Stability,Bifurcation And Chaos In Evolutionary Game Dynamics

Evolutionary game dynamics is an interdisciplinary subject,which combines evolutionary game theory with dynamic system,and it is one of the hot topics in game theory in recent years.Imitative dynamics is an evolutionary game dynamics used to describe the propagation of successful strategy in biological evolution,economic dynamics,network evolution and other evolutionary games.At present,the challenge in the research is how to establish the relationship between dynamical behavior in dynamical system and Nash equilibrium and evolutionary stability strategy in game.In this thesis,we will discuss this problem in-depth.Based on previous researches,we will research the problem from two aspects.On the one hand,we will study the dynamical behavior of evolutionary game in many strategies,such as the stability of the equilibrium point in the dynamic system,bifurcation phenomena and chaotic phenomena,etc.On the other hand,we will discuss the existence and persistence of Nash equilibrium and evolutionary stable strategy in game model.Through analysis of the combination of two parts,we can obtain the relationship between dynamic behavior and the Nash equilibrium and evolutionary stable strategy of game strategy.The main work of this paper can be summarized as the following four parts.(1)We study the stability of imitative dynamics with discrete distributed delays.First of all,we build the two classes of evolutionary game dynamics models with discrete delays.Secondly,we discuss the stability of the game of Nash equilibrium in a single community and two communities,and obtain the sufficient conditions for the stability of the Nash equilibrium.Finally,the theoretical result in single community and two communities are verified by a snowdrift game.(2)We study the bifurcation phenomenon of three-strategy imitative dynamics with mutations.Firstly,the existence of Hopf bifurcation in the three-strategy game simulation dynamics without mutation is analyzed.Secondly,the corresponding stability of the three-strategy game simulation dynamics with mutation and the generation of limit cycles are discussed,and the critical value of subcritical Hopf bifurcation is obtained.Finally,the theoretical results are verified by numerical simulations through the classic rockscissors-paper game.(3)We study the complex dynamical behavior in four-strategy game.We research and build the ‘tiger-rod-chicken-worm' four-strategy replicator dynamical game model.Then by analysis of the complex dynamical behavior in the replicator dynamics,the results show that complex dynamic behaviors such as limit cycle and boundary polymorphic stability may be generated depending on the different values of the parameters in the payoff matrix of four-strategy game.(4)We study the chaotic behavior in asymmetric three-strategy game.Take asymmetric three-strategy game model as the research object,we discuss the complex dynamical behavior under the imitative dynamics,and explore the possibility of chaotic behavior.First of all,an asymmetric evolutionary game imitate dynamic model is established according to some social problems in the reality.Secondly,we use some mathematical methods to analyze the complex dynamic behavior in the dynamical model.Then,chaotic phenomenon is predicted through calculation of Lyapunov exponent and entropy in high-dimensional dynamical system.Finally,the chaotic behavior of the asymmetric model in zero-sum and positive-sum games is confirmed by numerical simulation.Finally,we make a summary of the work,and give some future research direction of this thesis.