Chaotic Behavior and Chaos Control for Three Dimensional Dynamical Systems

Abstract

In this dissertation, we investigated the dynamics of a chaotic and upgraded chaotic behavior to control the complexities and damages efficiently by using non linear tools for chaotic systems with applications to specific problems, including, Robbins disc dynamo and Magnetoconvection systems. The Robbins disc dynamo inherited from Geodynamics, which represents the mag netic field generation phenomena. Chaotic responses of three dimensional Rob bins disc dynamo are considered through boundedness and Lyapunov exponents to specify the place where the controller needed to be applied. In order to get the beneficial aspects in engineering of the Robbins disc dynamo, the controller is calculated by using the concept of Lie derivative which is the most significant facet of control analytical technique. The necessary and sufficient conditions are confirmed and a single input function controller is constructed for linear and non linear output function. The Robbins disc dynamo system is totally controlled for the required goals in both cases. To show the effectiveness of proposed control technique, comparison with other techniques is presented in this article. Numeri cal simulation results confirm our analytical results. In magnetoconvection, the magnetic field, the induced current and the electro magnetic interaction strongly affect the motion of fluid and play an important role in engineering applications, such as, astrophysical and geophysical area of study. The switching of behavior, from the upgraded chaotic to fully controlled xx magnetoconvection model, is studied by a feedback control technique. The mag netoconvection model shows hyperchaotic oscillations for different parametric val ues: Chandrasekhar number Q, Rayleigh number r, and diffusivity ratio. The controller for the magnetoconvection model is estimated by using the concept of the Lie derivative. Speed and dislocated feedback techniques are also utilized with the examination of stability analysis through feedback gains. Advantages of the feedback control technique are highlighted through comparison, with other control techniques such as, speed and dislocated feedback techniques. Simulation results show that the analytical strategy for controlling the oscillation is effective and controlled within a small duration of time. Apart from above mentioned chaotic systems (Robbins disc dynamo and Magneto convection), hyperchaotic behavior has been constructed in the three dimensional Pan model. The local dynamics of the three dimensional Pan system and it’s induced upgraded chaotic behavior, including stability analysis, type of attractors and the Hopf bifurcation are analyzed by using a center manifold and the normal form theories. A detailed set of conditions is derived,which guarantee the exis tence of subcritical Hopf bifurcation in three dimensional and supercritical Hopf bifurcation in upgraded chaotic, Pan system. The proposed upgraded chaotic be havior in Pan system undergoes periodic, chaotic and hyperchaotic orbit with the variation of bifurcation parameter and control parameter.