Convergence of a Subclass of Cohen–Grossberg Neural Networks via the Łojasiewicz Inequality
Mauro Forti · IEEE Transactions on Systems Man and Cybernetics Part B (Cybernetics) · 2008
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 Łojasiewicz 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.