We consider a piecewise smooth 2 × 2 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.

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

BERARDI, MARCO;D'ABBICCO, MARCELLO
2016-01-01

Abstract

We consider a piecewise smooth 2 × 2 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11586/176658
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