Estimating the Dimension of an Inertial Manifold from Unstable Periodic Orbits.

We provide numerical evidence that a finite-dimensional inertial manifold on which the dynamics of a chaotic dissipative dynamical system lives can be constructed solely from the knowledge of a set of unstable periodic orbits. In particular, we determine the dimension of the inertial manifold for the Kuramoto-Sivashinsky system and find it to be equal to the "physical dimension" computed previously via the hyperbolicity properties of covariant Lyapunov vectors.

Newton's descent method for the determination of invariant tori.

We formulate a fictitious-time-flow equation which drives an initial guess torus to a torus invariant under a given dynamics, provided such a torus exists. The method is general and applies in principle to continuous time flows and discrete time maps in arbitrary dimension and to both Hamiltonian and dissipative systems.

Deterministic chaos and noise in three in vitro hippocampal models of epilepsy.

Recent reports have suggested that chaos control techniques may be useful for electrically manipulating epileptiform bursting behavior in neuronal ensembles. Because the dynamics of spontaneous in vitro bursting had not been well determined previously, analysis of this behavior in the rat hippocampus was performed. Epileptiform bursting was induced in transverse rat hippocampal slices using three experimental methods. Slices were bathed in artificial cerebrospinal fluid containing: (1) elevated potassium ([K+]o= 10.5 mM), (2) zero magnesium, or (3) the GABAA-receptor antagonists bicuculline (20 microM) and picrotoxin (250 microM). The existence of chaos and determinism was assessed using two different analytical techniques: unstable periodic orbit (UPO) analysis and a new technique for estimating Lyapunov exponents. Significance of these results was assessed by comparing the calculations for each experiment with corresponding randomized surrogate data. UPOs of multiple periods were highly prevalent in experiments from all three epilepsy models: 73% of all experiments contained at least one statistically significant period-1 or period-2 orbit. However, the expansion rate analysis did not provide any evidence of determinism in the data. This suggests that the system may be globally stochastic but contains local pockets of determinism. Thus, manipulation of bursting behavior using chaos control algorithms may yet hold promise for reverting or preventing epileptic seizures.

Spectrum of stochastic evolution operators: local matrix representation approach.

A matrix representation of the evolution operator associated with a nonlinear stochastic flow with additive noise is used to compute its spectrum. In the weak noise limit a perturbative expansion for the spectrum is formulated in terms of local matrix representations of the evolution operator centered on classical periodic orbits. The evaluation of perturbative corrections is easier to implement in this framework than in the standard Feynman diagram perturbation theory. The results are perturbative corrections to a stochastic analog of the Gutzwiller semiclassical spectral determinant computed to several orders beyond what has so far been attainable in stochastic and quantum-mechanical applications.