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- Donald J. Albers
- 1998

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)

- Donald J. Albers, Julien Clinton Sprott, James P. Crutchfield
- Physical review. E, Statistical, nonlinear, and…
- 2006

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)

This paper examines the most probable route to chaos in high-dimensional dynamical systems function space (time-delay neural networks) endowed with a probability measure in a computational setting . The most probable route to chaos (relative to the measure we impose on the function space) as the dimension is increased is observed to be a sequence of… (More)

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… (More)

The dynamical stability conjectures of Palis and Smale, and Pugh and Shub are investigated from the stand-point of numerical observation, and a new stability conjecture is proposed. As the dimension of a dissipative dynamical system is increased, the number of positive Lyapunov exponents increases monotonically, the number of observable periodic windows… (More)

- Donald J. Albers
- 2006

A space of time-delay dynamical systems known to be universal approximators (neural networks) is investigated qualitatively with respect to increasing dimension and number of parameters. The space of mappings is partitioned with a bifurcation parameter according to the qualitative dynamic type (fixed points, chaos, etc). Scaling laws are then investigated… (More)

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)

- Donald J. Albers
- 2004

Every researcher using a computer to model any sort of phenomena is selecting their model equations from some set of abstract mappings. Further, nearly all of these models contain parameters whose variation provides important and interesting insight. In fact, often the very point of setting up the model is to discover what can happen upon parameter… (More)

- Donald J. Albers, Lynn A. Steen
- Annals of the History of Computing
- 1982

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)

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