The Space Of Non-linear Convective Systems Chaotic Behavior

2-D discrete dynamical systems, particularly 2-D delayed discrete dynamical systems, constitute an important part of delayed large-scale systems, belonging to the field of multivariable discrete-time sequences or spatial sequences in control theory. As a part of 2-D discrete dynamical systems, the nonlinear convection system is a new research field developed in recent years. Due to its importance in solving practical problems and its wide-ranged applications, it has become an attractive academic research area in recent years. In this thesis, basic research is carried out on spatial dynamical theory, spatial chaos, and control of nonlinear convection systems. The main contents of the thesis include the following aspects:Ⅰ The Definition of Spatial Chaos in the Sense of Rotation Number IntervalBased on the overall analyses of 1-D discrete dynamical systems, a series of new concepts are introduced reasonably in the space, including mapping circle, rotation number, rotation number interval, a shift on the sequence space, and so on, so that we have some new research works done well. For example, based on the construction of spatial k periodic orbits, and the theorem of rotation number interval in 2-D nonlinear discrete dynamical systems, qualitative analysis of boundary stability is studied. Also, the behavior of spatial chaos in the sense of rotation number interval in nonlinear convection systems is discussed.Ⅱ Perturbation Control on Spatial ChaosFor 2-D discrete dynamical systems, the control of spatially chaotic behaviors is investigated, and the perturbation control on spatial chaos is studied. By extending the method of the perturbation control applied to 1-D systems to 2-D discrete dynamical systems, we study its control on spatial chaos in the sense of space. Some sufficient conditions on the stability of the fixed surface are also obtained.Ⅲ Feedback Control on Spatial ChaosFor 2-D discrete dynamical systems, the control of spatially chaotic behaviors is investigated, and the duality nonlinear feedback control and prediction-basedfeedback control of spatially chaotic behaviors are also studied.1. The method of duality nonlinear feedback control overcomes the limitation that the convergence control district of the perturbation control is small, difficult to regulate, and easy to pass across the control district and escape from the expected periodical orbits under the circumstance of noises.2. The method of prediction-based feedback control overcomes the limitation of the DFC theoretically. We do not have to calculate target periodic orbits beforehand. Combining prediction with nonlinear estimation techniques, we have more general prediction-based feedback control.