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In this paper, a necessary and sufficient condition for an invariant subset of a flow to be an attractor is given. At the same time a criterion for the existence of strongly gradient-like flows is also given.

In this paper, several results involving m-chain are proved. At the same time, a criterion for the existence of trajectories joining singular points is given.

Some properties of attractivity, strong repellors and uniform persistence for continuous maps in metric spaces are presented. A result in [J. Dynam. Differential Equations 13 (2001) 107] on uniform persistence is improved.

Some definitions such as m-chain recurrent set, weakly gradient flow and generalized Morse decomposition for a flow defined on a topological space are introduced in this paper. Some conclusions, include the chain recurrent set contain the m-chain recurrent set; the m-chain recurrent set contain the non-wandering set, are proved. In some flows the non-wandering set is proper subset of the m-chain recurrent set; in the meantime the m-chain recurrent set is proper subset of the chain recurrent set. Moreover some criterions for the existence of trajectories joining singular points and a necessary and sufficient condition of the weakly gradient flow are also given here. At last the generalized Morse decomposition of the invariant set are discussed in the paper.

In this paper, we discuss the autonomous differential systems in higher dimensional spaces R n (n ⩾ 3) . By using the index of vector field and fixed point theorem, we prove the existence of periodic solutions for these kind of systems.

In this paper, we studied the periodic solution, connecting orbits between singular points and global structure in Oregonator.