Research On Several Problems Of Complex Dynamics And Control Of Dynamical Systems

Complex dynamics and control of dynamical systems is an active research field with many unsolved problems,which are very challenging.This paper focuses on some prob-lems of the field and is divided into two parts.In the first part,we study complex dynamic behaviours of two kinds of three dimensional continuous-time polynomial systems,mainly involving the chaos of three-dimensional nonautonomous systems and the fractal structure of the boundary of the attraction basin in a three-dimensional autonomous system.In the second part,we study properties of several control systems,mainly involving the indexes of n-dimensional linear control systems with saturated state feedback and the controllability of the dual inverted pendulum and the parallel n inverted pendulum.The contents are as follows:(1)We study a kind of Lorenz systems with periodically excited parameters and demon-strate a new mechanism of generating chaos.In the case that the original Lorenz system has only two asymptotically stable equilibrium points,a saddle and no chaotic dynamics,the concepts of time-varying equilibrium point and the pesudo-stable manifold of the Lorenz systems with periodically excited parameters are given to describe the changes of the bound-aries+ of attraction basins.When the parametric excitation changes periodically,the time-varying stable equilibrium points and the pesudo-stable manifold also changes changes pe-riodically.The mechanism of generating chaos in the periodically parametrically excited Lorenz system is demonstrated by showing that some trajectories can visit different attrac-tion basins of time-varying stable equilibrium points.(2)For a class of three-dimensional ordinary differential equations without linear terms,it is proved that the origin is a asymptotically stable equilibrium point under certain parameter conditions,and the complexity of its attractive basin is further discussed.We prove that the boundary of the attraction basin of the origin contains an invariant set with fractal structure,which is homeomorphic to the strange attractor in the Lorenz system.(3)We investigate the relationship between the number of equilibrium points and the feedback in n-dimensional linear control systems with saturated state feedback.By using the Poincaré-Hopf index theorem,we provide an index formula for equilibrium points and discuss its relation to boundaries of attraction basins in feedback systems with single input.In addition,when the dimension n = 3,for the conjecture that the attraction basin of the origin is a convex set,a counterexample is provided.(4)We study the controllability and stabilizability for the dual inverted pendulum.The sufficient and necessary conditions of the controllability of the system with frictions are given.We also discuss the relationship between the stabilized feedback gains and the lengths of pendulum,and the convexity of the parameter region of feedback gains.For the parallel n inverted pendulum system with n different inverted pendulums,the mathematical model is established.Based on the linearized model,we obtain the sufficient and necessary condi-tions of(local)controllability of the parallel n inverted pendulum.