Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps

Abstract

We show that maps with homoclinic tangencies of arbitrarily high orders and, as a consequence, with arbitrarily degenerate periodic orbits are dense in the Newhouse regions in spaces of real-analytic area-preserving two-dimensional maps and general real-analytic two-dimensional maps (the result was earlier known only for the space of smooth non-conservative maps). Based on this, we show that a generic area-preserving map from the Newhouse region is 'universal' in the sense that its iterations approximate the dynamics of any other area-preserving map with arbitrarily good accuracy. In fact, we show that every dynamical phenomenon which occurs generically in any open set of symplectic diffeomorphisms of a two-dimensional disc, or in any open set of finite-parameter families of such diffeomorphisms, can be encountered at a perturbation of any area-preserving two-dimensional map with a homoclinic tangency.

Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps

“…(c) Similar results related to finite-smooth normal forms of saddle maps were established in [8,14,17,20] for general, near-conservative and conservative maps. In this paper we, in fact, modify the corresponding proofs adapting them to the reversible case.…”

“…Precisely, it will be shown that in some regions of the space of parameters (c,M ) the map T km possesses chaotic dynamics and has four saddle fixed points, two of them symmetric conservative and a symmetric couple of fixed points (that is, symmetric one to each other and with Jacobian greater and less than 1, respectively). According to [12,23,20], the following result holds:…”

“…Moreover, if the initial T 0 is C r with respect to coordinates and parameters, then the normal form (3.2) is C r−1 with respect to coordinates and C r−2 with respect to parameters (see [20]). (b) It follows from Lemma 1 that the involution L(x, y) = (y, x) is very convenient for the construction of symmetric saddle maps.…”

“…The situation becomes drastically different in the non-transversal case. One can say even that the corresponding system is infinitely degenerate, since its bifurcations can produce homoclinic tangencies of arbitrary high orders and, as a consequence, arbitrary degenerate periodic orbits [10,13,20].…”

Arnold Diffusion in A Priori Chaotic Symplectic Maps

“…Then for any sufficiently C 3 -close approximation of H (1) by an analytic Hamiltonian [the analiticity of H (1) and H (2) is needed for the family Φ μ to be analytic, i.e. lie in V N ] conditions (75) and (78) will still be satisfied (the idea of constructing analytic perturbations by approximating smooth parametric families of perturbations can be traced back to [16], it was also used in [51]). …”