### Abstract

We consider the existence of homoclinic orbits for a first order Hamiltonian system z ̇ = JH_{z}(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 (z_{T}(t))_{Tε{lunate}N} (i.e., 2πT-periodic solutions) of (HS) such that Z_{T}(t) → Z_{∞}(t) in C_{loc}
^{1}(R,R^{2N}) as T → ∞.

Original language | English |
---|---|

Pages (from-to) | 315-339 |

Number of pages | 25 |

Journal | Journal of Differential Equations |

Volume | 94 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1991 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Analysis

### Cite this

**Homoclinic orbits in a first order superquadratic hamiltonian system : Convergence of subharmonic orbits.** / Tanaka, Kazunaga.

Research output: Contribution to journal › Article

}

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 -