| Chaos being one of the most fascinating phenomenon in nonlinear science has evolved the study of nonlinear sciences into a new age. Chaos has redesigned various sciences by its applications wherever those sciences dealt with nonlinear behavior. The study of chaos is supposed to deal with nonlinear and complex behaviors that are far away from prediction such as weather, turbulence, the stock market, human psychology and so on.The first mathematical model of chaos was coined accidentally by E. N. Lorenz while he was studying a weather model. An approach to pursue the research on Lorenz system laid down a foundation of a whole new science of chaos theory. So far some great discoveries have been achieved yet extremely complex dynamics of chaotic and hyperchaotic systems are required to explore. In fact, it’s a very challenging task and one reason that chaos and hyperchaos are now hot research topics in the nonlinear sciences.This dissertation is engaged to the study of chaos and hyperchaos, and mainly focused on stability, bifurcation and singularly degenerate heteroclinic orbits of three dimensional chaotic system. Moreover, it reveals different dynamics of a four dimensional hyperchaotic system with no equilibria. The main research work accomplished in this dissertation is as follows.Chapter 1 is dedicated to the study of research background, history, achievements,definitions and basic knowledge of chaos theory. Applications of chaos in the real world are rapidly increasing which further reveal the importance of chaos theory. Therefore,some of the applications of chaos have been stated. The classical Lorenz system, Lorenztype systems and hyperchaotic systems are recapitulated.In Chapter 2, an unusual three dimensional chaotic system with two stable nodefoci has been presented. The system contains different chaotic systems as its special case, such as diffusionless-Lorenz system, Burke-Shaw system and Yang-Wei-Chen system. The algebraic structure of the system is very similar to Lorenz-type systems but the topological non-equivalency of the system has been shown effectively. The system is thoroughly studied using center manifold theory and Hopf bifurcation theory, the local dynamics including the stability, bifurcation of equilibrium is studied. In addition,singularly degenerate heteroclinic orbits in the system are also investigated.Chapter 3 contributes to the study of a three-dimensional autonomous generalized Lorenz-type chaotic system with seven parameters. Dynamical behaviours of this system are revealed through bifurcation diagrams, Lyapunov exponents spectrum and Poincar′e projections. Further, using the parameter-dependent center manifold theory and Hopf bifurcation theory, the local dynamics of this chaotic system, such as the stability and bifurcation of equilibrium, are investigated. Moreover through numerical and analytical techniques the chaotic and periodic regions have been specified in the system.In Chapter 4, a new fractional order chaotic system with two stable node-foci is investigated. Some sufficient conditions for the local stability of equilibria considering both commensurate and incommensurate cases are given. In addition, with the effective dimension less than three, the minimum effective dimension of the system is approximated as 2.8485 and is verified numerically. Furthermore, combination synchronization of the system is analyzed with the help of feedback control method. Theoretical results are verified by performing numerical simulations.In Chapter 5, a new simple in structure yet complex in dynamics four-dimensional Lorenz-type hyperchaotic system with no equilibria is proposed. Utilizing the techniques Lyapunov exponents spectrum and Poincar′e map, the dynamic behaviors of this hyperchaotic system are investigated thoroughly. Choosing proper parameter, this new system can exhibit hyperchaotic, chaotic, quasi-periodic and periodic dynamics. In particular,parameter dependent hyperchaotic, chaotic, quasi-periodic and periodic regions have been specified. |
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