Convergence of a Subclass of Cohen–Grossberg Neural Networks via the Łojasiewicz Inequality

This correspondence proves a convergence result for the Lotka-Volterra dynamical systems with symmetric interaction parameters between different species. These can be considered as a subclass of the competitive neural networks introduced by Cohen and Grossberg in 1983. The theorem guarantees that each forward trajectory has finite length and converges toward a single equilibrium point, even for those parameters for which there are infinitely many nonisolated equilibrium points. The convergence result in this correspondence, which is proved by means of a new method based on the Lojasiewicz inequality for gradient systems of analytic functions, is stronger than the previous result established by Cohen and Grossberg via LaSalle´s invariance principle, which requires, for convergence, the additional assumption that the equilibrium points be isolated.