Nonlinear Dynamics

, Volume 92, Issue 3, pp 1061–1078 | Cite as

Fractional-order PWC systems without zero Lyapunov exponents

  • Marius-F. DancaEmail author
  • Michal Fečkan
  • Nikolay V. Kuznetsov
  • Guanrong Chen
Original Paper


In this paper, it is shown numerically that a class of fractional-order piece-wise continuous systems, which depend on a single real bifurcation parameter, have no zero Lyapunov exponents but can be chaotic or hyperchaotic with hidden attractors. Although not analytically proved, this conjecture is verified on several systems including a fractional-order piece-wise continuous hyperchaotic system, a piece-wise continuous chaotic Chen system, a piece-wise continuous variant of the chaotic Shimizu-Morioka system and a piece-wise continuous chaotic Sprott system. These systems are continuously approximated based on results of differential inclusions and selection theory, and numerically integrated with the Adams-Bashforth-Moulton method for fractional-order differential equations. It is believed that the obtained results are valid for many, if not most, fractional-order PWC systems.


Fractional-order piece-wise continuous system Continuous approximation Lyapunov exponent Chaotic system Hyperchaotic system 



Fractional order


Fractional-order differential equation


Differential inclusion


Piece-wise continuous


Initial value problem



List of symbols


Integration step-size (for ABM method)


Normalization step-size (for Lyapunov exponents)


Caputo’s derivative with 0 as starting point



The study of hidden attractors was done by M.-F. Danca, N. Kuznetsov and G. Chen within the RSF Project (14-21-00041). M. Fečkan is also supported in part by the Slovak Research and Development Agency under the Contract No. APVV-14-0378 and by the Slovak Grant Agency VEGA Nos. 2/0153/16 and 1/0078/17. M.-F. Danca thanks Vlad–Marius Griguta of University of Manchester for helping with some computations.

Supplementary material


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Copyright information

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Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceAvram Iancu UniversityCluj-NapocaRomania
  2. 2.Romanian Institute for Science and TechnologyCluj-NapocaRomania
  3. 3.Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and InformaticsComenius University in Bratislava, Mlynská dolinaBratislavaSlovak Republic
  4. 4.Mathematical Institute, Slovak Academy of SciencesBratislavaSlovak Republic
  5. 5.Department of Applied CyberneticsSaint-Petersburg State UniversitySaint PetersburgRussia
  6. 6.Department of Mathematical Information TechnologyUniversity of JyväskyläJyväskyläFinland
  7. 7.Department of Electronic EngineeringCity University of Hong KongHong KongChina

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