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The Routh–Hurwitz conditions of fractional type in stability analysis of the Lorenz dynamical system

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Abstract

This paper discusses stability conditions and a chaotic behavior of the Lorenz dynamical system involving the Caputo fractional derivative of orders between 0 and 1. Contrary to some existing results on the topic, we study these problems with respect to a general (not specified) value of the Rayleigh number as a varying control parameter. Such a bifurcation analysis is known for the classical Lorenz system; we show that analysis of its fractional extension can yield different conclusions. In particular, we theoretically derive (and numerically illustrate) that nontrivial equilibria of the fractional Lorenz system become locally asymptotically stable for all values of the Rayleigh number large enough, which contradicts the behavior known from the classical case. As a main proof tool, we derive the optimal Routh–Hurwitz conditions of fractional type, i.e., necessary and sufficient conditions guaranteeing that all zeros of the corresponding characteristic polynomial are located inside the Matignon stability sector. Beside it, we perform other bifurcation investigations of the fractional Lorenz system, especially those documenting its transition from stability to chaotic behavior.

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Acknowledgments

The research has been supported by the Project LO1202 (funded by the MEYS under the National Sustainability Programme I). The authors are grateful to the referees for a careful reading of this manuscript and their useful recommendations.

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  1. Institute of Mathematics, Brno University of Technology, Technická 2, CZ-616 69, Brno, Czech Republic

    Jan Čermák & Luděk Nechvátal

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  1. Jan Čermák
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  2. Luděk Nechvátal
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Correspondence to Jan Čermák.

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Čermák, J., Nechvátal, L. The Routh–Hurwitz conditions of fractional type in stability analysis of the Lorenz dynamical system. Nonlinear Dyn 87, 939–954 (2017). https://doi.org/10.1007/s11071-016-3090-9

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  • DOI: https://doi.org/10.1007/s11071-016-3090-9

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