Study On Some Issues Of Generation And Control Of Chaotic And Hyperchaotic Systems

In1963, Lorenz discovered the first chaotic system when studying the atmospheric convection. The Lorenz system is the first chaotic model, which has become a touchstone for the developing of chaos theory, and led to further think about the complexity of nonlinear phenomena in the real world. In the pursuit of theory and application research on chaos, the generation and control for chaotic and hyperchaotic systems became two important research directions.This paper presented several chaotic and hyperchaotic systems, discussed stable control for the permanent magnet synchronous motor, investigated the issue of input-to-state stable control for a class of chaotic and hyperchaotic systems with undeterministic parameters and external interference, and investigated the tracking control of several hyperchaotic systems.The main research works of this dissertation are as follows.1. Based on the construction pattern of Chen and Liu chaotic systems, a new chaotic system is proposed by developing Lorenz chaotic system. The essential features of chaotic system are analyzed via equilibrium and stability, continuous spectrum, Poincare mapping. The different dynamic behaviors of the system are analyzed especially when changing each system parameter. It’s found that when the parameters d and e vary, the Lyapunov exponent spectrum keeps invariable, and there exist the functions of global nonlinear amplitude adjuster for d and partial nonlinear amplitude adjuster for e. Then, based on the model of multiplier modulation, we realize the amplitude modulation of the sinusoidal signal via the chaotic singals. Finally, by picking a suitable cross-section with respect to the attractor carefully, a topological horseshoe of the corresponding first-returned Poincare map is found, thus giving a rigorous confirmation of the existence of chaos in this system, and a practical circuit is designed to implement this new chaotic system, which confirms that the chaotic system can be achieved physically.2. A four-dimensional hyperchaotic system is proposed by adding a state feedback controller to a chaotic system. Numerical simulations and theoretical analysis show that this four-dimensional system will take on periodic, complex periodic, quasi-periodic, chaotic and hyperchaotic dynamical behaviours as parameters vary. Moreover, an electronic circuit diagram is designed for demonstrating the existence of the hyperchaos.3. A novel four-dimensional smooth autonomous system is proposed, which is special since it has only one equilibrium, but it can generate a four-wing chaotic or hyperchaotic attractor. By applying either analytical or numerical methods, some basic properties of the4D system, such as phase diagrams, bifurcation diagram and Lyapunov exponents are investigated to observe chaotic motions.4. A ring-scroll Chua chaotic system is proposed by introducing a generalized ring transformation. Some basic dynamical properties of this generalized ring transformation are discussed. The parameter regions and the periodic orbits, which are embedded in Chua chaotic attractor mapping to those in ring-scroll Chua chaotic attractor, are investigated, too. Finally, the ring-scroll Chua chaotic attractor is physically implemented by using digital signal processors.5. Based on the LaSalle’s invariant set theorem, an adaptive controller is developed to acquire chaos control for the permanent magnet synchronous motor. And then an extended adaptive controller is developed by introducing a control strength factor. The chaotic behaviour of the fractional-order permanent magnet synchronous motor is investigated, and an adaptive controller is developed based on the stability theory for fractional systems.6. The issue of input-to-state stable control for a class of chaotic and hyperchaotic systems with undeterministic parameters and external interference is investigated. Based on the Lyapunov stable theory, a linear state feedback controller is presented to guarantee the asymptotic stability and achieve the bounded state variable for any bounded disturbance. The control strength matrix can be obtained by solving the linear matrix inequality (LMI). Furthermore, with a nonsingular coordinate transformation of the dynamics equation, a simplified linear state feedback controller is obtained. Finally, the illustration is given by using two different chaotic and hyperchaotic systems with numerical simulations to verify the proposed ISS control scheme.7. The generalized tracking control is investigated. First, a nonlinear controller is proposed to acquire generalized projective synchronization of full states of a fifth-order circuit’s hyperchaotic system based on adjusting the scaling factor and the accelerated factor. It allows one to drive the fifth-order system to arbitrary periodic orbits or fixed point rapidly. Based on this, hyperchaotic systems, chaotic systems, periodic signal, and constant signal are taken as examples respectively. Second, based on constructing a new Lyapunov function, an adaptive controller is proposed to acquire adaptive tracking control for a hyperchaotic system. Finally, the function projective synchronization and tracking control of a class of hyperchaotic systems with unknown parameters and disturbance are investigated. Based on the Lyapunov’s stability theory, a robust controller is designed to drive the hyperchaotic system to any desired reference signals up to the scaling function factor, and a parameter update law for identifying the unknown parameters of the hyperchaotic system is gained simultaneously.