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Differential Equations, Dynamical Systems, and Linear Algebra
Morris W. Hirsch,Steve Smale +1 more
TLDR
In this article, the structure theory of linear operators on finite-dimensional vector spaces has been studied and a self-contained treatment of that subject is given, along with a discussion of the relations between dynamical systems and certain fields outside pure mathematics.Abstract:
This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by the structure theory of linear operators on finite-dimensional vector spaces; the authors have included a self-contained treatment of that subject.read more
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Ordinary differential equations
TL;DR: The fourth volume in a series of volumes devoted to self-contained and up-to-date surveys in the theory of ODEs was published by as discussed by the authors, with an additional effort to achieve readability for mathematicians and scientists from other related fields so that the chapters have been made accessible to a wider audience.
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Simple mathematical models with very complicated dynamics
Robert M. May,Robert M. May +1 more
TL;DR: This is an interpretive review of first-order difference equations, which can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations.
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Evolutionary stable strategies and game dynamics
Peter D. Taylor,Leo Jonker +1 more
TL;DR: In this article, the authors consider a class of matrix games in which successful strategies are rewarded by high reproductive rates, so become more likely to participate in subsequent playings of the game, thus, over time, the strategy mix should evolve to some type of optimal or stable state.
Book
Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems
TL;DR: Monotone dynamical systems Stability and convergence Competitive and cooperative differential equations Irreducible cooperative systems Cooperative systems of delay differential equations Nonquasimonotone delay differential equation Quasimonoteone systems of parabolic equations A competition model Appendix Bibliography as discussed by the authors
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Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons.
TL;DR: The dynamics of networks of sparsely connected excitatory and inhibitory integrate-and-fire neurons are studied analytically, revealing a rich repertoire of states, including synchronous states in which neurons fire regularly; asynchronous states with stationary global activity and very irregular individual cell activity; andStates in which the global activity oscillates but individual cells fire irregularly.