DYNAMICAL BEHAVIOR OF KAUFFMAN NETWORKS WITH AND-OR GATES

2000 ◽  
Vol 08 (02) ◽  
pp. 151-175 ◽  
Author(s):  
ERIC GOLES ◽  
GONZALO HERNÁNDEZ
Keyword(s):  
Dynamical Behavior ◽  
Transition Function ◽  
Basins Of Attraction ◽  
Diffusion Problem ◽  
Transient Time ◽  
Binary State ◽  
Strongly Connected ◽  
Parallel Dynamics ◽  
Necessary And Sufficient ◽  
Local Transition

We study the parallel dynamics of a class of Kauffman boolean nets such that each vertex has a binary state machine {AND, OR} as local transition function. We have called this class of nets AON. In a finite, connected and undirected graph, the transient length, attractors and its basins of attraction are completely determined in the case of only OR (AND) functions in the net. For finite, connected and undirected AON, an exact linear bound is given for the transient time using a Lyapunov functional. Also, a necessary and sufficient condition is given for the diffusion problem of spreading a one all over the net, which generalizes the primitivity notion on graphs. This condition also characterizes its architecture. For finite, strongly connected and directed AON a non-polynomial time bound is given for the transient time and for the period on planar graphs, together with an example where this transient time and period are attained. Furthermore, on infinite but finite connected, directed and non planar AON we simulate an universal two-register machine, which allows us to exhibit universal computing capabilities.

scholarly journals A Characterization of the Dynamics of Schröder’s Method for Polynomials with Two Roots

2021 ◽  
Vol 5 (1) ◽  
pp. 25
Author(s):  
Víctor Galilea ◽  
José M. Gutiérrez
Keyword(s):  
Nonlinear Equations ◽  
Complex Plane ◽  
Iterative Process ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Schröder’S Method

The purpose of this work is to give a first approach to the dynamical behavior of Schröder’s method, a well-known iterative process for solving nonlinear equations. In this context, we consider equations defined in the complex plane. By using topological conjugations, we characterize the basins of attraction of Schröder’s method applied to polynomials with two roots and different multiplicities. Actually, we show that these basins are half-planes or circles, depending on the multiplicities of the roots. We conclude our study with a graphical gallery that allow us to compare the basins of attraction of Newton’s and Schröder’s method applied to some given polynomials.

ON TWO-DIMENSIONAL DYNAMICAL SYSTEMS ASSOCIATED WITH MEMBRANE KINETICS UNDERLYING CARDIAC ARRHYTHMIAS

2009 ◽  
Vol 19 (05) ◽  
pp. 1709-1732 ◽  
Author(s):  
B. M. BAKER ◽  
M. E. KIDWELL ◽  
R. P. KLINE ◽  
I. POPOVICI
Keyword(s):  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Two Dimensional ◽  
Stimulus Period ◽  
Smooth Maps ◽  
Periodic Stimulation ◽  
Piecewise Smooth Maps ◽  
Dimensional Version ◽  
Bifurcation Law ◽  
The One

We study the orbits, stability and coexistence of orbits in the two-dimensional dynamical system introduced by Kline and Baker to model cardiac rhythmic response to periodic stimulation — as a function of (a) kinetic parameters (two amplitudes, two rate constants) and (b) stimulus period. The original paper focused mostly on the one-dimensional version of this model (one amplitude, one rate constant), whose orbits, stability properties, and bifurcations were analyzed via the theory of skew-tent (hence unimodal) maps; the principal family of orbits were so-called "n-escalators", with n a positive integer. The two-dimensional analog (motivated by experimental results) has led to the current study of continuous, piecewise smooth maps of a polygonal planar region into itself, whose dynamical behavior includes the coexistence of stable orbits. Our principal results show (1) how the amplitude parameters control which escalators can come into existence, (2) escalator bifurcation behavior as the stimulus period is lowered — leading to a "1/n bifurcation law", and (3) the existence of basins of attraction via the coexistence of three orbits (two of them stable, one unstable) at the first (largest stimulus period) bifurcation. We consider the latter result our most important, as it is conjectured to be connected with arrhythmia.

Allee’s Effect Bifurcation in Generalized Logistic Maps

2019 ◽  
Vol 29 (03) ◽  
pp. 1950039
Author(s):  
J. Leonel Rocha ◽  
Abdel-Kaddous Taha
Keyword(s):  
Allee Effect ◽  
Dynamical Behavior ◽  
Complete Classification ◽  
Unimodal Maps ◽  
New Class ◽  
New Type ◽  
Local And Global Bifurcations ◽  
One Dimensional Maps ◽  
Necessary And Sufficient

This paper concerns the study of the Allee effect on the dynamical behavior of a new class of generalized logistic maps. The fundamentals of the dynamics of this 4-parameter family of one-dimensional maps are presented. A complete classification of the nature and stability of its fixed points is provided. The main results relate to the Allee effect bifurcation: a new type of bifurcation introduced for this class of unimodal maps. A necessary and sufficient condition so that the Allee fixed point is a snap-back repeller is established. In addition, in the parameters space is defined an Allee’s effect region, which determines the existence of an essential extinction for the generalized logistic maps. Local and global bifurcations of generalized logistic maps are investigated.

scholarly journals Dynamical Behaviour in Two Prey-Predator System with Persistence

2016 ◽  
Vol 16 ◽  
pp. 20-37
Author(s):  
V. Madhusudanan ◽  
S. Vijaya
Keyword(s):  
Limit Cycle ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Behaviour ◽  
Positive Equilibrium ◽  
Predator Population ◽  
Interior Equilibrium ◽  
Trivial Equilibrium ◽  
Necessary And Sufficient

In this work, the dynamical behavior of the system with two preys and one predator population is investigated. The predator exhibits a Holling type II response to one prey which is harvested and a Beddington-DeAngelis functional response to the other prey. The boundedness of the system is analyzed. We examine the occurrence of positive equilibrium points and stability of the system at those points. At trivial equilibrium E0and axial equilibrium (E1); the system is found to be unstable. Also we obtain the necessary and sufficient conditions for existence of interior equilibrium point (E6) and local and global stability of the system at the interior equilibrium (E6): Depending upon the existence of limit cycle, the persistence condition is established for the system. The numerical simulation infer that varying the parameters such as e and λ1it is possible to change the dynamical behavior of the system from limit cycle to stable spiral. It is also observed that the harvesting rate plays a crucial role in stabilizing the system.

scholarly journals Two-Parameter Dynamics of An Autonomous Mechanical Governor System With Time Delay

2021 ◽  
Author(s):  
Shuning Deng ◽  
Jinchen Ji ◽  
Guilin Wen ◽  
Huidong Xu
Keyword(s):  
Time Delay ◽  
Dynamical Behavior ◽  
Evolutionary Process ◽  
Basins Of Attraction ◽  
Poincaré Section ◽  
Poincare Section ◽  
Interesting Phenomenon ◽  
Computing Technique ◽  
Intersection Points ◽  
Two Parameter

Abstract Understanding of dynamical behavior in the parameter-state space plays a vital role in the optimal design and motion control of mechanical governor systems. By combining the GPU parallel computing technique with two determinate indicators, namely, the Lyapunov exponents and Poincaré section, this paper presents a detailed study on the two-parameter dynamics of a mechanical governor system with different time delays. By identifying different system responses in two-parameter plane, it is shown that the complexity of evolutionary process can increase significantly with the increase of time delay. The path-following strategy and the time domain collocation method are used to explore the details of the evolutionary process. An interesting phenomenon is found in the dynamical behavior of the delayed governor system, which can cause the inconsistency between the number of intersection points of certain periodic response on Poincaré section and the actual period characteristic. For example, the commonly exhibited period-1 orbit may have two or more intersection points on the Poincaré section instead of one point. Furthermore, the variations of basins of attraction are also discussed in the plane of initial history conditions to demonstrate the observed multistability phenomena and chaotic transitions.

scholarly journals An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior

2019 ◽  
Vol 17 (1) ◽  
pp. 1567-1598
Author(s):  
Tianbao Liu ◽  
Xiwen Qin ◽  
Qiuyue Li
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Second Derivative ◽  
Third Order ◽  
Numerical Tests ◽  
The Family ◽  
Theoretical Results ◽  
High Computational Efficiency

Abstract In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.

scholarly journals Dynamical Tangles in Third-Order Oscillator with Single Jump Function

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Jiří Petržela ◽  
Tomas Gotthans ◽  
Milan Guzan
Keyword(s):  
Differential Equation ◽  
Vector Field ◽  
Nonlinear Oscillator ◽  
Invariant Manifolds ◽  
Order Differential Equation ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Third Order ◽  
Newtonian Dynamics ◽  
Jump Function

This contribution brings a deep and detailed study of the dynamical behavior associated with nonlinear oscillator described by a single third-order differential equation with scalar jump nonlinearity. The relative primitive geometry of the vector field allows making an exhaustive numerical analysis of its possible solutions, visualizations of the invariant manifolds, and basins of attraction as well as proving the existence of chaotic motion by using the concept of both Shilnikov theorems. The aim of this paper is also to complete, carry out and link the previous works on simple Newtonian dynamics, and answer the question how individual types of the phenomenon evolve with time via understandable notes.

scholarly journals Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 322 ◽  
Author(s):  
Yanlin Tao ◽  
Kalyanasundaram Madhu
Keyword(s):  
Dynamical Behavior ◽  
Computational Cost ◽  
Basins Of Attraction ◽  
Order Method ◽  
Eighth Order ◽  
Projectile Motion ◽  
First Derivative ◽  
Iteration Functions ◽  
Optimal Launch Angle ◽  
Fourth Order Method

The principal objective of this work is to propose a fourth, eighth and sixteenth order scheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourth order method uses two evaluations of the function and one evaluation of the first derivative; the eighth order method uses three evaluations of the function and one evaluation of the first derivative; and sixteenth order method uses four evaluations of the function and one evaluation of the first derivative. So these all the methods have satisfied the Kung-Traub optimality conjecture. In addition, the theoretical convergence properties of our schemes are fully explored with the help of the main theorem that demonstrates the convergence order. The performance and effectiveness of our optimal iteration functions are compared with the existing competitors on some standard academic problems. The conjugacy maps of the presented method and other existing eighth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane. We apply the new scheme to find the optimal launch angle in a projectile motion problem and Planck’s radiation law problem as an application.

scholarly journals Basins of Attraction for Various Steffensen-Type Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Alicia Cordero ◽  
Fazlollah Soleymani ◽  
Juan R. Torregrosa ◽  
Stanford Shateyi
Keyword(s):  
Computer Algebra System ◽  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Numerical Tests ◽  
Derivative Free ◽  
Iteration Functions ◽  
The Stability ◽  
Programming Package ◽  
The Basin Of Attraction

The dynamical behavior of different Steffensen-type methods is analyzed. We check the chaotic behaviors alongside the convergence radii (understood as the wideness of the basin of attraction) needed by Steffensen-type methods, that is, derivative-free iteration functions, to converge to a root and compare the results using different numerical tests. We will conclude that the convergence radii (and the stability) of Steffensen-type methods are improved by increasing the convergence order. The computer programming package MATHEMATICAprovides a powerful but easy environment for all aspects of numerics. This paper puts on show one of the application of this computer algebra system in finding fixed points of iteration functions.

scholarly journals An Optimal Fourth Order Iterative Method for Solving Nonlinear Equations and Its Dynamics

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Rajni Sharma ◽  
Ashu Bahl
Keyword(s):  
Fourth Order ◽  
Dynamical Behavior ◽  
Computational Cost ◽  
Efficiency Index ◽  
Basins Of Attraction ◽  
Optimal Order ◽  
Extraneous Fixed Points ◽  
Fourth Order Method ◽  
Jarratt Method ◽  
Better Than

We present a new fourth order method for finding simple roots of a nonlinear equation f(x)=0. In terms of computational cost, per iteration the method uses one evaluation of the function and two evaluations of its first derivative. Therefore, the method has optimal order with efficiency index 1.587 which is better than efficiency index 1.414 of Newton method and the same with Jarratt method and King’s family. Numerical examples are given to support that the method thus obtained is competitive with other similar robust methods. The conjugacy maps and extraneous fixed points of the presented method and other existing fourth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane.

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