Olga Gordeeva

We consider a one-parameter family $f_\mu$ of multidimensional diffeomorphisms such that for $\mu=0$ the diffeomorphism $f_0$ has a transversal homoclinic orbit to a nonhyperbolic fixed point of arbitrary finite order $n\geqslant 1$ of degeneracy, and for $\mu>0$ the fixed point becomes a hyperbolic saddle. In the paper, we give a complete description of the structure of the set $N_\mu$ of all orbits entirely lying in a sufficiently small fixed neighborhood of the homoclinic orbit. Moreover, we show that for $\mu\geqslant 0$ the set $N_\mu$ is hyperbolic (for $\mu=0$ it is nonuniformly hyperbolic) and the dynamical system $f_\mu\bigl|_{N_\mu}$ (the restriction of $f_\mu$ to $N_\mu$) is topologically conjugate to a certain nontrivial subsystem of the topological Bernoulli scheme of two symbols.

We study the main bifurcations of multidimensional diffeomorphisms having a nontransversal homoclinic orbit to a saddle-node fixed point. On a parameter plane we build a bifurcation diagram for single-round periodic orbits lying entirely in a small neighborhood of the homoclinic orbit. Also, a relation of our results to the well-known codimension one bifurcations of a saddle fixed point with a quadratic homoclinic tangency and a saddle-node fixed point with a transversal homoclinic orbit is discussed.