April  1997, 3(2): 289-303. doi: 10.3934/dcds.1997.3.289

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

1. 

Dipartimento di Energetica, Facoltà di Ingegneria-Università, 67040 Monteluco Roio, L'Aquila, Italy

2. 

Facoltà di Scienze - Università, Via Saffi 2, 61029 Urbino, Italy

Received  May 1996 Revised  October 1996 Published  January 1997

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.
Citation: Flaviano Battelli, Claudio Lazzari. On the bifurcation from critical homoclinic orbits in n-dimensional maps. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 289-303. doi: 10.3934/dcds.1997.3.289
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