The Stable Basin Of Attracting Set In Control Chaos And Some Global Bifurcations Of The Domains Of Feasible Trajectories In Nonlinear Ecological Systems

In this thesis, we first introduce the least Lipschiz constant and the nonlinear measure into nonlinear dynamical system, and verify that the least Lipschiz constant and the nonlinear measure can characterize the stability and the size of stable basin. We also, based on them and the polar coordinates transformation, derive general and precise algorithms for determining the radius of stable basin in nonlinear dynamical system. Then, many dynamical systems in control of chaos, such as Henon map, Ikeda map, Lorenz system, Rossler system and Chua system, are taken as examples to illustrate the implementation of our theory. Furthermore, for special two-dimensional discrete dynamics (in fact the recurrence equations), some global bifurcations that change feasible basin structure of ecological systems are given by a computer-assisted study. Four recurrence equations, such as a delayed regulation model, an extended logistic model, a model considering survival of the adults for two breeding seasons and a predator-prey model, are analyzed. The first two models belong to the in-vertible maps of degeneration analyzed by the use of the singular set and focal point and prefocal curve, a powerful tool for the analysis of global bifurcations of the planar maps having at least one component being fractional rational function with a denominator can vanish; and the others belong to noninvert-ible maps investigated by the use of the critical curve, a powerful tool for the analysis of global properties of two-dimensional mapsIn chapter 2, we apply OPCL control to discrete system, and based on the least Lipschiz constant, give an algorithm for estimating the radius of stable basin. We rigorously prove that the basins is bound to be of existencefor nonlinear discrete system, whose goal dynamics is either periodic orbits or fixed point. We also, in particular, investigate the stable basin in a quadratic polynomial map system, and present that the stable basin is irrelevant to the goal orbits with a negative Jacobian gain matrix. Furthermore, we take the well-known Henon system and Ikeda system as examples to illustrate the implementation of our theory, and give the corresponding simulations to reinforce our method.In chapter 3 we introduce the nonlinear measure of time-continuous system into the control of chaos, and verify that nonlinear measure can characterize the exponential stability and the size of stable basin. We also, based on it and the polar coordinates transformation, derive a general and precise algorithm for determining the radius of stable basin in controlling time-continuous chaotic dynamical system and for estimating the exponential decay of the controlled system converging to the desired goal dynamics. Furthermore, we take the well-known Lorenz , Rosslor system and Chua system as examples to illustrate the implementation of our theory.Chapter 4 is an attempt to give new results, by a computer-assisted study, on some global bifurcations that change structure of the domain of feasible trajectories (bounded discrete trajectories having an ecological sense) which can be obtained by the union of all rank preimages of axes. Two planar maps (or two-dimensional recurrence equations) being degenerated invertible maps, are analyzed. The basins of attractor for these maps are obtained by the backward iteration of a stable manifold of a saddle fixed point belonging to the basin boundary, and the interior domains of feasible trajectories are given by the intersection between the basin of attractor and the first quadrant.Chapter 5 concerns the two non-invertible maps, such as a model considering survival of the adults for two breeding seasons and a predator-preymodel which are investigated by the use of critical curves, a powerful tool for the analysis of global properties of two-dimensional maps. The global bifurcation that cause qualitative changes of the attractors and the basin of attraction, are obtained for these two planar maps.We end this thesis in chapter 6, where we give some summaries and comments on our res...