Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ahmad W, El-Khazali R, El-Wakil A (2001) Fractional-order Wien-bridge oscillator. Electr Lett 37:1110–1112
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):1291–1301
Tavazoei MS, Haeri M (2008) Regular oscillations or chaos in a fractional order system with any effective dimension. Nonlinear Dynam 54(3):213–222
Radwan AG, El-Wakil AS, Soliman AM (2008) Fractional-order sinusoidal oscillators: design procedure and practical examples. IEEE Trans Circ Syst I 55(7):2051–2063
Tavazoei MS, Haeri M (2008) Chaotic attractors in incommensurate fractional order systems. Physica D 237(20):2628–2637
Hartley TT, Lorenzo CF, Qammer HK (1995) Chaos in a fractional order Chua’s system. IEEE Trans Circ Syst I 42:485–490
Tavazoei MS, Haeri M (2007) A necessary condition for double scroll attractor existence in fractional order systems. Phys Lett A 367(1–2):102–113
Podlubny I (1999) Fractional differential equations. Academic, San Diego
Matignon D (1996) Stability result on fractional differential equations with applications to control processing. In: IMACS-SMC Proceedings. Lille, France, pp 963–968
Daftardar-Gejji V, Jafari H (2007) Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives. J Math Anal Appl 328:1026–1033
Deng W, Li C, L J (2007) Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynam 48:409–416
Adams RA (2006) Calculus: a complete course, 6th edn. Addison Wesley, Toronto
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Bolouki, S., Haeri, M., Tavazoei, M. ., Siami, M. (2010). Some Bounds on Maximum Number of Frequencies Existing in Oscillations Produced by Linear Fractional Order Systems. In: Baleanu, D., Guvenc, Z., Machado, J. (eds) New Trends in Nanotechnology and Fractional Calculus Applications. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3293-5_17
Download citation
DOI: https://doi.org/10.1007/978-90-481-3293-5_17
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-3292-8
Online ISBN: 978-90-481-3293-5
eBook Packages: EngineeringEngineering (R0)