Discrete dichotomies and bifurcations from critical homoclinic orbits (Q1269657)

The authors consider discrete time dynamical systems of the form \(x_{n+1} = f(x_n) + \mu g(x_n,\mu)\) with \(x \in \mathbb R^n\). The unperturbed map for \(\mu= 0\) has an orbit \(\{ \gamma_n \}\) homoclinic to a hyperbolic fixed point \(p\). Moreover, at one point \(\gamma_0\) in the orbit, \(Df (\gamma_0)\) is not invertible. Bifurcation equations for homoclinic orbits that are close to \(\{ \gamma_n\}\) if \(\mu\) is small are derived.