The authors consider the following linearly coupled system of ordinary differential equations
where , is the coupling matrix with for , and with . The main result of the paper is a number of criteria for synchronization of the above system, i.e., for the stability of the invariant subspace . Using Lyapunov function approach, conditions for global synchronization are given. Presented criteria for local synchronization are mathematically rigorous reformulation of the so-called master stability function approach [L. M. Pecora, T. L. Carroll, G. A. Johnson, D. J. Mar and J. F. Heagy, Chaos 7, 520–543 (1997; Zbl 0933.37030)].
The paper presents two examples of linearly coupled neural networks.