*(English)*Zbl 0553.92009

Considered is a class of n-dimensional dynamical systems

where the matrix $C=\left[{c}_{ik}\right]$ is symmetric and the system as a whole is competitive. Several examples of applications of this type of equations are indicated as nonlinear neural networks and, in general, global pattern formation.

A global Lyapunov function for the system discussed is introduced. Its absolute stability with infinite but totally disconnected equilibrium points is studied by the LaSalle invariance principle. Decomposition of equilibria of the system into suprathreshold and subthreshold variables is also presented $({x}_{i}\left(t\right)$ is called suprathreshold at t if ${x}_{i}\left(t\right)>{{\Gamma}}_{i}^{-}$ where ${{\Gamma}}_{i}^{-}$ stands for inhibitory threshold of ${d}_{i})$.

##### MSC:

92Cxx | Physiological, cellular and medical topics |

93D05 | Lyapunov and other classical stabilities of control systems |

92F05 | Applications of mathematics to other natural sciences |

37-99 | Dynamic systems and ergodic theory (MSC2000) |

93C15 | Control systems governed by ODE |