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Algebraic and Symbolic Computation Methods in Dynamical Systems pp 239–258Cite as

A Review on Multiple Purely Imaginary Spectral Values of Time-Delay Systems

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Part of the book series: Advances in Delays and Dynamics ((ADVSDD,volume 9))

Abstract

A standard framework in analyzing time-delay systems consists first, in identifying the associated crossing roots and secondly, then, in characterizing the local bifurcations of such roots with respect to small variations of the system parameters. Moreover, the dynamics of such spectral values are strongly related to their multiplicities (algebraic/geometric). This chapter review some new results by the authors from Boussaada and Niculescu (IEEE Trans Autom Control 61:1601–1606, [1]), Boussaada and Niculescu (Acta Applicandæ Mathematicæ 145(1):47–88, [2]), Boussaada and Niculescu (Proceeding of the 21st International Symposium on Mathematical Theory of Networks and Systems, pp. 1–8, [3]) allowing one to characterize the algebraic multiplicity of a quasipolynomial’s crossing imaginary roots. First, we emphasize the link between the multiplicity characterization and functional Birkhoff matrices. Secondly, we elaborate a constructive bound for the multiplicity of a given crossing imaginary root. It is shown that Pólya-Szegő generic bound is never reached when the crossing frequency is different from zero.

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Notes

  1. 1.

    An equilibrium point is called a Hopf point if the Jacobian at that point has a conjugate pair of purely imaginary spectral values \(\pm i\omega \), \(\omega > 0\). If there are two such pairs \(\pm i\omega _1, \pm i\omega _2\) then it is called a double Hopf point. If additionally, \(\omega _1=\omega _2\) then it is called a 1:1 resonant double Hopf point.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

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Authors and Affiliations

  1. IPSA & Laboratoire des Signaux et Systèmes, CNRS-CentraleSupélec-Université Paris Sud, 3 rue Joliot-Curie, 91192, Gif-sur-Yvette cedex, France

    Islam Boussaada

  2. Laboratoire des Signaux et Systèmes, CNRS-CentraleSupélec-Université Paris Sud, 3 rue Joliot-Curie, 91192, Gif-sur-Yvette cedex, France

    Silviu-Iulian Niculescu

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  1. Islam Boussaada
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  2. Silviu-Iulian Niculescu
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Correspondence to Islam Boussaada .

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Editors and Affiliations

  1. Inria Paris, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne University, Paris, France

    Alban Quadrat

  2. Lehrstuhl D fuer Mathematik, RWTH Aachen University, Aachen, Germany

    Eva Zerz

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Boussaada, I., Niculescu, SI. (2020). A Review on Multiple Purely Imaginary Spectral Values of Time-Delay Systems. In: Quadrat, A., Zerz, E. (eds) Algebraic and Symbolic Computation Methods in Dynamical Systems. Advances in Delays and Dynamics, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-030-38356-5_9

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