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This report investigates the dynamical stability conjectures of Palis and Smale and Pugh and Shub from the standpoint of numerical observation and lays the foundation for a stability conjecture. As the dimension of a dissipative dynamical system is increased, it is observed that the number of positive Lyapunov exponents increases monotonically, the Lyapunov… (More)
Neural networks are dense in the space of dynamical systems. We present a Monte Carlo study of the dynamic properties along the route to chaos over random dynamical system function space by randomly sampling the neural network function space. Our results show that as the dimension of the system (the number of dynamical variables) is increased, the… (More)
An extensive statistical survey of universal approximators shows that as the dimension of a typical dissipative dynamical system is increased, the number of positive Lyapunov exponents increases monotonically and the number of parameter windows with periodic behavior decreases. A subset of parameter space remains where noncatastrophic topological change… (More)
The MAA Notes and Reports Series, started in 1982, addresses a broad range of topics and themes of interest to all who are involved with undergraduate mathematics. The volumes in this series are readable, informative, and useful, and help the mathematical community keep up with developments of importance to mathematics.
Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties as universal approximators (neural networks). The eigenvalue spectrum is considered both numerically and analytically… (More)
The Annals is reprinting this interview, more a monologue than a conversation, because Donald E. Knuth is one of the giants of computing. We would like to understand how he developed, what he thought, and how he came by this idea or that as he created his many contributions, not the least of which is the brilliant clarity and comprehensiveness with which he… (More)