Chaotic Dynamical Systems: Extension And Analysis

The chaos-generating system is an important part of chaos theory and application study. The nonlinearity in dynamical systems is compulsory to generate chaotic phenomena and its characteristics and composition forms directly determine the chaotic behaviors and chaos-generating mechanisms. With advances in these areas, this dissertation conducts a thorough investigation on the nonlinear composition characteristics of the chaotic systems and extends some typical chaotic systems. Some new chaotic phenomena are found and new chaos-generating mechanisms are revealed.This dissertation is divided into two parts of the discrete time systems and the continuous time systems.In the study on discrete time systems, we pay attention on a class of one-dimensional and two-dimensional discrete maps with exponential quadratic terms and the current mode controlled switching DC-DC converters with ramp compensation.Generalized square map and DOG wavelet map are constructed and their complex dynamical behaviors are revealed. With the evolving orbits of the fixed points and the iterative curves with corresponding degree, it is found that (1) one-dimensional generalized square map has physical phenomenon similar to single-peak square map and its dynamical behavior distributes in the unit region. (2) DOG wavelet map. whose the number of the fixed points varies with the parameters, exhibits rich nonlinear phenomena, such as period-doubling bifurcation, tangent bifurcation, boundary crisis bifurcation, period-window, and imperfect Feigenbaum-tree. and so on. (3) two-dimensional generalized square map has the phenomena of Hopf bifurcation and locked-frequency with complex, flexible, strange shaped limit cycles and chaotic attractors.Switching DC-DC converter is a kind of special time-varying discrete chaotic systems and has two boundaries in the wide ranges of circuit parameters. It exhibits the phenomenon of border collision bifurcation. With the variations of circuit parameters, the switching DC-DC converter has two routes to chaos through period-doubling bifurcation and border collision bifurcation. The border collision bifurcation will lead to state shifts between the stable period-one state, robust chaos state in continuous conduction mode (CCM) and weak chaos state with strong intermittency in discontinuous conduction mode (DCM). By introducing suitable ramp compensation, the operation modes of the switching DC-DC converter can shift from DCM to CCM, and can also be controlled at the stable period-1 region. Furthermore, two borderline equations of the orbit state shifts are derived and the circuit parameters corresponding to the operation state regions are presented. The results are confirmed by circuit experiments.In the study on continuous time systems, we pay attention on three-dimensional chaotic systems, four-dimensional hyperchaotic systems, as well as multi-scroll chaotic systems and smooth memristor chaotic circuits.A robust chaotic system with an exponential quadratic term and a modified generalized Lorenz system in a canonical form are presented. The robust chaotic system has simple algebraic structure, can generate 2-scroll chaotic attractors and has chaotic phenomena in a very wide parameter ranges. The modified generalized Lorenz system with a folded factor can display complex 2-scroll and 1-scroll folded attractors and exhibits complicated nonlinear dynamical phenomena. In addition, different hyperchaotic systems or their corresponding circuits are obtained by adding linear or nonlinear state controllers to the different three-dimensional chaotic systems or circuits. Their hyperchaotic dynamics are comfirmed by simulations and experiments.A general guide to design multi-scroll chaotic systems is proposed. With Colpitts oscillator model, by modifying the nonlinear term in model state equations, grid-scroll and multi-scroll chaotic and hyperchaotic systems are constructed, which can generate a (2N+1)-scroll chaotic attractor and a (2M+1)x(2N+1)-grid scroll hyperchaotic attractor. By designing two different saw-tooth functions in the linear system, some multi-scroll chaotic attractors with different scroll-location distributions are obtained and the designs of the scroll number and scroll-location distributions of the chaotic attractors are realized.In the study on smooth memristor chaotic circuits, a smooth flux-controlled memristor having a cubic nonlinearity is presented and a smooth memristor chaotic circuit is derived by replacing Chua's diode in Chua's chaotic circuit with the memristor. The dynamical behaviors are analysed under different circuit parameters and initial states. The research results demonstrate that the dynamics of the smooth memristor oscillator, different from general chaotic systems, depend not only on circuit parameters, but also closely on the initial states. With different initial states, the circuit orbits will transfer among chaotic oscillation or periodic oscillation or stable sink. In addition, some transient chaos and state transitions in the smooth memristor oscillator are revealed and analyzed.