Study On Qualitative Theory Of Dynamical Systems

Dynamical system is a mathematical discipline on the laws of evolution of systems. Roughly speaking, dynamical systems consist of differential dynamical system, topological dynamical system, infinite dimensional dynamical system, complex dynamical system and ergodic theory etc. This paper deals with the dynamical behaviors concerning the qualitative analysis of differential equations, differential dynamical system, topological dynamical system and infinite dimensional dynamical system. The connecting orbits between two critical points play a fundamental role in the study of qualitative analysis of differential equations. For gradient-like systems, using the characteristic of connecting orbits, this paper gives some criteria on the non-existence of connecting orbits in R n. With the help of half-bounded connecting arc, some criteria for the existence of trajectories connecting a pair of critical points of planar dynamitic systems are given. When studying the chain-recurrent attributes and Ω-limit sets of flows in compact metric space, C. C. Conley gave the conception of Ω-limit sets and its basic properties. Then, W. Chen and others gave more properties on it. In this paper, some other properties of Ω-limit sets of flows are investigated. In differential dynamical system, the Closing Lemma for non-wandering point is very important. We proved a C~1 closing lemma for chain recurrent point, and from this closing lemma, we proved that chain recurrent is always preperiodic. The existence of global attractor and its dimensions are the central problems in the study of infinite dimensional dynamical systems. In this paper, we define a stability of semigroups in a complete space, and prove that the collection of continuous semigroups possessing global attractors is in general not stable.