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Stability of Solutions to Systems of Nonlinear Differential Equations with Discontinuous Right-Hand Sides: Applications to Hopfield Artificial Neural Networks

Author

Listed:
  • Ilya Boykov

    (Department of Higher and Applied Mathematics, Penza State University, 40, Krasnaya Str., 440026 Penza, Russia)

  • Vladimir Roudnev

    (Department of Computational Physics, Saint Petersburg State University, 1 Ulyanovskaya Str., 198504 Saint Petersburg, Russia)

  • Alla Boykova

    (Department of Higher and Applied Mathematics, Penza State University, 40, Krasnaya Str., 440026 Penza, Russia)

Abstract

In this paper, we study the stability of solutions to systems of differential equations with discontinuous right-hand sides. We have investigated nonlinear and linear equations. Stability sufficient conditions for linear equations are expressed as a logarithmic norm for coefficients of systems of equations. Stability sufficient conditions for nonlinear equations are expressed as the logarithmic norm of the Jacobian of the right-hand side of the system of equations. Sufficient conditions for the stability of solutions of systems of differential equations expressed in terms of logarithmic norms of the right-hand sides of equations (for systems of linear equations) and the Jacobian of right-hand sides (for nonlinear equations) have the following advantages: (1) in investigating stability in different metrics from the same standpoints, we have obtained a set of sufficient conditions; (2) sufficient conditions are easily expressed; (3) robustness areas of systems are easily determined with respect to the variation of their parameters; (4) in case of impulse action, information on moments of impact distribution is not required; (5) a method to obtain sufficient conditions of stability is extended to other definitions of stability (in particular, to p-moment stability). The obtained sufficient conditions are used to study Hopfield neural networks with discontinuous synapses and discontinuous activation functions.

Suggested Citation

  • Ilya Boykov & Vladimir Roudnev & Alla Boykova, 2022. "Stability of Solutions to Systems of Nonlinear Differential Equations with Discontinuous Right-Hand Sides: Applications to Hopfield Artificial Neural Networks," Mathematics, MDPI, vol. 10(9), pages 1-16, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1524-:d:807516
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    References listed on IDEAS

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    1. Kaihong Zhao & Yongkun Li, 2009. "Robust Stability Analysis of Fuzzy Neural Network with Delays," Mathematical Problems in Engineering, Hindawi, vol. 2009, pages 1-13, February.
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    Cited by:

    1. Ilya Boykov & Vladimir Roudnev & Alla Boykova, 2022. "Approximate Methods for Solving Problems of Mathematical Physics on Neural Hopfield Networks," Mathematics, MDPI, vol. 10(13), pages 1-22, June.
    2. Vikneswari Someetheram & Muhammad Fadhil Marsani & Mohd Shareduwan Mohd Kasihmuddin & Nur Ezlin Zamri & Siti Syatirah Muhammad Sidik & Siti Zulaikha Mohd Jamaludin & Mohd. Asyraf Mansor, 2022. "Random Maximum 2 Satisfiability Logic in Discrete Hopfield Neural Network Incorporating Improved Election Algorithm," Mathematics, MDPI, vol. 10(24), pages 1-29, December.
    3. Arsen Palestini, 2022. "Preface to the Special Issue “Mathematical Modeling with Differential Equations in Physics, Chemistry, Biology, and Economics”," Mathematics, MDPI, vol. 10(10), pages 1-2, May.

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