# zbMATH — the first resource for mathematics

A new chaotic system with fractional order and its projective synchronization. (English) Zbl 1204.37035
Summary: Based on Rikitake system, a new chaotic system is discussed. Some basic dynamical properties, such as equilibrium points, Lyapunov exponents, fractal dimension, Poincaré map, bifurcation diagrams and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed is a new chaotic system. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the new fractional-order three-dimensional system with order less than 3. The lowest order to yield chaos in this system is 2.733. The results are validated by the existence of one positive Lyapunov exponent and some phase diagrams. Further, based on the stability theory of the fractional-order system, projective synchronization of the new fractional-order chaotic system through designing the suitable nonlinear controller is investigated. The proposed method is rather simple and need not compute the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the presented synchronization scheme.

##### MSC:
 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 34C28 Complex behavior and chaotic systems of ordinary differential equations 34A08 Fractional ordinary differential equations 37N35 Dynamical systems in control 93D15 Stabilization of systems by feedback
##### Software:
Sprott's Software
Full Text:
##### References:
 [1] Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963) · Zbl 1417.37129 [2] Sparrow, C.: The Lorenz Equations: Bifurcation, Chaos, and Strange Attractors. Springer, New York (1982) · Zbl 0504.58001 [3] Stewart, I.: The Lorenz attractor exists. Nature 406, 948–949 (2002) [4] Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976) · Zbl 1371.37062 [5] Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50, R647 (1994) [6] Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999) · Zbl 0962.37013 [7] Lü, J.H., Chen, G.R.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 3, 659–661 (2002) · Zbl 1063.34510 [8] Lü, J.H., Chen, G.R., Cheng, D.Z., Čelikovský, S.: Bridge the gap between the Lorenz and the Chen system. Int. J. Bifurc. Chaos 12, 2917–2926 (2002) · Zbl 1043.37026 [9] Liu, C., Liu, T., Liu, L., Liu, K.: A new chaotic attractor. Chaos Solitons Fractals 22, 1031–1038 (2004) · Zbl 1060.37027 [10] Wang, G., Zhang, X., Zheng, Y., Li, X.: A new modified hyperchaotic Lü system. Physica A 371, 260–272 (2006) [11] Chen, C.H., Sheu, L.J., Chen, H.K., Chen, J.H., Wang, H.C., Chao, Y.C., Lin, Y.K.: A new hyper-chaotic system and its synchronization. Nonlinear Anal. Real World Appl. 10, 2088–2096 (2009) · Zbl 1163.65337 [12] Rikitake, T.: Oscillations of a system of disk dynamos. Proc. Camb. Philos. Soc. 54, 89–95 (1958) · Zbl 0087.23703 [13] Keisuke, I.: Chaos in the Rikitake two-disk dynamo system. Earth Planet. Sci. Lett. 51, 451–457 (1980) [14] Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999) · Zbl 0924.34008 [15] Hifer, R.: Applications of Fractional Calculus in Physics. World Scientific, New Jersey (2001) [16] Koeller, R.C.: Application of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 299–307 (1984) · Zbl 0544.73052 [17] Sun, H.H., Abdelwahad, A.A., Onaral, B.: Linear approximation of transfer function with a pole of fractional order. IEEE Trans. Automat. Control 29, 441–444 (1984) · Zbl 0532.93025 [18] 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) [19] Heaviside, O.: Electromagnetic Theory. Chelsea, New York (1971) · JFM 30.0801.03 [20] Podlubny, I.: Fractional-order systems and PI {$$\lambda$$} D {$$\mu$$} -controllers. IEEE Trans. Automat. Control 44, 208–214 (1999) · Zbl 1056.93542 [21] Li, C.G., Chen, G.: Chaos and hyperchaos in the fractional-order Rössler equations. Physica A 341, 55–61 (2004) [22] 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 [23] Deng, W., Li, C.P.: Chaos synchronization of the fractional Lü system. Physica A 353, 61–72 (2005) [24] Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990) · Zbl 0938.37019 [25] Li, G.H.: Modified projective synchronization of chaotic system. Chaos Solitons Fractals 32, 1786–1790 (2007) · Zbl 1134.37331 [26] Hung, M.L., Yan, J.J., Liao, T.L.: Generalized projective synchronization of chaotic nonlinear gyros coupled with dead-zone input. Chaos Solitons Fractals 35, 181–187 (2008) [27] Li, C.G., Liao, X.F., Yu, J.B.: Synchronization of fractional order chaotic systems. Phys. Rev. E 68, 067203 (2003) [28] Lu, J.G.: Chaotic dynamics and synchronization of fractional-order Arneodo’s systems. Chaos Solitons Fractals 26, 1125–1133 (2005) · Zbl 1074.65146 [29] Li, C., Yan, J.: The synchronization of three fractional differential systems. Chaos Solitons Fractals 32, 751–757 (2007) [30] Tavazoei, M.S., Haeri, M.: Synchronization of chaotic fractional-order systems via active sliding mode controller. Physica A 387, 57–70 (2008) [31] Hardy, Y., Steeb, W.H.: The Rikitake two-disk dynamo system and domains with periodic orbits. Int. J. Theor. Phys. 38, 2413–2417 (1999) · Zbl 0980.86005 [32] Vaněcěk, A., Čelikovský, S.: Control Systems: From Linear Analysis to Synthesis of Chaos. Prentice-Hall, London (1996) · Zbl 0874.93006 [33] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Geophys. J. R. Astron. Soc. 13, 529–539 (1967) [34] Samko, S.G., Klibas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Amsterdam (1993) [35] Keil, F., Mackens, W., Werther, J.: Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties. Springer, Heidelberg (1999) [36] Butzer, P.L., Westphal, U.: An Introduction to Fractional Calculus. World Scientific, Singapore (2000) · Zbl 0987.26005 [37] Ahmad, W.M., Sprott, J.C.: Chaos in fractional-order autonomous nonlinear systems. Chaos Solitons Fractals 16, 339–351 (2003) · Zbl 1033.37019 [38] Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order. Electron. Trans. Numer. Anal. 5, 1–6 (1997) · Zbl 0890.65071 [39] 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) · Zbl 1009.65049 [40] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002) · Zbl 1014.34003 [41] Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31–52 (2004) · Zbl 1055.65098 [42] Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: IEEE-SMC Proceedings, Computational Engineering in Systems and Application Multi-Conference, IMACS, vol. 2, pp. 963–968. Lille, France (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.