Fractional-order Chua’s system: discretization, bifurcation and chaos. (English) Zbl 1391.39026
Summary: In this paper we are interested in the fractional-order form of Chua’s system. A discretization process will be applied to obtain its discrete version. Fixed points and their asymptotic stability are investigated. Chaotic attractor, bifurcation and chaos for different values of the fractional-order parameter are discussed. We show that the proposed discretization method is different from other discretization methods, such as predictor-corrector and Euler methods, in the sense that our method is an approximation for the right-hand side of the system under study.
MSC:
| 39A33 | Chaotic behavior of solutions of difference equations |
| 39A30 | Stability theory for difference equations |
| 34A08 | Fractional ordinary differential equations |
| 34C28 | Complex behavior and chaotic systems of ordinary differential equations |
Keywords:
Chua’s system; fractional-order differential equations; fixed points; asymptotic stability; chaotic attractor; bifurcation; chaosReferences:
| [1] | Bhalekara, S; Daftardar-Gejjib, V; Baleanuc, D; Magine, R, Transient chaos in fractional Bloch equations, Comput. Math. Appl, 64, 3367-3376, (2012) · Zbl 1268.34009 · doi:10.1016/j.camwa.2012.01.069 |
| [2] | Faieghi, M; Kuntanapreeda, S; Delavari, H; Baleanu, D, LMI-based stabilization of a class of fractional-order chaotic systems, Nonlinear Dyn, 72, 301-309, (2013) · Zbl 1268.93121 · doi:10.1007/s11071-012-0714-6 |
| [3] | Golmankhaneh, AK; Arefi, R; Baleanu, D, The proposed modified Liu system with fractional order, No. 2013, (2013) · Zbl 1298.37017 |
| [4] | Das S: Functional Fractional Calculus for System Identification and Controls. Springer, Berlin; 2007. |
| [5] | El-Sayed, AMA; El-Mesiry, A; El-Saka, H, On the fractional-order logistic equation, Appl. Math. Lett, 20, 817-823, (2007) · Zbl 1140.34302 · doi:10.1016/j.aml.2006.08.013 |
| [6] | El-Sayed, AMA, On the fractional differential equations, J. Appl. Math. Comput, 49, 205-213, (1992) · Zbl 0757.34005 · doi:10.1016/0096-3003(92)90024-U |
| [7] | El-Sayed, AMA, Nonlinear functional-differential equations of arbitrary orders, Nonlinear Anal, 33, 181-186, (1998) · Zbl 0934.34055 · doi:10.1016/S0362-546X(97)00525-7 |
| [8] | Podlubny I: Fractional Differential Equations. Academic Press, London; 1999. · Zbl 0924.34008 |
| [9] | Matouk, AE, Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system, Phys. Lett. A, 373, 2166-2173, (2009) · Zbl 1229.34099 · doi:10.1016/j.physleta.2009.04.032 |
| [10] | Matouk, AE, Chaos synchronization between two different fractional systems of Lorenz family, No. 2009, (2009) · Zbl 1181.37047 |
| [11] | Matouk, AE, On some stability conditions and hyperchaos synchronization in the new fractional order hyperchaotic Chen system, (2009) |
| [12] | Sun, H; Abdelwahed, A; Onaral, B, Linear approximation for transfer function with a pole of fractional order, IEEE Trans. Autom. Control, 29, 441-444, (1984) · Zbl 0532.93025 · doi:10.1109/TAC.1984.1103551 |
| [13] | Diethelm, K; Ford, NJ; Freed, AD, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn, 29, 3-22, (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341 |
| [14] | Tavazoei, MS; Haeri, M, Unreliability of frequency domain approximation in recognizing chaos in fractional order systems, IET Signal Process, 1, 171-181, (2007) · doi:10.1049/iet-spr:20070053 |
| [15] | Tavazoei, MS; Haeri, M, Limitation of frequency domain approximation for detecting chaos in fractional order systems, Nonlinear Anal., Theory Methods Appl, 69, 1299-1320, (2008) · Zbl 1148.65094 · doi:10.1016/j.na.2007.06.030 |
| [16] | Wang, Z, A numerical method for delayed fractional-order differential equations, No. 2013, (2013) · Zbl 1266.65118 |
| [17] | Erjaee, GH, On analytical justification of phase synchronization in different chaotic systems, Chaos Solitons Fractals, 3, 1195-1202, (2009) · Zbl 1197.37027 · doi:10.1016/j.chaos.2007.06.006 |
| [18] | Yan, JP; Li, CP, On chaos synchronization of fractional differential equations, Chaos Solitons Fractals, 32, 725-735, (2007) · Zbl 1132.37308 · doi:10.1016/j.chaos.2005.11.062 |
| [19] | Wu, GC, Baleanu, D: Discrete fractional logistic map and its chaos. Nonlinear Dyn. (2013). doi:10.1007/s11071-013-1065-7. 10.1007/s11071-013-1065-7 |
| [20] | Akhmet, MU, Stability of differential equations with piecewise constant arguments of generalized type, Nonlinear Anal, 68, 794-803, (2008) · Zbl 1173.34042 · doi:10.1016/j.na.2006.11.037 |
| [21] | Akhmet, MU, Altntana, D, Ergenc, T: Chaos of the logistic equation with piecewise constant arguments. arXiv:1006.4753 (2010). e-printatarXiv.org |
| [22] | El-Sayed, AMA; Salman, SM, Chaos and bifurcation of discontinuous dynamical systems with piecewise constant arguments, Malaya J. Mat, 1, 14-18, (2012) · Zbl 1369.39016 |
| [23] | El-Sayed, AMA; Salman, SM, Chaos and bifurcation of the logistic discontinuous dynamical systems with piecewise constant arguments, Malaya J. Mat, 3, 14-20, (2013) · Zbl 1369.39017 |
| [24] | El-Sayed, AMA; Salman, SM, The unified system between Lorenz and Chen systems: a discretization process, Electron. J. Math. Anal. Appl, 1, 318-325, (2013) · Zbl 1417.37291 |
| [25] | El-Sayed, AMA; Salman, SM, On a discretization process of fractional order riccati’s differential equation, J. Fract. Calc. Appl, 4, 251-259, (2013) · Zbl 1488.65210 |
| [26] | El-Sayed, AMA, Salman, SM: On a discretization process of fractional-order Logistic differential equation. J. Egypt. Math. Soc. (accepted) · Zbl 1302.65271 |
| [27] | Stegemann, C; Albuquerque, HA; Rech, PC, Some two dimensional parameter space of a Chua system with cubic nonlinearity, Chaos, Interdiscip. J. Nonlinear Sci, 20, 817-823, (2007) |
| [28] | Hartely, TT; Lorenzo, CF; Qammer, HK, Chaos in a fractional order chau’s system, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl, 42, 485-490, (1995) · doi:10.1109/81.404062 |
| [29] | Elaidy, SN, Undergraduate texts in mathematics, (2005), New York |
| [30] | Holmgren R: A First Course in Discrete Dynamical Systems. Springer, New York; 1994. · Zbl 0797.58001 · doi:10.1007/978-1-4684-0222-3 |
| [31] | Udwadia, FE; von Bremen, H, A note on the computation of the largest \(p\)-Lyapunov characteristic exponents for nonlinear dynamical systems, J. Appl. Math. Comput, 114, 205-214, (2000) · Zbl 1112.93316 · doi:10.1016/S0096-3003(99)00113-7 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
