Results 1  10
of
32
Universal coalgebra: a theory of systems
, 2000
"... In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certa ..."
Abstract

Cited by 408 (42 self)
 Add to MetaCart
In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (NonWellFounded sets, CSLI Leethre Notes, Vol. 14, center for the study of Languages and information, Stanford, 1988) on a theory of nonwellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages. Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras. Thus, the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken
PeriodDoublings to Chaos in A Simple Neural Network: An Analytical Proof
 COMPLEX SYSTEMS
, 1991
"... The dynamics of discretetime neural networks with the sigmoid function as neuron activation function can be extraordinarily complex as some authors have displayed in numerical simulations. Here we consider a simple neural network of only two neurons, one excitatory and the other inhibitory, with no ..."
Abstract

Cited by 32 (3 self)
 Add to MetaCart
(Show Context)
The dynamics of discretetime neural networks with the sigmoid function as neuron activation function can be extraordinarily complex as some authors have displayed in numerical simulations. Here we consider a simple neural network of only two neurons, one excitatory and the other inhibitory, with no external inputs and no time delay as a parameterized family of two dimensional maps, and give an analytical proof for existence of perioddoublings to chaos and strange attractors in the network.
Chaos in highdimensional neural and gene networks
, 1996
"... Abstract Neural and gene networks are often modeled by differential equations. If the continuous threshold functions in the differential equations are replaced by step functions, the equations become piecewise linear (PL equations). The flow through the state space is represented schematically by p ..."
Abstract

Cited by 20 (1 self)
 Add to MetaCart
(Show Context)
Abstract Neural and gene networks are often modeled by differential equations. If the continuous threshold functions in the differential equations are replaced by step functions, the equations become piecewise linear (PL equations). The flow through the state space is represented schematically by paths and directed graphs on an ndimensional hypercube. Closed pathways, called cycles, may/effect periodic orbits with associated fixed points in a chosen Poincar6 section. A return map in the Poincar6 section can be constructed by the composition of fractional linear maps. The stable and unstable manifolds of the fixed points can be determined analytically. These methods allow us to analyze the dynamics in higherdimensional networks as exemplified by a fourdimensional network that displays chaotic behavior. The threedimensional Poincar6 map is projected to a twodimensional plane. This much simpler piecewise linear twodimensional map conserves the important qualitative features of the flow.
Fractal and Chaotic Dynamics in Nervous Systems
 Prog. Neurobiol
, 1991
"... : This paper presents a review of fractal and chaotic dynamics in nervous systems and the brain, exploring mathematical chaos and its relation to processes, from the neurosystems level down to the molecular level of the ion channel. It includes a discussion of parallel distributed processing models ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
: This paper presents a review of fractal and chaotic dynamics in nervous systems and the brain, exploring mathematical chaos and its relation to processes, from the neurosystems level down to the molecular level of the ion channel. It includes a discussion of parallel distributed processing models and their relation to chaos and overviews reasons why chaotic and fractal dynamics may be of functional utility in central nervous cognitive processes. Recent models of chaotic pattern discrimination and the chaotic electroencephalogram are considered. A novel hypothesis is proposed concerning chaotic dynamics and the interface with the quantum domain. Contents : 0 : Introduction 2 1 : Concepts and Techniques in Chaos 2 (a) Chaotic Systems 2 (b) Indicators of Chaos 4 (i) Liapunov Exponent and Entropy 4 (ii) Power Spectrum 6 (iii) Hausdorff dimension and Fractals 7 (iv) Correlation Integral 8 (c) Iterations as Examples of Chaos 10 (i) The Logistic Map 10 (ii) The Transition from Q...
Group classification of systems of nonlinear reaction– diffusion equations
 Ukrainian Mathematical Bulletin
, 2005
"... Group classification of systems of two coupled nonlinear reactiondiffusion equation with a diagonal diffusion matrix is carried out. Symmetries of diffusion systems with singular diffusion matrix and additional first order derivative terms are described. 1 ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
(Show Context)
Group classification of systems of two coupled nonlinear reactiondiffusion equation with a diagonal diffusion matrix is carried out. Symmetries of diffusion systems with singular diffusion matrix and additional first order derivative terms are described. 1
Dynamical Analysis Of DensityDependent Selection In A Discrete OneIsland Migration Model
, 2000
"... . A system of nonlinear difference equations is used to model the effects of densitydependent selection and migration in a population characterized by two alleles at a single gene locus. Results for the existence and stability of polymorphic equilibria are established. Properties for a genetically ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
. A system of nonlinear difference equations is used to model the effects of densitydependent selection and migration in a population characterized by two alleles at a single gene locus. Results for the existence and stability of polymorphic equilibria are established. Properties for a genetically important class of equilibria associated with complete dominance in fitness are described. The birth of an unusual chaotic attractor is also illustrated. This attractor is produced when migration causes chaotic dynamics on a boundary of phase space to bifurcate into the interior of phase space, resulting in bistable genetic polymorphic behavior. 3 1. INTRODUCTION The combined effects of natural selection and migration can significantly influence the genetic composition and demographic properties of populations. Of particular interest in evolutionary theory are ways that these processes concurrently act to maintain genetic polymorphisms and regulate population growth. The simplest migration...
Semigroups of chaotic operators
, 2009
"... We prove the existence of chaotic semigroups of operators that do not contain any chaotic operator. In particular, we obtain a chaotic operator T such that λT is not chaotic for some unimodular complex number λ. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We prove the existence of chaotic semigroups of operators that do not contain any chaotic operator. In particular, we obtain a chaotic operator T such that λT is not chaotic for some unimodular complex number λ.
Županović, Box dimension of trajectories of some discrete dynamical systems
 Chaos Solitons Fractals
"... ar ..."
(Show Context)
Polynomiography: A New Intersection between Mathematics and Art
, 2000
"... Polynomiography is defined to be “the art and science of visualization in approximation of the zeros of complex polynomials, via fractal and nonfractal images created using the mathematical convergence properties of iteration functions.” An individual image is called a “polynomiograph.” The word po ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
Polynomiography is defined to be “the art and science of visualization in approximation of the zeros of complex polynomials, via fractal and nonfractal images created using the mathematical convergence properties of iteration functions.” An individual image is called a “polynomiograph.” The word polynomiography is a combination of the word “polynomial” and the suffix “graphy.” It is meant to convey the idea that it represents a certain graph of polynomials, but not in the usual sense of graphing, say a parabola for a quadratic polynomial. Polynomiographs are obtained using algorithms requiring the manipulation of thousands of pixels on a computer monitor. Depending upon the degree of the underlying polynomial, it is possible to obtain beautiful images on a laptop computer in less time than a TV commercial. Polynomials form a fundamental class of mathematical objects with diverse applications; they arise in devising algorithms for such mundane task as multiplying two numbers, much faster than the ordinary way we have all learned to do this task (FFT). According to the Fundamental Theorem of Algebra, a polynomial of degree n, with real or complex coefficients, has n zeros (roots) which may or may not be distinct. The task of approximation of the zeros of polynomials is a problem that was known to Sumerians (third millennium B.C.). This problem
DiscreteTime Dynamics of Coupled QuasiPeriodic and Chaotic
 Neural Network Oscillators, International Joint Conference on Neural Networks
, 1992
"... ..."
(Show Context)