Effect of Delay on the Boundary of the Basin of Attraction in a Self-Excited Single Graded-Response Neuron

1997 ◽  
Vol 9 (2) ◽  
pp. 319-336 ◽  
Author(s):  
K. Pakdaman ◽  
C. P. Malta ◽  
C. Grotta-Ragazzo ◽  
J.-F. Vibert
Keyword(s):  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Transmission Delays ◽  
The Past ◽  
Graded Response ◽  
The Stability ◽  
Delay Dependent ◽  
The Basin Of Attraction

Little attention has been paid in the past to the effects of interunit transmission delays (representing a xonal and synaptic delays) ontheboundary of the basin of attraction of stable equilibrium points in neural networks. As a first step toward a better understanding of the influence of delay, we study the dynamics of a single graded-response neuron with a delayed excitatory self-connection. The behavior of this system is representative of that of a family of networks composed of graded-response neurons in which most trajectories converge to stable equilibrium points for any delay value. It is shown that changing the delay modifies the “location” of the boundary of the basin of attraction of the stable equilibrium points without affecting the stability of the equilibria. The dynamics of trajectories on the boundary are also delay dependent and influence the transient regime of trajectories within the adjacent basins. Our results suggest that when dealing with networks with delay, it is important to study not only the effect of the delay on the asymptotic convergence of the system but also on the boundary of the basins of attraction of the equilibria.

Finding Method and Analysis of Hidden Chaotic Attractors for Plasma Chaotic System From Physical and Mechanistic Perspectives

2020 ◽  
Vol 30 (05) ◽  
pp. 2050072 ◽  
Author(s):  
Yingjuan Yang ◽  
Guoyuan Qi ◽  
Jianbing Hu ◽  
Philippe Faradja
Keyword(s):  
Chaotic Attractor ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
The Self ◽  
Numerical Continuation ◽  
Chaotic Attractors ◽  
Phase Portrait Analysis ◽  
The Stability ◽  
Two Parameters ◽  
The Basin Of Attraction

A method for finding hidden chaotic attractors in the plasma system is presented. Using the Routh–Hurwitz criterion, the stability distribution associated with two parameters is identified to find the region around the equilibrium points of the stable nodes, stable focus-nodes, saddles and saddle-foci for the purpose of investigating hidden chaos. A physical interpretation is provided of the stability distribution for each type of equilibrium point. The basin of attraction and parameter region of hidden chaos are identified by excluding the self-excited chaotic attractors of all equilibrium points. Homotopy and numerical continuation are also employed to check whether the basin of chaotic attraction intersects with the neighborhood of a saddle equilibrium. Bifurcation analysis, phase portrait analysis, and basins of different dynamical attraction are used as tools to distinguish visually the self-excited chaotic attractor and hidden chaotic attractor. The Casimir power reflects the error power between the dissipative energy and the energy supplied by the whistler field. It explains physically, analytically, and numerically the conditions that generate the different dynamics, such as sinks, periodic orbits, and chaos.

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 Basin of Attraction through Invariant Curves and Dominant Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Ziyad AlSharawi ◽  
Asma Al-Ghassani ◽  
A. M. Amleh
Keyword(s):  
Stable Equilibrium ◽  
Basin Of Attraction ◽  
Invariant Curves ◽  
Equilibrium Solutions ◽  
Invariant Region ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Stability ◽  
The Basin Of Attraction

We study a second-order difference equation of the formzn+1=znF(zn-1)+h, where bothF(z)andzF(z)are decreasing. We consider a set of invariant curves ath=1and use it to characterize the behaviour of solutions whenh>1and when0<h<1. The caseh>1is related to the Y2K problem. For0<h<1, we study the stability of the equilibrium solutions and find an invariant region where solutions are attracted to the stable equilibrium. In particular, for certain range of the parameters, a subset of the basin of attraction of the stable equilibrium is achieved by bounding positive solutions using the iteration of dominant functions with attracting equilibria.

The Qualitative Analysis of Two Populations Commensalisms Model

2012 ◽  
Vol 524-527 ◽  
pp. 3705-3708
Author(s):  
Guang Cai Sun
Keyword(s):  
Qualitative Analysis ◽  
Equilibrium Point ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Mathematics Model ◽  
Stable Equilibrium Point ◽  
The Stability ◽  
Two Populations

This paper deals with the mathematics model of two populations Commensalisms symbiosis and the stability of all equilibrium points the system. It has given the conclusion that there is only one stable equilibrium point the system. This paper also elucidates the biology meaning of the model and its equilibrium points.

Basin of Attraction of the Simplest Walking Model

2001 ◽  
Author(s):  
A. L. Schwab ◽  
M. Wisse
Keyword(s):  
Equations Of Motion ◽  
Basin Of Attraction ◽  
Mapping Method ◽  
Walking Robots ◽  
Natural Energy ◽  
Walking Motion ◽  
The Stability ◽  
The Basin Of Attraction ◽  
General Method ◽  
Small Disturbances

Abstract Passive dynamic walking is an important development for walking robots, supplying natural, energy-efficient motions. In practice, the cyclic gait of passive dynamic prototypes appears to be stable, only for small disturbances. Therefore, in this paper we research the basin of attraction of the cyclic walking motion for the simplest walking model. Furthermore, we present a general method for deriving the equations of motion and impact equations for the analysis of multibody systems, as in walking models. Application of the cell mapping method shows the basin of attraction to be a small, thin area. It is shown that the basin of attraction is not directly related to the stability of the cyclic motion.

scholarly journals Stability and Multistability of a Bounded Rational Mixed Duopoly Model with Homogeneous Product

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16 ◽  
Author(s):  
Wei Zhou ◽  
Na Zhao ◽  
Tong Chu ◽  
Ying-Xiang Chang
Keyword(s):  
Joint Venture ◽  
Fractal Structure ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Mixed Duopoly ◽  
Formation Mechanisms ◽  
Multiple Attractors ◽  
Homogeneous Product ◽  
The Stability

In this paper, a mixed duopoly dynamic model with bounded rationality is built, where a public-private joint venture and a private enterprise produce homogeneous products and compete in the same market. The purpose of this research is to study the stability and the multistability of the established model. The local stability of all the equilibrium points is discussed by using Jury condition, and the stability region of the Nash equilibrium point has been given. A special fractal structure called “hub of periodicity” has been found in the two-parameter space by numerical simulation. In addition, the phenomena of multistability (also called coexistence of multiple attractors) are also studied using basins of attraction and 1-D bifurcation diagrams with adiabatic initial conditions. We find that there are two different coexistences of multiple attractors. And, the fractal structure of the attracting basin is also analyzed, and the formation mechanisms of “holes” and “contact” bifurcation have been revealed. At last, the long-term profits of the enterprises are studied. We find that some enterprises can even make more profits under a chaotic circumstance.

scholarly journals Robustness of Supercavitating Vehicles Based on Multistability Analysis

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Yipin Lv ◽  
Tianhong Xiong ◽  
Wenjun Yi ◽  
Jun Guan
Keyword(s):  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Small Range ◽  
Practical Application ◽  
Stable States ◽  
Supercavitating Vehicles ◽  
Control Gain ◽  
Stable Periodic ◽  
Setting Parameters

Supercavity can increase speed of underwater vehicles greatly. However, external interferences always lead to instability of vehicles. This paper focuses on robustness of supercavitating vehicles. Based on a 4-dimensional dynamic model, the existence of multistability is verified in supercavitating system through simulation, and the robustness of vehicles varying with parameters is analyzed by basins of attraction. Results of the research disclose that the supercavitating system has three stable states in some regions of parameters space, namely, stable, periodic, and chaotic states, while in other regions it has various multistability, such as coexistence of two types of stable equilibrium points, coexistence of a limit cycle with a chaotic attractor, and coexistence of 1-periodic cycle with 2-periodic cycle. Provided that cavitation number varies within a small range, with increase of the feedback control gain of fin deflection angle, size of basin of attraction becomes smaller and robustness of the system becomes weaker. In practical application, robustness of supercavitating vehicles can be improved by setting parameters of system or adjusting initial launching conditions.

scholarly journals Dynamics in Bank Crisis Model

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Tianshu Jiang ◽  
Mengzhe Zhou ◽  
Bi Shen ◽  
Wendi Xuan ◽  
Sijie Wen ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Complex Networks ◽  
Critical Point ◽  
Dynamic Systems ◽  
Degree Distribution ◽  
Equilibrium Points ◽  
Modified Model ◽  
The Past ◽  
Banking Market ◽  
The Stability

Bank crisis is grabbing more serious attention as several financial turmoils have broken out in the past several decades, which leads to a number of researches in this field. Comparing with researches carried out on basis of degree distribution in complex networks, this paper puts forward a mathematical model constructed upon dynamic systems, for which we mainly focus on the stability of critical point. After the model is constructed to describe the evolution of the banking market system, we devoted ourselves to find out the critical point and analyze its stability. However, to refine the stability of the critical point, we add some impulsive terms in the former model. And we discover that the bank crisis can be controlled according to the analysis of equilibrium points of the modified model, which implies the interference from outside may modify the robustness of the bank network.

Research on the Complexity of Dual-Channel Supply Chain Model in Competitive Retailing Service Market

2017 ◽  
Vol 27 (07) ◽  
pp. 1750098 ◽  
Author(s):  
Junhai Ma ◽  
Ting Li ◽  
Wenbo Ren
Keyword(s):  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Chain Model ◽  
Service Market ◽  
Optimal Decisions ◽  
Dual Channel ◽  
Supply Chain Model ◽  
The Stability ◽  
Dual Channel Supply Chain ◽  
Theoretical Results

This paper examines the optimal decisions of dual-channel game model considering the inputs of retailing service. We analyze how adjustment speed of service inputs affect the system complexity and market performance, and explore the stability of the equilibrium points by parameter basin diagrams. And chaos control is realized by variable feedback method. The numerical simulation shows that complex behavior would trigger the system to become unstable, such as double period bifurcation and chaos. We measure the performances of the model in different periods by analyzing the variation of average profit index. The theoretical results show that the percentage share of the demand and cross-service coefficients have important influence on the stability of the system and its feasible basin of attraction.

On-Off Intermittency and Riddled Basins of Attraction in a Coupled Map System

1998 ◽  
Vol 01 (02n03) ◽  
pp. 161-180 ◽  
Author(s):  
J. Laugesen ◽  
E. Mosekilde ◽  
Yu. L. Maistrenko ◽  
V. L. Maistrenko
Keyword(s):  
Periodic Orbits ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Chaotic State ◽  
One Dimensional ◽  
Type Iii ◽  
Different Types ◽  
One Dimensional Maps ◽  
The Basin Of Attraction

The paper examines the appearance of on-off intermittency and riddled basins of attraction in a system of two coupled one-dimensional maps, each displaying type-III intermittency. The bifurcation curves for the transverse destablilization of low periodic orbits embeded in the synchronized chaotic state are obtained. Different types of riddling bifurcation are discussed, and we show how the existence of an absorbing area inside the basin of attraction can account for the distinction between local and global riddling as well as for the distinction between hysteric and non-hysteric blowout. We also discuss the role of the so-called mixed absorbing area that exists immediately after a soft riddling bifurcation. Finally, we study the on-off intermittency that is observed after a non-hysteric blowout bifurcaton.

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