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A study on semiflows generated by cooperative full‐range CNNs
Author(s) -
Di Marco Mauro,
Forti Mauro,
Grazzini Massimo,
Nistri Paolo,
Pancioni Luca
Publication year - 2012
Publication title -
international journal of circuit theory and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.364
H-Index - 52
eISSN - 1097-007X
pISSN - 0098-9886
DOI - 10.1002/cta.1797
Subject(s) - mathematics , monotone polygon , monotonic function , generalization , irreducibility , limit (mathematics) , range (aeronautics) , differential inclusion , ordinary differential equation , pure mathematics , cellular neural network , convergence (economics) , discrete mathematics , differential equation , mathematical analysis , artificial neural network , computer science , geometry , materials science , machine learning , economic growth , economics , composite material
The paper considers the full‐range (FR) model of cellular neural networks (CNNs) characterized by ideal hard‐limiter nonlinearities with two vertical segments in the current–voltage characteristic. It is shown that when the FRCNNs are cooperative, i.e., there are excitatory interconnections between distinct neurons, the generated solution semiflow is monotone and that monotonicity implies some fundamental restrictions on the geometry of omega‐limit sets. The result on monotonicity is a generalization to the class of differential inclusions describing the dynamics of FRCNNs of a classic result due to Kamke for cooperative ordinary differential equations. The paper also points out difficulties to use the standard theory of eventually strongly monotone (ESM) semiflows for addressing convergence of FRCNNs. By means of counterexamples, it is shown that, even assuming the irreducibility of the interconnections, the semiflow generated by a cooperative FRCNN is not ESM; furthermore, also the limit set dichotomy can be violated. Copyright © 2012 John Wiley & Sons, Ltd.