Neutral networks: a combinatorial perspective

1999 ◽  
Vol 02 (03) ◽  
pp. 283-301 ◽  
Author(s):  
Stephan Kopp ◽  
Christian M. Reidys
Keyword(s):  
Combinatorial Optimization ◽  
Dynamical Systems ◽  
Computer Simulations ◽  
Evolutionary Change ◽  
Rna Molecules ◽  
New Class ◽  
Neutral Networks ◽  
Threshold Phenomenon ◽  
Theoretical Investigations ◽  
Sequential Dynamical Systems

The existence of neutral networks in genotype-phenotype maps has provided significant insight in theoretical investigations of evolutionary change and combinatorial optimization. In this paper we will consider neutral networks of two particular genotype-phenotype maps from a combinatorial perspective. The first map occurs in the context of folding RNA molecules into their secondary structures and the second map occurs in the study of sequential dynamical systems, a new class of dynamical systems designed for the analysis of computer simulations. We will prove basic properties of neutral nets and present an error threshold phenomenon for evolving populations of simulation schedules.

Chaotic Dynamics of Partial Difference Equations with Polynomial Maps

2021 ◽  
Vol 31 (09) ◽  
pp. 2150133
Author(s):  
Haihong Guo ◽  
Wei Liang
Keyword(s):  
Dynamical Systems ◽  
Computer Simulations ◽  
Difference Equations ◽  
Chaotic Dynamics ◽  
Discrete Dynamical Systems ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
Polynomial Maps ◽  
Expansion Theory ◽  
Theoretical Results

In this paper, chaotic dynamics of a class of partial difference equations are investigated. With the help of the coupled-expansion theory of general discrete dynamical systems, two chaotification schemes for partial difference equations with polynomial maps are established. These controlled equations are proved to be chaotic either in the sense of Li–Yorke or in the sense of both Li–Yorke and Devaney. One example is provided to illustrate the theoretical results with computer simulations for demonstration.

scholarly journals Global Dynamical Systems Involving Generalized -Projection Operators and Set-Valued Perturbation in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Yun-zhi Zou ◽  
Xi Li ◽  
Nan-jing Huang ◽  
Chang-yin Sun
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Banach Spaces ◽  
Equilibrium Points ◽  
Projection Operators ◽  
Initial Value ◽  
New Class ◽  
The Fixed Point Theorem ◽  
Generalized Projection Operators ◽  
Generalized Dynamical Systems

A new class of generalized dynamical systems involving generalizedf-projection operators is introduced and studied in Banach spaces. By using the fixed-point theorem due to Nadler, the equilibrium points set of this class of generalized global dynamical systems is proved to be nonempty and closed under some suitable conditions. Moreover, the solutions set of the systems with set-valued perturbation is showed to be continuous with respect to the initial value.

Elements of a theory of simulation II: sequential dynamical systems

2000 ◽  
Vol 107 (2-3) ◽  
pp. 121-136 ◽  
Author(s):  
C.L. Barrett ◽  
H.S. Mortveit ◽  
C.M. Reidys
Keyword(s):  
Dynamical Systems ◽  
Sequential Dynamical Systems

CHAOS, BIFURCATION AND ROBUSTNESS OF A CLASS OF HOPFIELD NEURAL NETWORKS

2011 ◽  
Vol 21 (03) ◽  
pp. 885-895 ◽  
Author(s):  
WEN-ZHI HUANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Chaotic Behavior ◽  
Robustness Analysis ◽  
Original System ◽  
Hopfield Neural Networks ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Computer Assisted ◽  
New Class

Chaos, bifurcation and robustness of a new class of Hopfield neural networks are investigated. Numerical simulations show that the simple Hopfield neural networks can display chaotic attractors and limit cycles for different parameters. The Lyapunov exponents are calculated, the bifurcation plot and several important phase portraits are presented as well. By virtue of horseshoes theory in dynamical systems, rigorous computer-assisted verifications for chaotic behavior of the system with certain parameters are given, and here also presents a discussion on the robustness of the original system. Besides this, quantitative descriptions of the complexity of these systems are also given, and a robustness analysis of the system is presented too.

NEUTRAL EVOLUTION AND MUTATION RATES OF SEQUENTIAL DYNAMICAL SYSTEMS

2004 ◽  
Vol 07 (03n04) ◽  
pp. 395-418
Author(s):  
H. S. MORTVEIT ◽  
C. M. REIDYS
Keyword(s):  
Dynamical Systems ◽  
Mutation Rate ◽  
Distance Measure ◽  
Neutral Evolution ◽  
Mutation Rates ◽  
Combinatorial Properties ◽  
Labeled Graph ◽  
Sequential Dynamical Systems ◽  
Natural Way ◽  
Independent Probability

In this paper we study the evolution of sequential dynamical systems [Formula: see text] as a result of the erroneous replication of the SDS words. An [Formula: see text] consists of (a) a finite, labeled graph Y in which each vertex has a state, (b) a vertex labeled sequence of functions (Fvi,Y), and (c) a word w, i.e. a sequence (w1,…,wk), where each wi is a Y-vertex. The function Fwi,Y updates the state of vertex wi as a function of the states of wi and its Y-neighbors and leaves the states of all other vertices fixed. The [Formula: see text] over the word w and Y is the composed map: [Formula: see text]. The word w represents the genotype of the [Formula: see text] in a natural way. We will randomly flip consecutive letters of w with independent probability q and study the resulting evolution of the [Formula: see text]. We introduce combinatorial properties of [Formula: see text] which allow us to construct a new distance measure [Formula: see text] for words. We show that [Formula: see text] captures the similarity of corresponding [Formula: see text]. We will use the distance measure [Formula: see text] to study neutrality and mutation rates in the evolution of words. We analyze the structure of neutral networks of words and the transition of word populations between them. Furthermore, we prove the existence of a critical mutation rate beyond which a population of words becomes essentially randomly distributed, and the existence of an optimal mutation rate at which a population maximizes its mutant offspring.

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