Some Bounds on Maximum Number of Frequencies Existing in Oscillations Produced by Linear Fractional Order Systems



In this paper, it has been studied that how the inner dimension of a fractional order system influences the maximum number of frequencies which may exist in oscillations produced by this system. Both commensurate and incommensurate systems have been considered to clarify the relationship between the inner dimension and maximum number of frequencies. It has been shown that although in commensurate fractional order systems, like integer order systems, the maximum number of frequencies is half of the inner dimension, in incommensurate systems the problem is significantly more complicated and the relationship between the inner dimension and maximum frequencies can not precisely determined. However, in this article, some upper and lower bounds, depending on the inner dimension, have been provided for the maximum frequencies of incommensurate systems.


Stability Boundary Order System Fractional Order System Undamped Oscillation Integer Order System 
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  1. 1.
    Ahmad W, El-Khazali R, El-Wakil A (2001) Fractional-order Wien-bridge oscillator. Electr Lett 37:1110–1112CrossRefGoogle Scholar
  2. 2.
    Barbosa RS, Machado JAT, Vingare BM, Calderon AJ (2007) Analysis of the Van der Pol oscillator containing derivatives of fractional order. J Vib Control 13(9–10):1291–1301zbMATHCrossRefGoogle Scholar
  3. 3.
    Tavazoei MS, Haeri M (2008) Regular oscillations or chaos in a fractional order system with any effective dimension. Nonlinear Dynam 54(3):213–222zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Radwan AG, El-Wakil AS, Soliman AM (2008) Fractional-order sinusoidal oscillators: design procedure and practical examples. IEEE Trans Circ Syst I 55(7):2051–2063CrossRefGoogle Scholar
  5. 5.
    Tavazoei MS, Haeri M (2008) Chaotic attractors in incommensurate fractional order systems. Physica D 237(20):2628–2637zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hartley TT, Lorenzo CF, Qammer HK (1995) Chaos in a fractional order Chua’s system. IEEE Trans Circ Syst I 42:485–490CrossRefGoogle Scholar
  7. 7.
    Tavazoei MS, Haeri M (2007) A necessary condition for double scroll attractor existence in fractional order systems. Phys Lett A 367(1–2):102–113CrossRefGoogle Scholar
  8. 8.
    Podlubny I (1999) Fractional differential equations. Academic, San DiegozbMATHGoogle Scholar
  9. 9.
    Matignon D (1996) Stability result on fractional differential equations with applications to control processing. In: IMACS-SMC Proceedings. Lille, France, pp 963–968Google Scholar
  10. 10.
    Daftardar-Gejji V, Jafari H (2007) Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives. J Math Anal Appl 328:1026–1033zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Deng W, Li C, L J (2007) Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynam 48:409–416Google Scholar
  12. 12.
    Adams RA (2006) Calculus: a complete course, 6th edn. Addison Wesley, TorontoGoogle Scholar

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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Advanced Control System Lab., Electrical Engineering DepartmentSharif University of TechnologyTehranIran

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