Nonlinear Dyn (2012) 69:275–284 DOI 10.1007/s11071-011-0263-4 ORIGINAL PAPER Hopf bifurcation and chaos in fractional-order modified hybrid optical system Mohammed-Salah Abdelouahab · Nasr-Eddine Hamri · Junwei Wang Received: 27 July 2011 / Accepted: 1 November 2011 / Published online: 23 November 2011 © Springer Science+Business Media B.V. 2011 Abstract In this paper, a chaotic fractional-order modified hybrid optical system is presented. Some ba- sic dynamical properties are further investigated by means of Poincaré mapping, parameter phase por- traits, 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 numer- ical simulations are also carried out to illustrate our results. Keywords Fractional system · Stability · Hopf bifurcation · Chaos 1 Introduction The idea of fractional calculus has been known since the development of the regular calculus, and it means a generalization of integration and differentiation to M.-S. Abdelouahab ( ) · N.-E. Hamri Institute of Science and Technology, University Center of Mila, Mila 43000, Algeria e-mail: medsalah3@yahoo.fr J. Wang Cisco School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510006, P.R. China arbitrary order. It has been found that many systems in interdisciplinary fields can be described by the frac- tional differential equations, such as viscoelastic sys- tems, dielectric polarization, electrode-electrolyte po- larization, electromagnetic waves, and quantum evo- lution of complex systems [1–5]. Optics is a field in which the use of conventional calculus plays a major role, and it is of interest to see how fractional calculus may offer useful mathemati- cal tools in this field. For example; fractionalization of Gaussian beams is given in [6], fractionalization of the Fourier transform and its applications has been already studied by several researchers [7–9], a fractional vari- ational optical flow model is introduced in [10], and a new class of nondiffracting fractional vortex beams that connect Bessel beams of successive order in a smooth transition is introduced in [11]. On the other hand, memory effect has been observed in optical sys- tems [12, 13]; this fact makes fractional modeling ap- propriate for dynamic behaviors in optical systems. Based on the above motivations, one might be tempted to introduce the fractional-order version of the modi- fied hybrid optical system presented in our previous work [14]. There are several definitions of fractional deriva- tives [15–18]. In this paper, we use the Caputo-type fractional derivative defined in [15] by