# Simple Models of Complex Chaotic Systems

J. C. Sprott

*Department of Physics, University
of Wisconsin, Madison, Wisconsin 53706*

(Received 28 August 2007; accepted 8 December 2007)

### ABSTRACT

Many^{ }phenomena in the real world are inherently complex and
involve^{ }many dynamical variables interacting nonlinearly
through feedback loops and exhibiting^{ }chaos,
self-organization, and pattern formation. It is useful to ask^{ }if
there
are generic features of such systems, and if^{ }so, how
simple can such systems be and still display^{ }these
features. This paper describes several such systems that are^{ }accessible
to
undergraduates and might serve as useful examples of^{ }complexity.
Ref: J. C. Sprott,
Am. J. Phys.
**76**, 474-480 (2008)

The complete paper is available in PDF
format.

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Fig. 1. Largest Lypunov exponent for a collection of 472 fully
connected artificial neural networks in Eq. (5) with N = 101 and random Gaussian weights.

Fig. 2. Largest Lyapunov exponent
for the sparse circulant neural network in Eq. (7) with N = 101 and four-neighbor
interactions.

Fig. 3. Spatiotemporal plot of
chaos in a sparse circulant neural network with N = 101 and b = 0.25.

Fig. 4. Largest Lyapunov exponent
for the hyperlabyrinth system in Eq. (8) with N = 101.

Fig. 5. Spatiotemporal plot of
hyperlabyrinth chaos for N =
101 and b = 0.25.

Fig. 6. Largest Lyapunov exponent
for the Lotka-Volterra model in Eq. (9) with N = 101.

Fig. 7. Spatiotemporal plot of
the Lotka-Volterra model for N
= 101 and b = 0.8.

Fig. 8. Largest Lyapunov exponent
for the delayed differential equation model in Eq. (10) with f(x)
=
sin x and N = 100.

Fig. 9. Strange attractor for the
delayed differ4ential equation model in Eq. (10) with f(x)
=
sin x, tau = 8 and N = 100.

Fig. 10. Largest Lyapunov
exponent for the PDE model in Eq. (13) with N = 101.

Fig. 11. Spatiotemporal plot of
the PDE model in Eq. (12) for N
= 101 and b = 0.