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A Critical Case for the Spiral Stability for \({2\times2}\) Discontinuous Systems and an Application to Recursive Neural Networks

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Abstract

We consider a piecewise smooth \({2\times2}\) 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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Authors and Affiliations

  1. Istituto di Ricerca sulle Acque, Consiglio Nazionale delle Ricerche, Viale F. de Blasio, 5, 70132, Bari, Italy

    Marco Berardi

  2. Departamento de Computação e Matemática, Universidade de São Paulo (USP), FFCLRP, Av. dos Bandeirantes, 3900, Ribeirão Preto, SP, CEP 14040-901, Brazil

    Marcello D’Abbicco

Authors
  1. Marco Berardi
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  2. Marcello D’Abbicco
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Correspondence to Marcello D’Abbicco.

Additional information

M. D’Abbicco is supported by São Paulo Research Foundation (FAPESP), Grants 2013/15140-2 and 2014/02713-7.

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Berardi, M., D’Abbicco, M. A Critical Case for the Spiral Stability for \({2\times2}\) Discontinuous Systems and an Application to Recursive Neural Networks. Mediterr. J. Math. 13, 4829–4844 (2016). https://doi.org/10.1007/s00009-016-0778-5

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  • DOI: https://doi.org/10.1007/s00009-016-0778-5

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