Recent results describing non-trivial dynamical phenomena in systems with homoclinic tangencies are represented. Such systems cover a large variety of dynamical models known from natural applications and it is established that so-called quasiattractors of these systems may exhibit rather non-trivial features which are in a sharp distinction with that one could expect in analogy with hyperbolic or Lorenz-like attractors. For instance, the impossibility of giving a finite-parameter complete description of dynamics and bifurcations of the quasiattractors is shown. Besides, it is shown that the quasiattractors may simultaneously contain saddle periodic orbits with different numbers of positive Lyapunov exponents. If the dimension of a phase space is not too low (greater than four for flows and greater than three for maps), it is shown that such a quasiattractor may contain infinitely many coexisting strange attractors. (c) 1996 American Institute of Physics.

We discuss a rather new phenomenon in chaotic dynamics connected with the fact that some three-dimensional diffeomorphisms can possess wild Lorenz-type strange attractors. These attractors persist for open domains in the parameter space. In particular, we report on the existence of such domains for a three-dimensional Hénon map (a simple quadratic map with a constant Jacobian which occurs in a natural way in unfoldings of several types of homoclinic bifurcations). Among other observations, we have evidence that there are different types of Lorenz-like attractor domains in the parameter space of the 3D Hénon map. In all cases the maximal Lyapunov exponent, Λ 1 , is positive. Concerning the next Lyapunov exponent, Λ 2 , there are open domains where it is definitely positive, others where it is definitely negative and, finally, domains where it cannot be distinguished numerically from zero (i.e. |Λ 2 | < ρ, where ρ is some tolerance ranging between 10 −5 and 10 −6 ). Furthermore, several other types of interesting attractors have been found in this family of 3D Hénon maps.

The phenomenon of the generic coexistence of infinitely many periodic orbits with different numbers of positive Lyapunov exponents is analysed. Bifurcations of periodic orbits near a homoclinic tangency are studied. Criteria for the coexistence of infinitely many stable periodic orbits and for the coexistence of infinitely many stable invariant tori are given. Mathematics Subject Classification: 37G25, 37D45, 37G15, 37C70L is called a saddle in the first case and a saddle-focus in the other cases.Define J = λ n s γ n u , i.e. J is the absolute value of the product of the leading multipliers. A system in the general position satisfies either of the two conditions: B. J < 1, and λγ = 1 in case (2, 1) or λγ 2 = 1 in case (2, 2) or B ′ . J > 1, and λγ = 1 in case (1, 2) or λ 2 γ = 1 in case (2, 2).In fact, by considering diffeomorphism f −1 instead of f , condition B is transformed to B ′ and vice versa. Therefore, it suffices to consider only the case where B holds.The meaning of the quantity J is quite simple: if L has no non-leading multipliers, then J is the Jacobian of the Poincaré map at L, so the volumes are contracted near L if J < 1 and expanded if J > 1.We will need more information about the volume-contraction properties near L. Assume condition B holds and introduce an 'effective dimension' d e [19]: d e = 1-in case (1, 1) and in case (2, 1) at λγ < 1; d e = 2-in case (2, 1) at λγ > 1, in case (1, 2), and in case (2, 2) at λγ 2 < 1; d e = 3-in case (2, 2) at λγ 2 > 1.By construction, since we assume J < 1, it follows that if L has no non-leading multipliers, then (d e + 1)-dimensional volumes are exponentially contracted near L while d e -dimensional volumes may be expanded by the iterations of f .

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.

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