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Hidden chaotic attractors in fractional-order systems

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Abstract

In this paper, we present a scheme for uncovering hidden chaotic attractors in nonlinear autonomous systems of fractional order. The stability of equilibria of fractional-order systems is analyzed. The underlying initial value problem is numerically integrated with the predictor-corrector Adams-Bashforth-Moulton algorithm for fractional-order differential equations. Three examples of fractional-order systems are considered: a generalized Lorenz system, the Rabinovich-Fabrikant system and a non-smooth Chua system.

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Notes

  1. Note that even fractional-order dynamics allow to describe a real object more accurately than classical “integer-order” dynamics, as proved recently for the existence of stable cycles in systems of fractional order to be impossible [35, 36].

  2. Recently, based on philosophical arguments rather than a mathematical point of view, some researchers questioned the appropriateness of using initial conditions of the classical form in the Caputo derivative [41]. However, it should be emphasized that, in practical (physical) problems, physically interpretable initial conditions are necessary and Caputo’s derivative is a fully justified tool [42].

  3. The geometric multiplicity represents the dimension of the eigenspace of the corresponding eigenvalues.

  4. In many cases, one can simplify this procedure and consider instead a path in the space of parameters, such that the starting point of the path corresponds to a self-excited attractor.

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Acknowledgements

MF Danca is supported by Tehnic B SRL.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Avram Iancu University, 400380, Cluj-Napoca, Romania

    Marius-F. Danca

  2. Romanian Institute for Science and Technology, 400487, Cluj-Napoca, Romania

    Marius-F. Danca

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Danca, MF. Hidden chaotic attractors in fractional-order systems. Nonlinear Dyn 89, 577–586 (2017). https://doi.org/10.1007/s11071-017-3472-7

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