The topological recurrence algorithm provides a fast and robust method for detecting the presence of unstable periodic orbits (UPO's) in short, noisy experimental data files. We present here a technique for improving this method by using a matrix fitting algorithm to extract dynamical information about the system from these UPO's. This method greatly increases the sensitivity of the algorithm, ...

We consider global attractors of in nite dimensional dynamical systems given by dissipative partial di erential equations u t = u xx + f(x; u; u x ) on the unit interval 0 < x < 1 under separated, linear, dissipative boundary conditions. Global attractors are called orbit equivalent, if there exists a homeomorphism between them which maps orbits to orbits. The global attractor class is the set ...

The trajectory of a billiard ball in motion on a frictionless billiards table is completely determined by its initial position, direction, and speed. When the ball strikes a bumper, we assume that the angle of incidence equals the angle of reflection. Once released, the ball continues indefinitely along its trajectory with constant speed unless it strikes a vertex, at which point it stops. If t...

Dynamical control of excitable biological systems is often complicated by the difficult and unreliable task of precontrol identification of unstable periodic orbits (UPO's). Here we show that, for both chaotic and nonchaotic systems, UPO's can be located, and their dynamics characterized, during control. Tracking of system nonstationarities emerges naturally from this approach. Such a method is...

A new nonlinear dynamical analysis is applied to complex behavior from neuronal systems. The conceptual foundation of this analysis is the abstraction of observed neuronal activities into a dynamical landscape characterized by a hierarchy of "unstable periodic orbits" (UPOs). UPOs are rigorously identified in data sets representative of three different levels of organization in mammalian brain....

We describe and characterize rigorously the homoclinic structure of the perturbed sine{ Gordon equation under periodic boundary conditions. The existence of invariant manifolds for a perturbed sine{Gordon equation is established. Mel'nikov method, together with geometric analysis are used to assess the persistence of the homoclinic orbits under bounded and time-periodic perturbations.

We study the asymptotic behavior of large data solutions in the energy space H := H(R) in very high dimension d ≥ 11 to defocusing Schrödinger equations iut + ∆u = |u| u + V u in R, where V ∈ C∞ 0 (R ) is a real potential (which could contain bound states), and 1 + 4 d < p < 1 + 4 d−2 is an exponent which is energy-subcritical and mass-supercritical. In the spherically symmetric case, we show t...

Chaotic diffusion on periodic orbits (POs) is studied for the perturbed Arnold cat map on a cylinder, in a range of perturbation parameters corresponding to an extended structural-stability regime of the system on the torus. The diffusion coefficient is calculated, using the following PO formulas: (1). the curvature expansion of the Ruelle zeta function; (2). the average of the PO winding-numbe...

We propose two periodic feedback schemes for the stabilization of periodic orbits for one dimensional discrete time chaotic systems. These schemes can be generalized to higher dimensional systems in a straightforward way. We show that the proposed schemes achieve stabilization of a wide range of periodic orbits. The proposed schemes are quite simple and we show that any hyperbolic periodic orbi...

We prove that chaotic flows (i.e. flows that satisfy the shadowing property and have a dense subset of periodic orbits) satisfy a reparametrized gluing orbit property similar to the one introduced in [7]. In particular, these are strongly transitive in balls of uniform radius. We also prove that the shadowing property for a flow and a generic time-t map, and having a dense subset of periodic or...