Schedule of Talks

Wednesday, Jan. 8

Titles and Abstracts of Invited Talks

A.-L. Barabasi, Notre Dame, Architecture of Complexity: From the topology of the WWW to the structure of the cell

Networks with complex topology describe systems as diverse as the cell or the World Wide Web. The emergence of these networks is driven by self-organizing processes that are governed by simple but generic laws. The analysis of the metabolic and protein network of various organisms shows that cells and complex man-made networks, such as the Internet or the world wide web, share the same large-scale topology. I will show that the scale-free topology of these complex webs have important consequences on their robustness against failures and attacks, with implications on drug design, the Internet's ability to survive attacks and failures, and our ability to understand the functional role of genes in model organisms. For further information and papers, see http://www.nd.edu/~networks.

This talk will identify a global mechanism that induces chaos-like behavior by stochastic perturbations, where chaos does not naturally occur. The bifurcation requires co-existing saddle periodic orbits in a multistable system, which we call a bi-instability. The noise facilitates dynamics by emulating a completed heteroclinic connection, therefore inducing a chaotic-like attractor. These claims are supported by an analysis of the stochastic transport between basins, approximated by the stochastic Frobenius-Perron operator. From this information, control strategies can also be readily derived. We illustrate examples of this generic mechanism from models used in optics and math biology. This work is in collaboration with Ira Schwartz, Erik Bollt, and David Morgan.

We discuss systems exhibiting a large number of coexisting attractors for a given set of parameters. A complex interwoven fractal basin boundary separates these attractors, leading to an extremely high sensitivity to the final state. Under the influence of external noise, the system's behavior results in a complex hopping dynamics between the different attractors, which can be interpreted as a chaotic itinerancy between various metastable states. The motion consists of two states, a regular motion in the neighborhood of stable states and a highly chaotic motion on the basin boundaries. These two phases alternate chaotically. We discuss the mechanism of the hopping process in terms of an interaction between the escape from the neighborhood of stable states and the motion on chaotic saddles embedded in the fractal basin boundaries. The concept of symbolic dynamics is used to quantify the complexity of the attractor hopping. In particular, we study the role of the chaotic saddles for the accessibility of different states and show how bifurcations of chaotic saddles lead to a change in the hopping dynamics. Furthermore we demonstrate an enhancement of the noise-induced escape from attractors due to the existence of chaotic saddles embedded in the open neighborhood of an attractor.

The nonlinear interactions of parametrically driven surface waves have been shown to yield a rich family of nonlinear states. When the system is driven by two commensurate frequencies, a number of interesting superlattice type states are generated via a number of different 3-wave resonant interactions. They occur either as symmetry-breaking bifurcations of hexagonal patterns composed of a single unstable mode or via nonlinear interactions between the two primary unstable modes generated by the two forcing frequencies. A particularly interesting state bifurcates from the featureless state in the close vicinity of the system's bi-critical point. This "unlocked" state exhibits highly disordered behavior in both space and time. The unlocked state exists in a wide range of phase space and is bordered on either side by superlattice patterns having different temporal parities. We will first present a characterization of the type of spatial-temporal disorder that is embodied in the unlocked state. We will then demonstrate that this highly disordered state can be rapidly stabilized to either of its neighboring nonlinear states by the application of a small-amplitude excitation at a third frequency. We will demonstrate that the spatial symmetry of the selected pattern is determined by the temporal symmetry of the third frequency used.

Molecular motors are protein molecules responsible for much active biological motion. In striking recent experiments single motor protein molecules, notably kinesin and myosin-V, have been observed in vitro pulling controlled loads of up to 7 pN along linear molecular tracks, specifically microtubules and actin filaments, while taking one step of 8.2 nm or 36 nm, respectively, for each 'fuel molecule' of ATP that is consumed and moving at average speeds as high as 900 nm/s. How should the stochastic motions of such motors be described? Recent work [1-3] invokes N-state periodic sequential models that include: the load-dependence of all the basic forward and reverse rates, indicating substeps and dwell times; death/detachment rates; branches and parallel processes; and waiting-time distributions embodied in 'mechancities' that vary from M = 0 for Poisson processes up to M = 1 for clock-work.

A coupled cell system is a collection of individual, but interacting, dynamical systems. Coupled cell models assume that the output from each cell is important---not just the dynamics considered as a whole. In these systems the signals from two or more cells can be compared and patterns of activity can emerge. We ask when can the cell dynamics in a subset of cells be identical (synchrony) or differ by a phase shift. In particular: How much of the qualitative dynamics observed in coupled cells is the product of network architecture and how much is related to the specific dynamics of cells and the way they are coupled? We illustrate the ideas through a series of examples and discuss two theorems. The first theorem classifies spatio-temporal symmetries of periodic solutions and the second gives necessary and sufficient conditions for synchrony in terms of network architecture and its symmetry groupoid. This is joint work with Ian Stewart.

Localized waves in one-dimensional structures can carry deformation, information, or energy. These waves are important in many different processes such as the twisting and untwisting of DNA molecules, the problem of prey detection and localization in spider webs, the motility of flagellar microorganisms and the alleged whipping of dinosaur tails. In this talk, I will review and study the propagation of (mostly localized) waves in one-dimensional elastic structures from a theoretical standpoint. I will start with some simple linear models of beams, string and wires and work my way up to nonlinear models of elastic rods. I will show exact and approximate traveling waves solutions and study their domain of existence and stability. Finally, I will consider the propagation and acceleration of localized waves in nonhomogeneous media with applications to whip cracking.

Many examples of chemical and biological processes take place in large-scale environmental flows. Such flows generate filamental patterns which are often fractal due to the presence of chaos in the underlying advection dynamics. In such processes, hydrodynamical stirring strongly couples into the reactivity of the advected species and might thus make the traditional treament of the problem through partial differential equations impossible. I will present a simple and universally valid approach for the activity in inhomogeneous stirred flows. I will derive a rate equation which is universal in the sense that is independent of the flow, of the particle properties, and of the details of the active process. I will show that the fractal patterns serve as skeletons and catalysts, and as a result, the production is singularly enhanced. One aspect of the universality of this approach is that it also applies to active particles of finite size (so called inertial particles). Moreover, I will argue that this approach holds even when the filamental patterns do not form a fractal in the traditional sense of fractality.

(*) T. Tel, T. Nishikawa, A. E. Motter, C. Grebogi, and Z. Toroczkai, PNAS, submitted.

Understanding the origin and properties of spatiotemporal chaos (states that are chaotic in time and disordered in space) remains a major frontier of nonlinear dynamics and is of great importance to researchers studying diverse scientific problems such as heterogeneous catalysis, transport of matter by non-laminar flows, and the onset and prevention of fibrillation. My talk will summarize some recent efforts by my group to understand these questions for the idealized case of approximately homogeneous nonequilibrium media. First, I will discuss the significance of extensive chaos and summarize an empirical study of extensive chaos for a two-dimensional model of an excitable medium. Second, I discuss an empirical investigation concerning to what extent high-dimensional spatiotemporal chaos can be characterized by some finite set of unstable periodic orbits. Finally, I discuss recent calculations that demonstrate the role of the so-called "mean flow" on the onset of spiral defect chaos, which is perhaps one of the most intensely studied experimental examples of spatiotemporal chaos. My talk will conclude with a list of my favorite theoretical questions regarding spatiotemporal chaos that I would like to see solved.

The Frenkel-Kontorova (FK) model is an old model first proposed in 1938. It describes a chain of atoms connected by springs in the presence of an external potential. This deceptively simple model however exhibits very rich and complex behaviors. The FK model has found applications in many physical systems such as adsorbed monolayers, Josephson junctions, charge density waves, magnetic spirals, tribilogy and DNA. In this talk we will report our studies of the many facets of the FK model: statics, dynamics, heat conduction, and quantum behavior.

This talk discusses the dynamical systems approach to developing closure models for the computational turbulence problem. The problem statement is: Develop a computable model for the resolvable fluid motions in numerical simulations of turbulent flow. (Hint: model the numerically resolvable effects of the unresolved scales.) For our purposes, the operative words in this problem statement are ``resolvable'' (which introduces a length scale) and ``model'' (not ``solve''). Regarding Dynamics Days, this talk is not specifically about chaos or fractal structure, although the approach does result in nonlinear PDEs that possess global attractors. And the fractal, or Hausdorf, dimension of these global attractors turns out to be finite.
Specifically, I shall speak about a new PDE turbulence closure model that is based on Lagrangian averages (following the fluid parcels). I shall review how well this model describes the mean effects of fluctuations in turbulent flows by comparing its predictions with experimental data and numerical simulations. Mathematically, the turbulence model has a global attractor whose fractal, or Hausdorf, dimension has a finite upper bound, proportional to the 3/2 power of the Reynolds number. So the dynamics on the attractor is computable as a finite degree-of-freedom dynamical system. I shall also discuss how the model affects the Kolmogorov scaling laws for turbulence at small scales and large scales. Finally, I shall discuss the potential advantages of using Lagrangian averaging in modeling GFD turbulence, such as occurs in global ocean circulation as a component of a global ocean-atmosphere climate model.

The failure of the semiclassical eigenfunction hypothesis will be demonstrated for systems where classical regular transport coexists with chaotic dynamics. All eigenstates, instead of being restricted to either a regular island or the chaotic sea, completely ignore these classical phase-space structures. We argue that this is true even in the semiclassical limit for extended systems with transporting regular islands. Examples are Hamiltonian ratchets and the paradigm model of quantum chaos - the standard map.
Hufnagel, Ketzmerick, Otto, Schanz, Phys. Rev. Lett. 89, 154101 (2002)

While it is generally agreed that the earth's atmosphere is chaotic, applications of the theory of chaotic dynamical systems are most successful for low-dimensional systems -- those that can be modeled with only a few state variables. By contrast, current numerical weather models keep track of around 1,000,000 variables. Nonetheless, we have found that in a local region (say the Eastern United States) over a time span of a few days, these models behave like low-dimensional systems. I will describe the current state of weather forecasting, our evidence for local low-dimensionality, and how we plan to use this perspective to improve weather forecasts.

Many spatially extended dynamical systems exhibit patterns that are chaotic in space and time. Various kinds of defects are often striking features of these patterns. A tantalizing question is to what extent the defects can be used to characterize and describe the patterns and their dynamics. I will discuss two systems that display defect-dominated spatio-temporal chaos. In a Swift-Hohenberg model for hexagon patterns in rotating non-Boussinesq convection we find a state of penta-hepta defect chaos that is characterized by the induced nucleation of defects, which leads to defect statistics that differ significantly from that found in other systems. In a Ginzburg-Landau model for parametrically driven waves we find a transition from a disordered to an ordered chaotic state. We characterize it in terms of the statistics of the defects' space-time trajectories. This approach provides a somewhat intuitive picture of the role of the defects in the break-down of the order.
(Work performed with G.D. Granzow, Y.-N. Young, and C. Huepe-Minoletti.)

Epilepsy is a common dynamical disease of the brain which has not received the benefit of a detailed description of the interactions of the neurons that create seizures, nor of methods of interacting with those dynamics. I will describe efforts to experimentally measure the dynamics of the interactions between neurons as neuronal networks transition from nonseizure to seizure activity, and modeling efforts to represent such dynamics in biophysically plausible models. Further, I will describe strategies to develop electrical control parameters and feedback strategies that have been applied both In Vitro and now In Vivo. A framework for the development of rational control strategy development will be proposed.

Filamentary organisms ranging from the prokaryotic actinomycetes to eukaryotic fungi are ubiquitous and important in Nature. During their complex growth cycle some actinomycetes can produce antibiotics, while some fungi are capable of mechanically degrading their local environment. This talk will describe biomechanical models of filamentary growth in which the cell is described, using large-deformation elasticity theory, as a bioelastic membrane incorporating a growth mechanism. The analysis shows that plausible modeling assumptions lead to self-similar filament growth.

This paper considers the stability and dynamics of a very flexible cable loop structure. There is a subtle interplay between gravitational loading (achieved through changing length and hence self-weight) and (linear) free vibrations. The cable stiffness exhibits a softening nonlinearity which leads to a subcritical bifurcation as the cable changes from an upright position and then droops heavily, and suddenly, to one side. The qualitative nature of the behavior is captured by a relatively simple model, and comparisons are made with an analysis of the continuous system. Both pre- and post-buckling dynamics are included in the study together with some preliminary experimental results.

We prove the emergence of chaotic behavior in the form of (a) horseshoes and (b) strange attractors with SRB measures and strong stochastic properties when certain simple dynamical systems are kicked at periodic time intervals. The settings considered include (arbitrary) limit cycles and stationary points undergoing Hopf bifurcations.