Homoclinic orbits in a first order superquadratic hamiltonian system

Convergence of subharmonic orbits

Research output: Contribution to journalArticle

78 Citations (Scopus)

Abstract

We consider the existence of homoclinic orbits for a first order Hamiltonian system z ̇ = JHz(t, z). We assume H(t, z) is of form H(t, z) = 1 2(Az, z) + W(t, z), where A is a symmetric matrix with δ(JA)∩iR = ∅ and W(t, z) is 2π-periodic in t and has superquadratic growth in z. We prove the existence of a nontrivial homoclinic solution z(t) and subharmonic solutions (zT(t))Tε{lunate}N (i.e., 2πT-periodic solutions) of (HS) such that ZT(t) → Z(t) in Cloc 1(R,R2N) as T → ∞.

Original languageEnglish
Pages (from-to)315-339
Number of pages25
JournalJournal of Differential Equations
Volume94
Issue number2
DOIs
Publication statusPublished - 1991
Externally publishedYes

Fingerprint

Hamiltonians
Subharmonics
First-order System
Homoclinic Orbit
Hamiltonian Systems
Orbits
Orbit
Subharmonic Solutions
Homoclinic Solutions
Symmetric matrix
Periodic Solution
Form

ASJC Scopus subject areas

  • Analysis

Cite this

@article{853a0b3e191a4276a689d1537c0718fb,
title = "Homoclinic orbits in a first order superquadratic hamiltonian system: Convergence of subharmonic orbits",
abstract = "We consider the existence of homoclinic orbits for a first order Hamiltonian system z ̇ = JHz(t, z). We assume H(t, z) is of form H(t, z) = 1 2(Az, z) + W(t, z), where A is a symmetric matrix with δ(JA)∩iR = ∅ and W(t, z) is 2π-periodic in t and has superquadratic growth in z. We prove the existence of a nontrivial homoclinic solution z∞(t) and subharmonic solutions (zT(t))Tε{lunate}N (i.e., 2πT-periodic solutions) of (HS) such that ZT(t) → Z∞(t) in Cloc 1(R,R2N) as T → ∞.",
author = "Kazunaga Tanaka",
year = "1991",
doi = "10.1016/0022-0396(91)90095-Q",
language = "English",
volume = "94",
pages = "315--339",
journal = "Journal of Differential Equations",
issn = "0022-0396",
publisher = "Academic Press Inc.",
number = "2",

}

TY - JOUR

T1 - Homoclinic orbits in a first order superquadratic hamiltonian system

T2 - Convergence of subharmonic orbits

AU - Tanaka, Kazunaga

PY - 1991

Y1 - 1991

N2 - We consider the existence of homoclinic orbits for a first order Hamiltonian system z ̇ = JHz(t, z). We assume H(t, z) is of form H(t, z) = 1 2(Az, z) + W(t, z), where A is a symmetric matrix with δ(JA)∩iR = ∅ and W(t, z) is 2π-periodic in t and has superquadratic growth in z. We prove the existence of a nontrivial homoclinic solution z∞(t) and subharmonic solutions (zT(t))Tε{lunate}N (i.e., 2πT-periodic solutions) of (HS) such that ZT(t) → Z∞(t) in Cloc 1(R,R2N) as T → ∞.

AB - We consider the existence of homoclinic orbits for a first order Hamiltonian system z ̇ = JHz(t, z). We assume H(t, z) is of form H(t, z) = 1 2(Az, z) + W(t, z), where A is a symmetric matrix with δ(JA)∩iR = ∅ and W(t, z) is 2π-periodic in t and has superquadratic growth in z. We prove the existence of a nontrivial homoclinic solution z∞(t) and subharmonic solutions (zT(t))Tε{lunate}N (i.e., 2πT-periodic solutions) of (HS) such that ZT(t) → Z∞(t) in Cloc 1(R,R2N) as T → ∞.

UR - http://www.scopus.com/inward/record.url?scp=38149144312&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38149144312&partnerID=8YFLogxK

U2 - 10.1016/0022-0396(91)90095-Q

DO - 10.1016/0022-0396(91)90095-Q

M3 - Article

VL - 94

SP - 315

EP - 339

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 2

ER -