In this paper, by discussing parameter conditions based on properties of activation functions, we decompose state space into positively invariant sets and establish sufficient conditions for the existence of locally stable equilibria for delayed Cohen-Grossberg neural networks (CGNNs) through Cauchy convergence principle. Some new criteria are derived for ensuring equilibria (periodic orbits) to be locally or globally exponentially stable in any designated region. Finally, our results are demonstrated by four numerical simulations.