A study on semiflows generated by cooperative full‐range CNNs

Abstract

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

Monotonicity of semiflows generated by cooperative delayed full-range CNNs

“…Theorem 1 can also be considered as an extension to the delayed case of a recent result on monotonicity of semiflows generated by cooperative FRCNNs without delays [25]. 2) Theorem 1 establishes monotonicity of the semiflow under the assumption of a cooperative interconnection matrix A and delayed interconnection matrix A τ of the D-FRCNNs.…”

“…2) Theorem 1 establishes monotonicity of the semiflow under the assumption of a cooperative interconnection matrix A and delayed interconnection matrix A τ of the D-FRCNNs. It can be shown, by means of counterexamples analogous to those in [25] for FRCNNs, that there are D-FRCNNs with irreducible A and A τ for which the semiflow of D-FRCNNs is not ESM nor SOP (cf. Sect.…”

Convergent Dynamics of Nonreciprocal Differential Variational Inequalities Modeling Neural Networks

“…A solution , , of the DVI ( 2) is an absolutely continuous function on any compact interval in such that for any and for almost all (a.a.) . It has been shown in [29] that, given any , there exists a unique solution , , of the DVI (2) with initial condition . Moreover, , , is a solution of (2) if and only if it satisfies the projected differential equation (3) It is also shown in [29] that there is at least an EP of (2).…”

“…It has been shown in [29] that, given any , there exists a unique solution , , of the DVI (2) with initial condition . Moreover, , , is a solution of (2) if and only if it satisfies the projected differential equation (3) It is also shown in [29] that there is at least an EP of (2). We denote by the set of EPs of (2).…”

“…In the paper we consider the relevant case where is cooperative or, equivalently, it satisfies the Kamke-Müller condition in an open and convex set containing the hypercube [28]. In this case it has been shown in [29] that the semiflow generated by the DVI is monotone and that monotonicity implies some restrictions on the geometry of omega-limit sets of the solutions. However, since we are interested in convergence of solutions, it is known from the monotone dynamical system theory that stronger properties with respect to monotonicity, as the property that the semiflow is strongly order preserving (SOP), are typically needed [28], [30].…”