Advertisement

Nonlinear Dynamics

, Volume 69, Issue 1–2, pp 275–284 | Cite as

Hopf bifurcation and chaos in fractional-order modified hybrid optical system

  • Mohammed-Salah Abdelouahab
  • Nasr-Eddine Hamri
  • Junwei Wang
Original Paper

Abstract

In this paper, a chaotic fractional-order modified hybrid optical system is presented. Some basic dynamical properties are further investigated by means of Poincaré mapping, parameter phase portraits, and the largest Lyapunov exponents. Fractional Hopf bifurcation conditions are proposed; it is found that Hopf bifurcation occurs on the proposed system when the fractional-order varies and passes a sequence of critical values. The chaotic motion is validated by the positive Lyapunov exponent. Finally, some numerical simulations are also carried out to illustrate our results.

Keywords

Fractional system Stability Hopf bifurcation Chaos 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bagley, R.L., Calico, R.A.: Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control Dyn. 14, 304–311 (1991) CrossRefGoogle Scholar
  2. 2.
    Sun, H.H., Abdelwahab, A.A., Onaral, B.: Linear approximation of transfer function with a pole of fractional order. IEEE Trans. Autom. Control 29, 441–444 (1984) zbMATHCrossRefGoogle Scholar
  3. 3.
    Ichise, M., Nagayanagi, Y., Kojima, T.: An analog simulation of noninteger order transfer functions for analysis of electrode process. J. Electroanal. Chem. 33, 253–265 (1971) CrossRefGoogle Scholar
  4. 4.
    Heaviside, O.: Electromagnetic Theory. Chelsea, New York (1971) Google Scholar
  5. 5.
    Kusnezov, D., Bulgac, A., Dang, G.D.: Quantum Levy processes and fractional kinetics. Phys. Rev. Lett. 82, 1136–1139 (1999) CrossRefGoogle Scholar
  6. 6.
    Julio, C., Gutiérrez, V., Sheng: Fractionalization of optical beams: I. Planar analysis. Opt. Lett. 32(11), 1521–1523 (2007) CrossRefGoogle Scholar
  7. 7.
    Namias, V.: The fractional Fourier transform and its application in quantum mechanics. J. Inst. Math. Appl. 25, 241–265 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ozaktas, H., Mendlovic, D.: Fractional Fourier transforms and their optical implementation: II. J. Opt. Soc. Am. A 10(9), 1975–1981 (1993) Google Scholar
  9. 9.
    Ozaktas, H., Alevsky, Z., Kutay, M.A.: The Fractional Fourier Transform. Wiley, New York (2001) Google Scholar
  10. 10.
    Chen, D., Chen, Y.Q., Sheng, H.: Fractional variational optical flow model for motion estimation. In: The 4th IFAC Workshop Fractional Differentiation and Its Application, Badajoz, Spain, Oct., pp. 18–20 (2010) Google Scholar
  11. 11.
    Julio, C., Gutiérrez, V., Carlos, L.-M.: Nondiffracting vortex beams with continuous orbital angular momentum order dependence. J. Opt. A, Pure Appl. Opt. 10, 1–8 (2008) Google Scholar
  12. 12.
    Chen, Y., Otis, L., Zhub, Q.: Polarization memory effect in optical coherence tomography and dental imaging application. J. Biomed. Opt. 16(8), 1–7 (2011) zbMATHCrossRefGoogle Scholar
  13. 13.
    Hui, K.C., Lai, C.W., Ong, H.C.: Electron-beam-induced optical memory effect in metallized ZnO thin films for the application of optical storage. Thin Solid Films 483(1–2), 222–225 (2005) CrossRefGoogle Scholar
  14. 14.
    Abdelouahab, M., Hamri, N.: A new chaotic attractor from hybrid optical bistable system. Nonlinear Dyn. (2011). doi: 10.1007/s11071-011-9994-5 Google Scholar
  15. 15.
    Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astron. Soc. 13, 529–539 (1967) CrossRefGoogle Scholar
  16. 16.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) zbMATHGoogle Scholar
  17. 17.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordan and Breach, Amsterdam (1993) zbMATHGoogle Scholar
  18. 18.
    Butzer, P.L., Westphal, U.: An introduction to fractional calculus. In: Hilfer, R. (ed.) Applications of Fractional Calculus in Physics, pp. 1–85. World Scientific, Singapore (2000) CrossRefGoogle Scholar
  19. 19.
    Matignon, D.: Stability results in fractional differential equation with applications to control processing. In: Proceedings of the Multiconference on Computational Engineering in Systems and Application IMICS. IEEE-SMC, Lile, France, vol. 2, pp. 963–968 (1996) Google Scholar
  20. 20.
    Moze, M., Sabatier, J.: LMI tools for stability analysis of fractional systems. In: Proceedings of ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Long Beach, CA, 24–28 Sep. 2011 (2005) Google Scholar
  21. 21.
    Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A.: On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems. Phys. Lett. A 358, 1–4 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Tavazoei, M.S., Haeri, M., Attari, M., Bolouki, S., Siami, M.: More details on analysis of fractional-order Van der Pol oscillator. J. Vib. Control 15(6), 803–819 (2009) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Tavazoei, M.S., Haeri, M., Attari, M.: A proof for non existence of periodic solutions in time invariant fractional order systems. Automatica 45(8), 1886–1890 (2009) zbMATHCrossRefGoogle Scholar
  24. 24.
    Tavazoei, M.S.: A note on fractional-order derivatives of periodic functions. Automatica 46, 945–948 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985) MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Mohammed-Salah Abdelouahab
    • 1
  • Nasr-Eddine Hamri
    • 1
  • Junwei Wang
    • 2
  1. 1.Institute of Science and TechnologyUniversity Center of MilaMilaAlgeria
  2. 2.Cisco School of InformaticsGuangdong University of Foreign StudiesGuangzhouP.R. China

Personalised recommendations