Complex Dynamics And Chaotic Synchronization In Some Discrete-time Dynamical Systems

Chaos refers to seemingly random irregular movements that occur in deterministic systems.Chaotic Dynamics is an important branch of complexity science and a popular subject in recent decades,which has developed into a relatively complete system and has shown its strong vitality in many fields.The study of the synchronization dynamics is one of the current research hotspots in the field of nonlinearity.Difference equations are widely used in daily life and in various fields.Using chaotic theory to analyze and study problems in traffic flows will do good to mathematical modeling of traffic systems,such as how to identify chaos and its practical significance,and make timely measures to prevent disorder,which can open new solutions to traffic flow problems.Fractional-order systems can better reflect the engineering and physical phenomena presented by actual systems.Many scholars have developed a strong interest in the study of discrete fractional-order chaotic systems in the past ten years.Following these trends,the main research results of this thesis include:(1)Based on the distribution of Eigen values and chaos theory,stability,multistability and chaos characteristics of the discrete excitable system with time delay are studied.(2)A class of single-lane discrete traffic car-following model is proposed,local stability analysis is carried out,and its rich dynamic behavior is explored,such as the existence of chaotic-fractional attractors.(3)An effective method for calculating the maximum Lyapunov exponent of discrete fractional order systems is proposed and used to determine whether chaos exists in discrete fractional order difference systems.At the same time,the bifurcation diagrams of discrete fractional order H non map and Logistic map that depend on given parameters are studied.(4)Using non-linear feedback method or parameter adaptive control method to study synchronization criteria of several types of fractional-order chaotic difference equations,establish a criterion for the distribution of a class of higher-order polynomial roots,and find that there exist some defects about stability analysis in the literature.(5)The thesis combines numerical simulation to verify the correctness of the theoretical analysis.