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Does Symmetry of the Operator of a Dynamical System Help Stability?

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Abstract

In this article, we address the question of relating the stability properties of an operator with the stability properties of its associate symmetric operator. The linear-algebra results of Bendixson and Hirsch indicate that the symmetric part of a matrix is always less stable than the matrix itself. We show that in a variety of cases, including infinite dimensional cases associated to systems of PDEs, the same result is valid. We also discuss the applicability to non-autonomous systems, and we show that, in general, this result is not valid. We also review some of the literature that in these years has appeared on the subject.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125, Catania, Italy

    P. Falsaperla & G. Mulone

  2. Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121, Padova, Italy

    A. Giacobbe

Authors
  1. P. Falsaperla
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  2. A. Giacobbe
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  3. G. Mulone
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Correspondence to P. Falsaperla.

Additional information

A. Giacobbe and P. Falsaperla have been partially supported by “Progetto Giovani Ricercatori 2011” of GNFM-INDAM, G.M. by the PRA of the University of Catania “Modelli in Fisica Matematica e Stabilità in Fluidodinamica, Termodinamica Estesa e Biomatematica”.

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Falsaperla, P., Giacobbe, A. & Mulone, G. Does Symmetry of the Operator of a Dynamical System Help Stability?. Acta Appl Math 122, 239–253 (2012). https://doi.org/10.1007/s10440-012-9740-0

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  • DOI: https://doi.org/10.1007/s10440-012-9740-0

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