Nonlinear Dynamics

, Volume 95, Issue 1, pp 479–493 | Cite as

Continuation of periodic solutions for systems with fractional derivatives

  • Pierre ViguéEmail author
  • Christophe Vergez
  • Bruno Lombard
  • Bruno Cochelin
Original Paper


This paper addresses the numerical computation of periodic solutions of nonlinear differential systems involving fractional derivatives. For this purpose, the Harmonic Balance Method and the Asymptotic Numerical Method are combined, generalizing an approach largely followed in non-fractional systems. This enables to perform the continuation of periodic solutions of fractional systems with respect to a system parameter or to the fractional order. In the particular case of a constant fractional order, the results are validated by a successful comparison with an alternative formulation based on the diffusive representation of fractional operators. The new numerical strategy presented here allows to simulate phenomena still lacking from theoretical foundations. For example, the numerical experiments proposed here lead to a bifurcation similar to the Hopf bifurcation, well known in the case of non-fractional systems. Throughout this article, the Weyl derivative is used; its link with the classic Caputo derivative is elucidated.


Nonlinear dynamics Fractional derivative Harmonic Balance Method Periodic solutions Numerical continuation Weyl fractional derivative Asymptotic Numerical Method 



This work has been carried out in the framework of the Labex MEC (ANR-10-LABX-0092) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the Investissements d’Avenir French Government program managed by the French National Research Agency (ANR).


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Pierre Vigué
    • 1
    Email author
  • Christophe Vergez
    • 1
  • Bruno Lombard
    • 1
  • Bruno Cochelin
    • 1
  1. 1.CNRS, Centrale Marseille, LMAAix Marseille UnivMarseilleFrance

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