Study On Chaos Dynamics Of Several Types Of Nonlinear Systems

In this paper,we investigate the dynamic behaviors of several kinds of nonlinear systems by numerical simulation,such as two dimensional random coupled Logistic map,two dimensional exponential Logistic map,two dimensional modulation coupled Logistic map and fractional cubic map.In Chapter 1,we introduce the development history,the basic theoretical and knowledge of chaos theory.Also,we briefly describe the purpose and significance of the research,as well as the innovation of the research content.In Chapter 2,the chaotic characteristics of two dimensional random coupled Logistic map are studied by numerical simulation.The coupling coefficients of the system are considered to obey two points distribution,and the phase diagram and Lyapunov exponents are exploited to judge the system state.Numerical results show that the system can lead to chaos based on periodic bifurcation and Hopf bifurcation when the coupling coefficients jump between the chaotic and non-chaotic regions according to certain probability.In Chapter 3,the chaotic properties of exponential Logistic map modulated by Gaussian function are studied.The map has many degrees of freedom,which makes it have different chaotic characteristics and increase the flexibility in Quantitative Finance and other models.Firstly,the stability of the fixed point is analyzed.Secondly,the chaotic behavior of the system is explored by using bifurcation diagram,phase diagram and Lyapunov exponent diagram.The experimental results show that the exponential Logistic map can produce chaos in a proper parameter space.In addition,the chaotic phenomenon of coupled exponential Logistic map is studied.In Chapter 4,the traditional chaotic maps have the narrow parameter range and poor ergodicity.On the other hand,most of the researches on the dynamic characteristics of chaotic maps only consider the change of control parameters in the constant space.From these characteristics,we propose a modulated coupled Logistic map,where we use the exponential Logistic map to modulate traditional Logistic map and couple the result together to obtain a two dimensional system.Firstly,we observe that the two dimensional modulated coupled Logistic map has the better ergodicity comparing with other existing chaotic maps.Secondly,the phase portraits,bifurcation diagrams and Lyapunov exponent are analyzed theoretically.The results reveal that the chaotic patterns of the map will emerge out of double-periodic bifurcation and Hopf bifurcation respectively.Finally,the dynamic behaviors of the system with parameter randomization are studied.In Chapter 5,first of all,we consider the cubic map with piecewise constant arguments and apply the proposed discretization process to solve the model numerically,thus obtain the fractional cubic map.By using bifurcation graphics and the Lyapunov exponent,it is concluded that the fractional cubic map may emerge out of double-periodic bifurcation.Compared with the cubic map in the form of integer order,its chaotic features are more abundant.Secondly,in order to better describe the law of motion of objective things,this chapter uses discretization method to obtain fractional delayed cubic maps based on the expression of cubic maps.The results show that the time of bifurcation nodes of chaotic phenomena produced by fractional time-delay cubic maps varies with the order.