S. V. Gonchenko

Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits

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.

On models with non-rough Poincare´ homoclinic curves

Abstract The possibility of an a priori complete description of finite-parameter models including systems with structurally unstable Poincare homoclinic curves is studied. The main result reported here is that systems having a countable set of moduli of ω-equivalence and systems having infinitely many degenerate periodic and homoclinic orbits are dense in the Newhouse regions of ω-non-stability. We discuss the question of correctly setting a problem for the analysis of models of such type.

THREE-DIMENSIONAL HÉNON-LIKE MAPS AND WILD LORENZ-LIKE ATTRACTORS

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 Henon 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 Henon map. In all cases the maximal Lyapunov exponent, Λ1, is positive. Concerning the next Lyapunov exponent, Λ2, 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. |Λ2| < 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 Henon maps.

Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone

We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.

On dynamical properties of multidimensional diffeomorphisms from Newhouse regions: I

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.

HOMOCLINIC TANGENCIES OF AN ARBITRARY ORDER IN NEWHOUSE DOMAINS

This paper is devoted to the study of complex and unexpected phenomena that are observed in twodimensional mappings (or in three-dimensional flows) with homoclinic tangencies. In particular, we show that, in the C-topology, for an arbitrary finite r, in any neighborhood of a system with a quadratic homoclinic tangency there are nonrough systems with homoclinic tangencies of arbitrarily high orders, i.e., systems of an arbitrarily high codimension. Such phenomena were not observed in bifurcation theory previously. The study of systems with homoclinic tangencies was initiated in [4]. First and foremost, three classes of such systems were distinguished in that paper. Namely, let L be a saddle periodic motion, and let Γ be a homoclinic trajectory along which the stable and unstable invariant manifolds of L are quadratically tangent to each other (see Fig. 1). Let λ and γ be multipliers of L, |λ| 1. Assume that |λγ| = 1; moreover, without loss of generality, we can assume that |λγ| < 1. Let U be a small neighborhood of the closure Γ ∪ L of the homoclinic trajectory, and let N be the set of all trajectories that lie entirely in U . Depending on the signs of multipliers and on the signs of certain coefficients that characterize the way in which the stable and unstable manifolds adjoin to Γ, the systems with homoclinic tangencies fall into one of the following three classes: (1) for systems of the first class, the set N is trivial: N = {L,Γ}; (2) for systems of the second class, N is a nontrivial, nonuniformly hyperbolic set that admits a complete description in the language of symbolic dynamics (via some quotient system of the topological Bernoulli scheme consisting of three symbols); (3) for systems of the third class, N still contains nontrivial, hyperbolic subsets, but, generally speaking, the set N is not exhausted by them; moreover, the everywhere dense nonroughness takes place on bifurcation films of systems of the third class. (In [10, 11], a similar classification was carried out for the multidimensional case, including the case of systems with homoclinic tangencies of an arbitrary finite order.) To be more specific, according to [4], systems that have nonrough periodic motions are dense in any one-parameter family of systems with homoclinic tangencies of the third class in which the quantity

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

We study the dynamics and bifurcations of two-dimensional reversible maps with non-transversal heteroclinic cycles containing symmetric saddle fixed points. We consider one-parameter families of reversible maps unfolding the initial heteroclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations and the birth of asymptotically stable, unstable and elliptic periodic orbits.

On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors

We study dynamics and bifurcations of three-dimensional diffeomorphisms with nontransverse heteroclinic cycles. We show that bifurcations under consideration lead to the birth of wild-hyperbolic Lorenz attractors. These attractors can be viewed as periodically perturbed classical Lorenz attractors, however, they allow for the existence of homoclinic tangencies and, hence, wild hyperbolic sets.

Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps

We study bifurcations of periodic orbits in two parameter general unfoldings of a certain type homoclinic tangency (called a generalized homoclinic tangency) to a saddle fixed point. We apply the rescaling technique to first return (Poincaré) maps and show that the rescaled maps can be brought to so-called generalized Hénon maps which have non-degenerate bifurcations of fixed points including those with multipliers e±iϕ. On the basis of this, we prove the existence of infinite cascades of periodic sinks and periodic stable invariant circles.

COMPLEXITY OF HOMOCLINIC BIFURCATIONS AND Ω-MODULI

Bifurcations of two-dimensional diffeomorphisms with a homoclinic tangency are studied in one-and two-parameter families. Due to the well-known impossibility of a complete study of such bifurcations, the problem is restricted to the study of the bifurcations of the so-called low-round periodic orbits. In this connection, the idea of taking Ω-moduli (continuous invariants of the topological conjugacy on the nonwandering set) as the main control parameters (together with the standard splitting parameter) is proposed. In this way, new bifurcational effects are found which do not occur at a one-parameter analysis. In particular, the density of cusp-bifurcations is revealed.