On the bifurcation from critical homoclinic orbits in n-dimensional maps

Flaviano Battelli; Claudio Lazzari

doi:10.3934/dcds.1997.3.289

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1997, Volume 3, Issue 2: 289-303. Doi: 10.3934/dcds.1997.3.289

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On the bifurcation from critical homoclinic orbits in n-dimensional maps

Flaviano Battelli^1, and
Claudio Lazzari^2,

1.
Dipartimento di Energetica, Facoltà di Ingegneria-Università, 67040 Monteluco Roio, L'Aquila
2.
Facoltà di Scienze - Università, Via Saffi 2, 61029 Urbino

Received: April 30, 1996

Revised: September 30, 1996

Published: March 1997

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Abstract

We consider the problem of existence of homoclinic orbits in systems like $x_{n+1}=f(x_n)+\mu g(x_n,\mu )$, $x\in \mathbb{R} ^N$, $\mu \in \mathbb{R}$, when the unperturbed system $x_{n+1}=f(x_n)$ has an orbit ${ \gamma _n } _{n\in \mathbb{Z}}$ homoclinic to an expanding fixed point (snap-back repeller) and such that $f'(\gamma _n)$ is invertible for any $n \ne 0$ but $f'(\gamma _0)$ is not. We show that, if a certain analytical condition is satisfied, homoclinic orbits of the perturbed equation occur in pair on one side of $\mu =0$ while are not present on the other side.

Keywords:

Homoclinic orbits,
perturbed equation.

Mathematics Subject Classification: 39A10, 58F14, 58F30.

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