Multistability of discrete-time delayed Cohen–Grossberg neural networks with second-order synaptic connectivity

• Investigate the multistability of second-order discrete-time Cohen–Grossberg neural networks with general activation functions.
• Prove the existence and local exponential stability of equilibrium point of the neural networks under some less conservative conditions.
• Obtain the local exponential stability of the equilibrium point in designated region.

This paper addresses the multistability problem of discrete-time delayed Cohen–Grossberg neural networks (DDCGNNs) with second-order synaptic connectivity. For the neural networks with nondecreasing saturated activation functions possessing 2 corner points, based on the partition space method and reduction ad absurdum, several sufficient conditions are derived to ensure that n  -neuron second-order DDCGNNs can have 2n2n locally exponentially stable equilibrium points. Then, the analyses are extended to nondecreasing saturated activation functions with 2r corner points and some sufficient conditions are given to guarantee that the n  -neuron DDCGNNs can have (r+1)n(r+1)n locally exponentially stable equilibrium points. Moreover, some conditions are obtained to ensure the existence of locally exponentially stable equilibrium point in a predesigned region. Finally, three examples are carried out to show the effectiveness of the proposed criteria.