Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits

Abstract

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.

Diffusion in randomly perturbed dissipative dynamics

“…On the other hand, when the non-hyperbolic component of the dynamics is increased, the parameter b gains importance and the diffusion of the random orbit in the support of the conditionally invariant measure [29] experiences a stickiness effect, resulting in a slower distribution of recurrence times to I ∂ with a power-law tail. Such an increase of non-hyperbolic characteristics under parameter change may be the result of homoclinic tangencies with highly non-uniformly hyperbolic properties [16,18]. Since we deal with dynamics under finite resolution, we cannot distinguish them from the other attractors.…”

“…In these cases, contrary to a rigorous mathematical framework and due to physical limitations one cannot ask for time going to infinity or infinitely small length intervals. By making such finite-size assumptions on the dynamics one may include among the detected invariant sets homoclinic tangencies and Newhouse attractors which support some invariant measure at least within finite scales, thus being indistinguishable under finite resolution from more general "real" attractors [2,16].…”

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“…However it is necessary to note that homoclinic tangencies arise inevitably under such perturbations. Consequently systems with periodically forced Lorenz attractors fall into Newhouse regions and demonstrate an extremely rich dynamics [GTS93], [GTS99]. Such attractors were called wild-hyperbolic attractors in [ST98]; 1 in this paper the theory of wild-pseudo-hyperbolic attractor was discussed and an example of a wild-hyperbolic spiral attractor was constructed.…”

“…It is well-known that quadratic maps appear naturally when studying bifurcations of quadratic homoclinic tangencies [GS72], [GS73], [TLY86], [GTS93], [GST03]. They appear as normal forms of rescaled first return maps to the neighborhood of a point on the homoclinic orbit.…”

“…The bifurcations in the general case, where P 0 = 1, were studied in [GST96], [GST03]. We will use their "rescaling method" to study our case as well.…”