword(s):  
Saddle Point ◽  
Existence Theorem ◽  
Equilibrium Problem ◽  
Existence Result ◽  
Existence Theorems ◽  
Saddle Points ◽  
Vector Equilibrium Problem ◽  
Saddle Point Theorem ◽  
Vector Equilibrium ◽  
Vector Valued

A new existence result ofε-vector equilibrium problem is first obtained. Then, by using the existence theorem ofε-vector equilibrium problem, a weaklyε-cone saddle point theorem is also obtained for vector-valued mappings.

scholarly journals Erratum to: Homoclinic orbits for an unbounded superquadratic Hamiltonian systems

2010 ◽  
Vol 18 (1) ◽  
pp. 115-115
Author(s):  
Jun Wang ◽  
Junxiang Xu ◽  
Fubao Zhang ◽  
Lei Wang
Keyword(s):  
Hamiltonian Systems ◽  
Homoclinic Orbits
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