A Critical Case for the Spiral Stability for 2 x 2 Discontinuous Systems and an Application to Recursive Neural Networks

Author(s)

Berardi, Marco ; D'Abbicco, Marcello

• Document type: Journal article
• Source: Mediterranean Journal of Mathematics; v. 13, n. 6, p. 4829-4844, DEC 2016.
• Web of Science Citations: 0

Abstract

We consider a piecewise smooth system, whose solutions locally spirally move around an equilibrium point which lies at the intersection of two discontinuity surfaces. We find a sufficient condition for the stability of this point, in the limit case in which a first-order approximation theory does not give an answer. This condition, depending on the vector field and its Jacobian evaluated at the equilibrium point, is trivially satisfied for piecewise-linear systems, whose first-order part is a diagonal matrix with negative entries. We show how our stability results may be applied to discontinuous recursive neural networks for which the matrix of self-inhibitions of the neurons does not commute with the connection weight matrix. In particular, we find a nonstandard relation between the ratio of the self-inhibition speeds and the structure of the connection weight matrix, which determines the stability. (AU)

• FAPESP's process: 13/15140-2 - Decay estimates for semilinear hyperbolic equations
• Grantee: Marcello Dabbicco
• Support Opportunities: Research Grants - Young Investigators Grants
• FAPESP's process: 14/02713-7 - Decay estimates for semilinear hyperbolic equations
• Grantee: Marcello Dabbicco
• Support Opportunities: Scholarships in Brazil - Young Researchers