We demonstrate that a fluidized bed of hard spheres during defluidization displays properties associated with formation of a glass. We observe that the volume fraction of the state achieved after defluidization is rate dependent but close to the hard sphere glass transition. As this state is approached, the bed exhibits dynamical heterogeneity, regions of mobile and immobile particles that are spatially and temporally correlated with increasing length and time scales. Such heterogeneities are thought to play a critical role in the formation of glasses. We propose that the loss of fluidization and the arrest of macroscopic particle motion is the result of the formation of a glass. Microscopic motion persists in the final state, but the bed can be jammed by a small increase in flow rate. Thus a fluidized bed can serve as a test system for studies of glass formation and jamming. Our proposal that the hard sphere glass transition controls defluidization could explain the results of Ohja, Menon and Durian [Phys. Rev. E., 2000], who showed that the state of the system after defluidization is independent of particle size and container aspect ratio.

We study the influence of coupling strength and network topology on synchronization behavior in excitatory and inhibitory networks of bursting neurons. First, we prove that the stability of the completely synchronous state in excitatory networks only depends on the number of signals each neuron receives, independent from all other details of the network topology. Through analysis and numerics, we show that the onset of synchrony in a network with any coupling topology admitting complete synchronization is ensured by one single condition (joint work with Martin Hasler and Enno de Lange ). Second, we analyze inhibitory networks of bursting neurons and discuss surprising effects of synaptic inhibition.We give the conditions on the individual neuron's properties under which weak common inhibition induces burst synchronization in strongly desynchronizing networks (joint work with Andrey Shilnikov).

Spatio-temporally chaotic dynamics of a classical field can be described by means of an infinite hierarchy of its unstable spatio-temporally periodic solutions. The periodic orbit theory yields the starting semiclassical approximation to the quantum theory. A new method for computing quantum corrections is proposed: a nonlinear field transformation yields the perturbative corrections in a form more compact than the Feynman diagram expansions.

We propose to use a hierarchy of exact unstable invariant solutions of the Navier-Stokes equations -- corresponding to the recurrent coherent structures observed in experiments -- to construct a description of the spatio-temporally chaotic dynamics of turbulent fluid flows as a walk through the space of such structures. This description should allow us to obtain quantitative predictions of transport properties of fluid flows such as bulk flow rate and mean wall drag. (Joint work with J. F. Gibson, J. Halcrow, and F. Waleffe)

The harmonic oscillator coherent states have several useful properties, including a means to explore and make use of the classical correspondence between quantum eigenstates and (ensembles of) classical trajectories. There has long been an interest in the generalization of these coherent states to other systems, but most such attempts have met with limited success. This talk outlines several difficulties inherent in the construction of generalized coherent states and proposes one solution to this problem, with particular emphasis on the coherent states of the hydrogen atom. A brief review of general classical correspondence, including Husimi-Wigner distributions, will also be provided.

The multiphoton ionization of hydrogen Rydberg atoms in strong microwave fields is a seemingly simple system with complex dynamics. Here we investigate ionization with bichromatic pulses using methodology of chaos.

1.Reducing multiphoton ionization by local control:

We present a control procedure to reduce the stochastic ionization of hydrogen atom in a strong microwave field by adding to the original Hamiltonian a comparatively small control term which might consist of an additional set of microwave fields. This modification restores select invariant tori in the dynamics and prevents ionization. We demonstrate the procedure on the one-dimensional model of microwave ionization.

2.How periodic orbit bifurcations drive multiphoton ionization:

The multiphoton ionization of hydrogen by a strong bichromatic microwave field is a complex process prototypical for atomic control research. Periodic orbit analysis captures this complexity: Through the stability of periodic orbits we can match qualitatively the variation of experimental ionization rates with a control parameter, the relative phase between the two modes of the field. Moreover, an empirical formula reproduces quantum simulations to a high degree of accuracy. This quantitative agreement shows how short periodic orbits organize the dynamics in multiphoton ionization.

In the first part of the talk I consider mixing via chaotic advection in microdroplets suspended at the free surface of a liquid substrate and driven along a straight line using the thermocapillary effect. I show that the mixing properties of the flow inside the droplet can vary dramatically as a function of the physical properties of the fluids and the imposed temperature profile. Proper characterization of the mixing quality requires introduction of two different metrics. The first metric determines the relative volumes of the domain of chaotic streamlines and the domain of regular streamlines. The second metric describes the time for homogenization inside the chaotic domain. In the second part of the talk I discuss a method to use capture into resonance to control the behavior of a certain class of dynamical systems. While resonance interaction can change invariants of the unperturbed system (e.g. energy), it is random in nature, and, consequently, is rather inefficient as a mechanism of regular transport. I discuss a method to structure the resonance interaction with little additional cost. As a model problem I consider dynamics of a charged particle in an electromagnetic field.

Bursting is a manifestation of the complex, multiple time scale dynamics observed in diverse neuronal models. A description list of the nonlocal bifurcations leading to its onset is far from being complete and presents a dare need for cross-disciplinary neuroscience and the dynamical systems theory.There has been a recent breakthrough in this direction that explains a few novel mechanisms of transitions between tonic spiking and bursting activity, as well as their co-existence in models of leech interneurons through homoclinic saddle-node bifurcations of periodic orbits including a blue sky catastrophe. We will discuss the bifurcation theory that underlies theses transitions, as well as one on a spike adding route: as a parameter shifting the membrane potential of half-inactivation slow potassium current is monotonically changed, a sequence of bifurcations occurs causing incremental change of the number of spikes in a burst. Of our special interest is the origin of the sequence, where each transition is accompanied by chaos. To figure out the transition dynamics we construct a one-parameter family of the onto Poincare return mappings on the central manifolds of slow motions. We show that the transitions in question are due to the bifurcations of homoclinics of a repelling point of the map setting a threshold between tonic spiking and hyperpolarized states of the neuron model.

To connect turbulence and dynamical systems theory, we must bridge a gap of dimensionality between the infinite-dimensional Navier-Stokes equations and the finite-dimensional manifolds on which their asymptotic dynamics are conjectured to lie. In this talk, we outline an approach for turbulence in low-Reynolds plane Couette flow. The PDE is transformed to an ODE in $\mathfrak{R}^N$ using an orthonormal basis set that incorporates the boundary and divergence constraints. The ODE fully resolves the flow at $N=10^5$. Fixed points and periodic orbits of the ODE and their linear stability are calculated efficiently with Krylov subspace methods. The few unstable modes provide local low-dimensional coordinate systems in which to view phase portraits, locate new invariant sets, and partition state space.

All physical systems are affected by some level of noise that limits the resolution that can be attained in partitioning their phase space. For chaotic, locally everywhere hyperbolic flows, such resolution depends on the balance between the contraction of regions in the stable directions and the smearing effect of noise. We determine the finest possible partition of the phase space for a given hyperbolic dynamical system affected by a given weak additive white noise. That is achieved by computing the local eigenfunctions of the Fokker-Planck evolution operator in linearized neighborhoods of periodic orbits of the corresponding deterministic system. The method is tested on a Lozi map, but applies in principle to both continuous- and discrete-time dynamical systems in arbitrary dimension.