Research Problems

J. C. Sprott
Department of Physics, University of Wisconsin, Madison, WI 53706, USA
September 27, 2000
(Revised September 30, 2003)

Here's a list of research problems that I'm interested in, mostly involving nonlinear dynamics. If anyone would like to collaborate on any of them, please contact me.

Probability of Chaos

Some earlier work indicates that the probability that a polynomial map with arbitrarily chosen coefficients is chaotic decreases with the dimension of the map. This result is counterintuitive and contradicts results for polynomial flows and for discrete-time neural networks. I'd like to understand the reason for the different behavior.

Scaling of Attractor Dimension

There is evidence at low dimensions (D < 10) in chaotic polynomial maps and flows that the average dimension of the resulting strange attractor is the order of the square root of the dimension of the system of equations that produced it.  I would like to extend this test to much higher dimensions (D ~100) using artificial neural networks and a Kaplan-Yorke dimension derived from the spectrum of Lyapunov exponents to remove bias resulting from errors in calculating large fractal dimensions.

Coupled Flow Lattices

As a crude model of a turbulent flow, one could construct large lattices whose elements are low-dimensional flows such as Lorenz attractors, coupled together.  One simple model would be a 1-dimensional ring with the attractors coupled to their nearest neighbors.  Other architectures in 2 and 3 dimensions with more global couplings are possible.  There would be one or more control parameters that correspond to the Reynolds number in fluid mechanics and that could be used to simulate the onset of turbulence.

Attracting 2-torus in 3-space

I would like to find the algebraically simplest example of an attracting 2-torus in a 3-dimensional autonomous system of ordinary differential equations and look at its bifurcation properties, especially the route to chaos.

Structural Stability of Attractors

It is claimed (Peixoto's theorem) that toruses with dimensions higher than 2 are structurally unstable and hence are not likely to occur in nature. I'd like to examine numerically the probability of n-toruses in systems of equations with randomly chosen coefficients along the lines of my studies of the probability of chaos. I'd also like to quantify the extent to which strange attractors are structurally stable as a function of their dimension.

Improved Kaplan-Yorke Dimension

I've had an idea of how to improve the Kaplan and Yorke estimate of the dimension of a strange attractor. It involves a higher order interpolation of the sum of the Lyapunov exponents. I'd like to compare this estimate with theirs for a number of standard attractors including ones where the dimension can be changed continuously from 2 to 3. If the method is not an improvement, I'd like to understand why.

Maximally Chaotic 3-D Jerk System

I would like to identify a simple autonomous jerk system (a third-order ODE) with parameters that can be varied to produce strange attractors with a Kaplan-Yorke dimension anywhere in the range of 2 to 3 and study its bifurcation properties, especially the route to chaos.  It would be an analog of a nearly Hamiltonian system with a very weak damping such as the Solar System or a particle accelerator.

Energetics of Walking and Running

I have a simple energetic model for the energy expended by a human who is walking and running and another dynamical model. The energy model predicts, among other things, that there is a transition around 2 m/s above which running is more efficient than walking. The dynamical model predicts a transition around 3 m/s. I'd like to compare these models with others and with experimental data, find their weaknesses and limitations, and publish the results.

Fitting the Topology of a Strange Attractor

A common problem is to find a mathematical model that mimics the apparently chaotic dynamics of an experimental system. Models that give good short-term predictability tend to give very inaccurate long-term behavior, even to the point of having unbounded or nonchaotic solutions. Is it possible to find models of data that give the right topology of their strange attractor at the expense of short-term predictability? My first attempts at this have not been very successful.

Basin of Attractions

Under some conditions (such as for the Hénon map) the boundary of the basin of attraction is smooth, and under other conditions (such as for the Mandelbrot set) it is fractal. What conditions determine the shape and size of the basin of attraction? Is there a correlation of its fractal dimension with the dimension of the attractor or other quantity? What role do the Cauchy-Reimann equations play, if any? Can two-dimensional maps that satisfy the Cauchy-Reimann equations have chaos on a set of nonzero measure in their parameter space?

Distinguishing Chaos from Colored Noise

Power spectrum analysis is not very useful for distinguishing chaos from noise since it appears possible to construct a chaotic system that produces an arbitrary power spectrum. For that purpose, people rely on the correlation dimension (Grassberger and Pracaccia, Phys. Rev. Lett. 50, 346-349 (1983)). However, Osborne & Provenzale (Physica D 35, 357, 1989) have shown that colored noise can give a spuriously low correlation dimension. Can it be shown analytically or numerically that an appropriately chosen noise spectrum can produce the same correlation integral as an arbitrary chaotic system?
 

Lotka-Volterra Models with Evolution

Lotka-Volterra models are extensions of the logistic differential equation to multi-species.  A simple example is the predator-prety problem.  One difficulty with these models when extended to systems with many species is that most species rapidly die out (the principle of competitive exclusion).  One way around this difficulty is to allow dying species to be replaced with new species as a crude model of biological evolution.  Interesting dynamics result that appear to be self-organized-critical (SOC) and chaotic, and they exhibit punctuated equilibria and leptokurtosis.  Such models might also be applicable to finance, and need to be more fully studied and characterized as described here.


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