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On the Probability of Chaos in Large Dynamical Systems: A Monte Carlo Study

W. Davis Dechert

*Department of Economics, University of Houston, Houston, USA*
Julien C. Sprott, David J. Albers

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

Received 29 August 1995; accepted 1 July 1997

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ABSTRACT

In this paper we report the result of a Monte Carlo study on the probability
of chaos in large dynamical systems. We use neural networks as the basis
functions for the system dynamics and choose parameter values for the networks
randomly. Our results show that as the dimension of the system and the
complexity of the network increase, the probability of chaotic dynamics
increases to 100%. Since neural networks are dense in the set of dynamical
systems, our conclusion is that most large systems are chaotic.
Ref: W. D. Dechert, J. C. Sprott, and D. J.
Albers, Journal of Economic Dynamics and Control **23**, 1197-1206 (1999)

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Fig. 1. Contour plot for *n* = 8 with increasing *d* and *s*
showing that the probability of chaos approaches unity as *d* becomes
large.

Fig. 2. Contour plot for *s* = 8 with increasing *n* and *d*
showing that the probability of chaos approaches unity as *d* becomes
large.

Fig. 3. Distributions of largest Lyapunov exponents for various *n*
and *d* with *s* = 8.

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