Nonlinear Map Resonances
Dynamical systems are often studied using maps; functions that determine the next state from the current state:
yn+1 = f(yn)
The map f "maps" the current state vector to the next one. We are most interested in nonlinear maps, specifically maps that result in chaotic dynamics (e.g. logistic map, Bournoulli shift). We study the effects of adding forcing:
xn+1 = f(xn) + Fn
We are interested in what patterns of forcing (subject to various constraints) cause this system to have a dynamical trajectory that is the most different from the original unforced trajectory, i.e. the greatest response. Relevant tools are the calculus of variations, difference equations, symbolic dynamics, and numerical modeling.
Principal investigators: Glenn Foster and Alfred Hübler.
Nonlinear Spectroscopy
We study the diffraction patterns of one-dimensional quasiperiodic scatterers from quasiperiodic pulse trains. We find a single sharp diffraction peak when the dynamics of the incident wave matches the arrangement of the scatterers, that is, when the pulse train and the scatterers are in resonance. The maximum diffraction angle and the resonant pulse train determine the positions of the scatterers. These results may provide a methodology for identifying quasicrystals with a very large signal to noise ratio.
Principal investigators: Jian Xu and Alfred Hübler.
Irregular Dynamics of Open One-Dimensional Mechanical Systems
We study the dynamics of an open one-dimensional discrete flow - a chain of moving point particles connected by ideal springs. These particles flow towards an inlet at constant velocity, pass into a region where they are free to move according to their nearest neighbor interactions, and then pass an outlet where they travel with a sinusoidally varying velocity. As the amplitude of the outlet oscillations is increased, we find that the resident time of particles in the chamber follows a bifurcating (Feigenbaum) route to chaos. This irregular dynamics may be related to the complex behavior of many particle discrete flows or is possibly a low-dimensional analogue of turbulence in continuous systems.
Principal investigators: Austin Gerig and Alfred Hübler.
Chain Formation and Network Self-Organization in an Open Dissipative Electromechanical System
In general, we study dendritic growth and the physical processes and conditions that underlie fractal formation. Specifically, we are performing experiments on an electromechanical open dissipative system, using macroscopic conducting particles, which relaxes into a branched network. We also search for the simplest computer algorithms and theoretical models that produce results consistent with empirical data.
Principal investigators: Joseph Jun and Alfred Hübler.
Faster Than the Wave Speed
We study the acceleration of particles by traveling waves. We find that certain particles can reach a speed that exceeds the wave speed. We discuss possible applications in astro-physics, particle accelerators, explosion physics.
Principal investigators: Joe Brewer and Alfred Hübler.
Relation Between Maximum Divergence and the Largest Liapunov Exponent of a Multidimensional Chaotic Dynamical System
We study the the divergence of neighboring trajectories in multi-dimensional chaotic dynamical systems. We find that in general maximum divergence is not related to the largest Liapunov exponent, except in special cases. We show that common numerical methods for computing the largest Liapunov exponent, determine the maximum divergence instead. We explore the possibility of periodic dynamics with positive Liapunov exponents, and chaotic dynamics where the largest Liapunov exponent is negative.
Principal investigators: Joe Muka and Alfred Hübler.
Conservation of Structure in Spatio-Temporal Chaos
We study the limiting dynamics of nonlinear spatially extended systems of type:
ψtt + ηψt + dV(φ)/dφ + dU(φx)/dφx = F(x,φ,φx,...)
We show that the limiting dynamics has a nontrivial conserved quantity H for F=0. We call this quantity structure since it is a measure for the maximum amplititude of the oscillations. We show that there is a structure work theorem:
H(x2) = H(x1) + Ix1x2F(x)dx
Further we show that structure has an addition property. If a system with structure H1 is coupled to a systems with structure H2 the structure of the combined system is H1+H2 plus a boundary term which is negegible small if the coupling is small. We apply these finding to chemical wave patterns, solitary water waves, and de Broglie waves.
Principal investigators: Davit Sivil and Alfred Hübler.
Quantization of Emergent Structures
We study the dissipative wave particle dynamics driven by band filtered noise. We show that particles have prefered locations if the main wavelength of the noise is comparable with the dimensions of the systems. We study transitions between the attractor locations.
Principal investigators: Davit Sivil and Alfred Hübler.
Control and Prediction of Singular Motion in the Presence of Noise
We study the relation between low dimensional macroscopic models (with constraints and singularities) and the corresponding smooth microscopic models without constraints. We find that the macroscopic models accurately predict the statistical properties, and are well suited for prediction and control.
Principal investigators: Joe Brewer and Alfred Hübler.
Percolation Behavior and Cascades in Networks
We are investigating propagation on networks (small-world graphs, lattices,
trees, random graphs, scale free graphs, socal neworks)
and robustness of networks.
This includes behavior like disease/viral infection
on social and computer networks and cascade failures in electronic and
physical systems. This has been an area of intense research in the last
few years by numerous researchers (notably Barabasi, Watts, Strogatz,
et. al.) with lots of interesting applications of percolation theory,
statistical mechanics, graph theory, and of course, computer simulation.
Principal investigators: Glenn Foster and Jian Xu.
Properties of Fractal Structures (Experimental Work)
We are investigating sponges, both natural (dendritic fractal structures) and man-made (usually a foam structure where voids exist on multiple length scales). In addition to general structural and percolation properties, we are specifically interested in the use of sponges for the storage of hazardous liquids.
Principal investigators: Glenn Foster and Alfred Hübler.