# 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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