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Applying Circulant Matrices Properties to Synchronization Problems

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Modern Mathematics and Mechanics

Part of the book series: Understanding Complex Systems ((UCS))

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Abstract

In this chapter, we use circulant matrices to study discrete dynamical systems of higher dimension than one. We show how these matrices are a common framework which is useful to investigate some dynamical properties of some models provided by natural and social sciences. In particular, discrete models from Biology, Economy and Chemistry are considered and analyzed with tools coming from the properties of circulant matrices. More precisely, the special shape of eigenvalues and eigenvectors of circulant matrices is very useful to check whether the dynamics of systems on phase spaces with dimension greater than two can be reduced to that of one dimensional systems.

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Notes

  1. 1.

    It is proved in [9] that the first two conditions in Devaney’s definition implies the third one. The definitions are presented in the original form because of the dynamical meaning of sensitive dependence on initial conditions.

  2. 2.

    Dinaburg [25] gave simultaneously a Bowen like definition for continuous maps on a compact metric space.

  3. 3.

    Two continuous maps f : X → X and g : Y → Y are said to be topologically conjugate if there is an homeomorphism φ : X → Y such that g ∘ φ = φ ∘ f. In general, conjugate maps share many dynamical properties.

  4. 4.

    See also [54] which almost simultaneously states the same result for C 2 diffeomorphisms on compact manifolds of dimension greater than one.

  5. 5.

    Since Smale’s work (see [52]), horseshoes have been in the core of chaotic dynamics, describing what we could call random deterministic systems.

  6. 6.

    A periodic orbit can be an attractor when spectral radius has modulus one, but in general the converse is not true.

  7. 7.

    From now on, we denote the production with the letter “q” instead of “x” because this is the usual notation for that.

  8. 8.

    Another alternatives which do not imply a optimization process can be see e.g. [10] as for instance

    $$\displaystyle \begin{aligned} f_{i}(q_{1},\ldots,q_{n})=q_{i}+\lambda _{i}\varPi _{i}(q_{1},\ldots,q_{n}), \end{aligned}$$

    or

    $$\displaystyle \begin{aligned} f_{i}(q_{1},\ldots,q_{n})=q_{i}+\lambda _{i} \frac{\partial \varPi _{i}}{\partial q_{i}} (q_{1},\ldots,q_{n}). \end{aligned}$$

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Acknowledgements

This work has been supported by the grants MTM2014-52920-P and MTM 2017-84079-P from Ministerio de Economía y Competitividad (Spain).

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  1. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain

    Jose S. Cánovas

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Correspondence to Jose S. Cánovas .

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  1. Lomonosov Moscow State University, Moscow, Russia

    Victor A. Sadovnichiy

  2. National Technical University of Ukraine, “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine

    Michael Z. Zgurovsky

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Cánovas, J.S. (2019). Applying Circulant Matrices Properties to Synchronization Problems. In: Sadovnichiy, V., Zgurovsky, M. (eds) Modern Mathematics and Mechanics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-96755-4_3

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  • DOI: https://doi.org/10.1007/978-3-319-96755-4_3

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