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- M. W. Hirsch, Hal Smith
- 2004

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher's permissions policies and our… (More)

The activation dynamics of nets are considered from a rigorous mathematical point of view. A net is identified with the dynamical system defined by a continuously differentiable vector field on the space of activation vectors, with fixed weights, biases, and inputs. Chaotic and oscillatory nets are briefly discussed, but the main goal is to find conditions… (More)

- M. W. Hirsch, Hal Smith
- 2004

The aim of this paper is to provide a brief review of the main results in the theory of discrete-time monotone dynamics.

- Michel Benäım, Olivier Raimond, Gerard Ben Arous, Thierry Coulhon, Morris W Hirsch, Josef Hofbauer +1 other
- 2005

Let M be a compact Riemannian manifold. A self-interacting diffusion on M is a stochastic process solution to dX t = dW t (X t) − 1 t t 0 ∇V Xs (X t)ds dt where {W t } is a Brownian vector field on M and V x (y) = V (x, y) a smooth function. Let µ t = 1 t t 0 δ Xs ds denote the normalized occupation measure of X t. We prove that, when V is symmetric, µ t… (More)

The theory of competitive and cooperative dynamical systems has had some remarkable applications to the biological sciences. The interested reader may consult the monograph [28] and lecture notes [29] of Smith, and to a forthcoming review by the authors [11] for a more in-depth treatment. In this brief review we give some of the main results in the theory… (More)

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a r t i c l e i n f o a b s t r a c t Attractors of cooperative dynamical systems are particularly simple; for example, a nontrivial periodic orbit cannot be an attractor, and orbits are nowhere dense. This paper provides characterizations of attractors for the wider class of coherent systems, defined by the property that all directed feedback loops are… (More)

In this paper we report on some new architectures for neural computation, motivated in part by biological considerations. One of our goals is to demonstrate that it is just as easy for a neural net to compute with arbitrary attractors--oscillatory or chaotic--as with the more usual asymptotically stable fixed points. The advantages (if any) of such… (More)