Dynamical Analysis Of Several Classes Of Systems Of Differential Equations

Chaos controlling has shown broad application potentials, hence leading to interests in their methodology development. This thesis investigates control methodologies for chaotic systems, along with the stochastic delay method to control a class of chaotic systems. It is found in this study that delay plays an important role in dynamical system. A dynamical system with delay realistically reveals the characteristics of the nature of certain processes. The thesis further investigates chaotic control problems and the Hopf bifurcation problem in several kinds of difference systems, proposes the relevant theory and methodologies, and presents the results from their corresponding numerical simulations. The primary contributions from this thesis are itemized as follows·It presents a stochastic delay methodology for controlling chaotic systems, based on realistic effects of stochastic factors and delay on dynamic systems, and the corresponding characteristics of chaotic system. This method, based on the Lyapunov exponent, utilizes white noise to control chaotic systems to reach their equilibrium. The equilibrium refers to the original point in the scenarios discussed in this study. For equilibriums other than the original point, linear transformation is applied to transform objective states into their corresponding original states. A special case, i.e. delay is equal to zero is also discussed.·It presents a methodology, based on delay feedback to control a class of hyper-chaotic systems to their periodic states; that is, the Hopf bifurcation analysis of hyperchaotic systems with delays is investigated. It is known that an increase of system dimensions and delays complicates the Hopf bifurcation analysis. Nevertheless a uniform delay for each state affects the system control adversely. The approach presented in this study deploys multiple delays to control hyperchaotic Lüsystems. It is further revealed that the feedback method based on the same number of delays cannot be applied to controlling hyperchaotic Lorenz system or hyperchaotic Chen systems to reach to their periodic states. ·This study analyzes Hopf bifurcation of two kinds of integro-differential delay equations. At present, the research on integro-differential delay equations is primarily focused on the stability analysis of numerical solutions, and the Hopf bifurcation is a topic that has yet to be thoroughly studied. When analyzing dynamical behaviors of difference delay systems, the key is to identify the distribution of the roots of system's characteristic equation. This study introduces a systematic approach, using numerical analysis and based on boundary points method, to identify the values of Hopf bifurcation points. The method for constant delay integro-differential equations varies from that for integro-differential equations with infinite delay. For constant delay integro-differential equations, the sensitivity analysis is used to identify the most suitable Hopf parameter, followed by identifying the existence of Hopf bifurcation points with characteristic root method. For integro-differential equations with infinite delay, they are transformed into ordinary differential equations prior to their investigation.