Fractional Rayleigh-Duffing-like system and its synchronization. (English) Zbl 1268.34089

Summary: This paper presents a periodically driven Rayleigh-Duffing-like system with the function \(x|x|\). It is proven via the Melnikov function method that the quadratic function \(x|x|\) induces Smale horseshoes to the Rayleigh-Duffing-like system. The Rayleigh-Duffing-like oscillator with fractional order is also discussed, and results of computer simulation demonstrate the chaotic dynamic behaviors of the system. Furthermore, two fractional Rayleigh-Duffing-like systems are synchronized by active control technology, the method based on state observer and nonlinear feedback method. Numerical results validate the effectiveness and applicability of the proposed synchronization schemes.


34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34D06 Synchronization of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34A08 Fractional ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
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[1] Ahmad, W.M., Sprott, J.C.: Chaos in fractional-order autonomous nonlinear systems. Chaos Solitons Fractals 16, 339–351 (2003) · Zbl 1033.37019
[2] Ahmed, E., EI-Sayed, A.M., EI-Saka, H.: Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J. Math. Anal. Appl. 325, 542–553 (2007) · Zbl 1105.65122
[3] Bhalekar, S., Daftardar-Gejji, V.: Synchronization of different fractional order chaotic systems using active control. Commun. Nonlinear Sci. Numer. Simul. 15, 3536–3546 (2010) · Zbl 1222.94031
[4] Butzer, P.L., Westphal, U.: An Introduction to Fractional Calculus. World Scientific, Singapore (2000) · Zbl 0987.26005
[5] Cafagna, D., Grassi, G.: Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic Rössler systems. Nonlinear Dyn. 68, 117–128 (2012) · Zbl 1243.93047
[6] Chen, D.Y., Liu, Y.X., Ma, X.Y., Zhang, R.F.: Control of a class of fractional-order chaotic systems via sliding mode. Nonlinear Dyn. 67, 893–901 (2012) · Zbl 1242.93027
[7] Deng, W., Li, C.P.: Chaos synchronization of the fractional Lü system. Physica A 353, 61–72 (2005)
[8] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002) · Zbl 1014.34003
[9] Diethelm, K., Freed, A.D., Ford, N.J.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1–4), 3–22 (2002) · Zbl 1009.65049
[10] Ge, Z.M., Zhang, A.R.: Chaos in a modified van der Pol system and in its fractional order systems. Chaos Solitons Fractals 32(5), 1791–1822 (2007)
[11] Hegazi, A.S., Matouk, A.E.: Dynamical behaviors and synchronization in the fractional order hyperchaotic Chen system. Appl. Math. Lett. 24, 1938–1944 (2011) · Zbl 1234.34036
[12] Hifer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2001)
[13] Li, C.P., Deng, W.H., Xu, D.: Chaos synchronization of the Chua system with a fractional order. Physica A 360, 171–185 (2006)
[14] Li, C.P., Peng, G.J.: Chaos in Chen’s system with a fractional order. Chaos Solitons Fractals 22, 443–450 (2004) · Zbl 1060.37026
[15] Li, C.G., Chen, G.: Chaos and hyperchaos in the fractional-order Rössler equations. Physica A 341, 55–61 (2004)
[16] Mahmouda, G.M., Mohameda, A.A., Alyb, S.A.: Strange attractors and chaos control in periodically forced complex Duffing’s oscillators. Physica A 292, 193–206 (2001) · Zbl 0972.37054
[17] Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems Applications, IMACS, IEEE-SMC, Lille, France, vol. 2, pp. 963–968 (1996)
[18] Pecora, L., Carroll, T.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(2), 821–824 (1990) · Zbl 0938.37019
[19] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[20] Siewe, M.S., Tchawoua, C., Woafo, P.: Melnikov chaos in a periodically driven Rayleigh–Duffing oscillator. Mech. Res. Commun. 37(4), 363–368 (2010) · Zbl 1272.70116
[21] Song, L., Yang, J.Y., Xu, S.Y.: Chaos synchronization for a class of nonlinear oscillators with fractional order. Nonlinear Anal. 72, 2326–2336 (2010) · Zbl 1187.34066
[22] Tang, K.-S., Man, K.F., Zhong, G.-Q., Chen, G.R.: Generating chaos via x|x|. IEEE Trans. Circuits Syst. I 48(5), 636–641 (2001) · Zbl 1010.34033
[23] Tavazoei, M.S., Haeri, M.: Synchronization of chaotic fractional-order systems via active sliding mode controller. Physica A 387, 57–70 (2008)
[24] Wu, X.J., Lu, H.T., Shen, S.L.: Synchronization of a new fractional-order hyperchaotic system. Phys. Lett. A 373, 2329–2337 (2009) · Zbl 1231.34091
[25] Wu, X.J., Lu, Y.: Generalized projective synchronization of the fractional-order Chen hyperchaotic system. Nonlinear Dyn. 57, 25–35 (2009) · Zbl 1176.70029
[26] Zeng, C.B., Yang, Q.G., Wang, J.W.: Chaos and mixed synchronization of a new fractional-order system with one saddle and two stable node-foci. Nonlinear Dyn. 65, 457–466 (2011) · Zbl 1286.34016
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