scholarly journals On Equilibrium Properties of the Replicator–Mutator Equation in Deterministic and Random Games

2019 ◽  
Vol 10 (3) ◽  
pp. 641-663 ◽  
Author(s):  
Manh Hong Duong ◽  
The Anh Han
Keyword(s):  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Evolutionary Games ◽  
Random Polynomial ◽  
Snow Drift ◽  
Random Games ◽  
Payoff Matrices ◽  
Number Of Equilibria ◽  
Polynomial Theory ◽  
Behavioural Diversity

AbstractIn this paper, we study the number of equilibria of the replicator–mutator dynamics for both deterministic and random multi-player two-strategy evolutionary games. For deterministic games, using Descartes’ rule of signs, we provide a formula to compute the number of equilibria in multi-player games via the number of change of signs in the coefficients of a polynomial. For two-player social dilemmas (namely the Prisoner’s Dilemma, Snow Drift, Stag Hunt and Harmony), we characterize (stable) equilibrium points and analytically calculate the probability of having a certain number of equilibria when the payoff entries are uniformly distributed. For multi-player random games whose pay-offs are independently distributed according to a normal distribution, by employing techniques from random polynomial theory, we compute the expected or average number of internal equilibria. In addition, we perform extensive simulations by sampling and averaging over a large number of possible payoff matrices to compare with and illustrate analytical results. Numerical simulations also suggest several interesting behaviours of the average number of equilibria when the number of players is sufficiently large or when the mutation is sufficiently small. In general, we observe that introducing mutation results in a larger average number of internal equilibria than when mutation is absent, implying that mutation leads to larger behavioural diversity in dynamical systems. Interestingly, this number is largest when mutation is rare rather than when it is frequent.

scholarly journals Non-Markov Stateful Evolutionary Games

2017 ◽  
Author(s):  
Mark Burgess
Keyword(s):  
Population Dynamics ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Evolutionary Game ◽  
Evolutionary Games ◽  
Complex Interactions ◽  
Evolving Populations

A new evolutionary game is introduced which incorporates states and actions into the strategies of the organisms of the evolving populations. The game centrally features actions that result in demographic flow between states that may not conserve organism numbers. It is by this feature that the game encapsulate a range of other evolutionary games, and can encode almost very complex interactions between organisms, species and populations. The game's formalism is expounded and the nature of the game's equilibrium is discussed. This discussion leads to an algorithm for numerically determining the stable equilibrium points which is exemplified in the context of a modified Hawk-Dove game. The game's flexibility for modeling population dynamics is evaluated and compared with other evolutionary games.

scholarly journals Non-Markov Stateful Evolutionary Games

2017 ◽  
Author(s):  
Mark Burgess
Keyword(s):  
Population Dynamics ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Evolutionary Game ◽  
Evolutionary Games ◽  
Complex Interactions ◽  
Evolving Populations

A new evolutionary game is introduced which incorporates states and actions into the strategies of the organisms of the evolving populations. The game centrally features actions that result in demographic flow between states that may not conserve organism numbers. It is by this feature that the game encapsulate a range of other evolutionary games, and can encode almost very complex interactions between organisms, species and populations. The game's formalism is expounded and the nature of the game's equilibrium is discussed. This discussion leads to an algorithm for numerically determining the stable equilibrium points which is exemplified in the context of a modified Hawk-Dove game. The game's flexibility for modeling population dynamics is evaluated and compared with other evolutionary games.

scholarly journals Research of Trajectory Optimization Approaches in Synthesized Optimal Control

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 336
Author(s):  
Askhat Diveev ◽  
Elizaveta Shmalko
Keyword(s):  
Optimal Control ◽  
Equilibrium Point ◽  
Trajectory Optimization ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Control Object ◽  
Symbolic Regression ◽  
Optimal Location ◽  
Stable Equilibrium Point ◽  
Stabilization System

This article presents a study devoted to the emerging method of synthesized optimal control. This is a new type of control based on changing the position of a stable equilibrium point. The object stabilization system forces the object to move towards the equilibrium point, and by changing its position over time, it is possible to bring the object to the desired terminal state with the optimal value of the quality criterion. The implementation of such control requires the construction of two control contours. The first contour ensures the stability of the control object relative to some point in the state space. Methods of symbolic regression are applied for numerical synthesis of a stabilization system. The second contour provides optimal control of the stable equilibrium point position. The present paper provides a study of various approaches to find the optimal location of equilibrium points. A new problem statement with the search of function for optimal location of the equilibrium points in the second stage of the synthesized optimal control approach is formulated. Symbolic regression methods of solving the stated problem are discussed. In the presented numerical example, a piece-wise linear function is applied to approximate the location of equilibrium points.

Chaos in the Periodically Parametrically Excited Lorenz System

2021 ◽  
Vol 31 (08) ◽  
pp. 2130024
Author(s):  
Weisheng Huang ◽  
Xiao-Song Yang
Keyword(s):  
Chaotic Dynamics ◽  
Lorenz System ◽  
Stable Equilibrium ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Time Varying ◽  
Asymptotically Stable ◽  
Parametrically Excited ◽  
The Lorenz System

We demonstrate in this paper a new chaotic behavior in the Lorenz system with periodically excited parameters. We focus on the parameters with which the Lorenz system has only two asymptotically stable equilibrium points, a saddle and no chaotic dynamics. A new mechanism of generating chaos in the periodically excited Lorenz system is demonstrated by showing that some trajectories can visit different attractor basins due to the periodic variations of the attractor basins of the time-varying stable equilibrium points when a parameter of the Lorenz system is varying periodically.

scholarly journals Memory Dynamics in Attractor Networks

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Guoqi Li ◽  
Kiruthika Ramanathan ◽  
Ning Ning ◽  
Luping Shi ◽  
Changyun Wen
Keyword(s):  
Energy Function ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Saddle Points ◽  
Memory Systems ◽  
Initial Stimulus ◽  
Attractor Network ◽  
Local Maxima ◽  
Attractor Networks ◽  
New Energy

As can be represented by neurons and their synaptic connections, attractor networks are widely believed to underlie biological memory systems and have been used extensively in recent years to model the storage and retrieval process of memory. In this paper, we propose a new energy function, which is nonnegative and attains zero values only at the desired memory patterns. An attractor network is designed based on the proposed energy function. It is shown that the desired memory patterns are stored as the stable equilibrium points of the attractor network. To retrieve a memory pattern, an initial stimulus input is presented to the network, and its states converge to one of stable equilibrium points. Consequently, the existence of the spurious points, that is, local maxima, saddle points, or other local minima which are undesired memory patterns, can be avoided. The simulation results show the effectiveness of the proposed method.

scholarly journals An Evolutionary Game Analysis of Internet Public Opinion Events at Universities: A Case from China

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Hongying Wen ◽  
Kairong Liang ◽  
Yiquan Li
Keyword(s):  
Public Opinion ◽  
University Students ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Evolutionary Game ◽  
Stable Equilibrium Point ◽  
University Administration ◽  
Negative Effects ◽  
Internet Medium ◽  
The University

Internet public opinion events at universities in China occurred frequently, creating painful repercussions for reputation and stability of colleges and universities. To better cope with the problem, this paper explores an evolutionary mechanism of the university Internet public opinion events. Firstly, we discuss the interactions and behavior of three key participants: an Internet medium, university students as a whole, and administration. Secondly, we construct a tripartite evolutionary game model consisting of an Internet medium, student group, and university administration and then analyze and obtain the differential dynamic equations and equilibrium points. Subsequently, the evolutionary stable equilibrium is further analyzed. Finally, we employ numerical studies to examine how the tripartite behavior choices affect evolutionary paths and evolutionary equilibrium strategies. Results are derived as follows: under certain conditions, there exists an asymptotically stable equilibrium point for the tripartite evolutionary game. On the one hand, appropriate penalties and rewards should be provided to foster objectives and fair behaviors of the network medium. On the other hand, university students should be educated and guided to deal rationally with negative effects of Internet public opinion events. Moreover, online real-name authentication is an important and necessary measure. Finally, the university administration should release truthful, timely, and comprehensive information of Internet public opinion events to mitigate potential negative impacts.

Numbers of stable equilibrium points of beams in vacuum optical tweezers based on Mento Carlo method

2021 ◽  
Author(s):  
Jingjing Su ◽  
Nan Li ◽  
Jiapeng Mou ◽  
Yishi Liu ◽  
Xingfan Chen ◽  
...  
Keyword(s):  
Optical Tweezers ◽  
Stable Equilibrium ◽  
Equilibrium Points

Jacobi analysis for an unusual 3D autonomous system

2020 ◽  
Vol 17 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Chunsheng Feng ◽  
Qiujian Huang ◽  
Yongjian Liu
Keyword(s):  
Chaotic System ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Chen System ◽  
Parameter Region ◽  
Stable Equilibria ◽  
The Lorenz System ◽  
Deviation Vector

Little seems to be known about the study of the chaotic system with only Lyapunov stable equilibria from the perspective of differential geometry. Therefore, this paper presents Jacobi analysis of an unusual three-dimensional (3D) autonomous chaotic system. Under certain parameter conditions, this system has positive Lyapunov exponents and only two linear stable equilibrium points, which means that chaotic attractor and Lyapunov stable equilibria coexist. The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The results show that the value of the deviation curvature tensor at equilibrium points is only related to parameters; the two equilibrium points of the system are Jacobi stable if the parameters satisfy certain conditions. Particularly, for a specific set of parameters, the linear stable equilibrium points of the system are always Jacobi unstable. A periodic orbit that is Lyapunov stable is also proven to be always Jacobi unstable. Next, Jacobi-stable regions of the Lorenz system, the Chen system and the system under study are compared for specific parameters. It can be found that although these three chaotic systems are very similar, their regions of Jacobi stable parameters are much different. Finally, by comparing Jacobi stability with Lyapunov stability, the obtained results demonstrate that the Jacobi stable parameter region is basically symmetric with the Lyapunov stable parameter region.

scholarly journals Sex ratio and dynamic behavior in populations of the exotic blowfly Chrysomya albiceps (Diptera, Calliphoridae)

2007 ◽  
Vol 67 (2) ◽  
pp. 347-353 ◽  
Author(s):  
H. Serra ◽  
WAC. Godoy ◽  
FJ. Von Zuben ◽  
CJ. Von Zuben ◽  
SF. Reis
Keyword(s):  
Sex Ratio ◽  
Dynamic Behavior ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Larval Density ◽  
Chaotic Regime ◽  
Demographic Parameters ◽  
Chrysomya Albiceps ◽  
Exponential Regression ◽  
The Impact

Sex ratio is an essential component of life history to be considered in population growth. Chrysomya albiceps is a blowfly species with a naturally biased sex ratio. In this study, we evaluated the impact of changes in sex ratio on the dynamic behavior of C. albiceps using a density-dependent mathematical model that incorporated demographic parameters such as survival and fecundity. These parameters were obtained by exponential regression, with survival and fecundity being estimated experimentally as a function of larval density. Bifurcation diagram of the results indicated the evolution of stable equilibrium points as a function of sex ratio. A continually increasing sex ratio yielded a hierarchy of bifurcating stable equilibrium points that evolved into a chaotic regime. The demographic parameters obtained by exponential regression were also changed to maximum and minimum values in order to analyze their influence on dynamic behavior with sex ratio being considered as an independent variable. Bifurcations with periodicity windows between chaos regimes were also found.

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.

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