Three-dimensional chaotic autonomous system with a circular equilibrium: analysis, circuit implementation and its fractional-order form. (English) Zbl 1345.93073
Summary: A three-dimensional autonomous chaotic system with a circular equilibrium is investigated in this paper. Some dynamical properties and behaviors of this system are described in terms of equilibria, eigenvalue structures, bifurcation diagrams, Lyapunov exponents, time series and phase portraits. For specific parameters, the system displays periodic and chaotic attractors. The physical existence of the chaotic behavior found in the proposed system is verified by using the Orcad-PSpice software and experimental verification. A good qualitative agreement is shown between the experimental results, PSpice and numerical simulations. Furthermore, the commensurate fractional-order version of the system with a circular equilibrium is numerically studied. It is found that chaos exists in this system with order less than three. By tuning the commensurate fractional order, the system with a circular equilibrium displays chaotic and periodic attractors, respectively. Finally, chaos synchronization of identical fractional-order chaotic systems with a circular equilibrium is achieved by using the unidirectional linear error feedback coupling. It is shown that the fractional-order chaotic system can achieve synchronization for appropriate coupling strength.
MSC:
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 93C10 | Nonlinear systems in control theory |
| 34H10 | Chaos control for problems involving ordinary differential equations |
| 34A08 | Fractional ordinary differential equations |
Keywords:
three-dimensional autonomous chaotic system; circular equilibrium; stability analysis; circuit implementation; fractional-order system; chaos synchronizationReferences:
| [1] | B. Blażejczyk-Okolewska, T. Kapitaniak, Co-existing attractors of impact oscillator. Chaos Solitons Fractals 9(8), 1439-1443 (1998) · Zbl 0942.37040 · doi:10.1016/S0960-0779(98)00164-7 |
| [2] | V. Bragin, V. Vagaitsev, N.V. Kuznetsov, G.A. Leonov, Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits. J. Comput. Syst. Sci. Int 50(4), 511-543 (2011) · Zbl 1266.93072 · doi:10.1134/S106423071104006X |
| [3] | R. Caponetto, Fractional Order Systems: Modeling and Control Applications, vol. 72 (World Scientific, Singapore, 2010) |
| [4] | G. Chen, T. Ueta, Yet another chaotic attractor. Int. J. Bifurc. Chaos 9(07), 1465-1466 (1999) · Zbl 0962.37013 · doi:10.1142/S0218127499001024 |
| [5] | A. Chudzik, P. Perlikowski, A. Stefanski, T. Kapitaniak, Multistability and rare attractors in van der Pol-Duffing oscillator. Int. J. Bifurc. Chaos 21(07), 1907-1912 (2011) · Zbl 1248.34039 · doi:10.1142/S0218127411029513 |
| [6] | W. Deng, Short memory principle and a predictor-corrector approach for fractional differential equations. J. Comput. Appl. Math. 206(1), 174-188 (2007) · Zbl 1121.65128 · doi:10.1016/j.cam.2006.06.008 |
| [7] | K. Diethelm, N.J. Ford, A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1-4), 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341 |
| [8] | A. Elwakil, Fractional-order circuits and systems: an emerging interdisciplinary research area. Circuits Syst. Mag. IEEE 10(4), 40-50 (2010) · doi:10.1109/MCAS.2010.938637 |
| [9] | I.V. Ermakov, S.T. Kingni, V.Z. Tronciu, J. Danckaert, Chaotic semiconductor ring lasers subject to optical feedback: applications to chaos-based communications. Opt. Commun. 286, 265-272 (2013) · doi:10.1016/j.optcom.2012.08.063 |
| [10] | S. Faraji, M. Tavazoei, The effect of fractionality nature in differences between computer simulation and experimental results of a chaotic circuit. Open Phys. 11(6), 836-844 (2013) · doi:10.2478/s11534-013-0255-8 |
| [11] | M. Feki, An adaptive feedback control of linearizable chaotic systems. Chaos Solitons Fractals 15(5), 883-890 (2003) · Zbl 1042.93510 · doi:10.1016/S0960-0779(02)00203-5 |
| [12] | A. Fichera, C. Losenno, A. Pagano, Clustering of chaotic dynamics of a lean gas-turbine combustor. Appl. Energy 69(2), 101-117 (2001) · doi:10.1016/S0306-2619(00)00067-2 |
| [13] | T.J. Freeborn, B. Maundy, A.S. Elwakil, Measurement of supercapacitor fractional-order model parameters from voltage-excited step response. Emerg. Sel. Top. Circuits Syst. IEEE J. 3(3), 1-10 (2013) · Zbl 1299.94121 · doi:10.1109/JETCAS.2013.2280430 |
| [14] | T. Gotthans, J. Petržela, New class of chaotic systems with circular equilibrium. Nonlinear Dyn. 81(3), 1143-1149 (2015) · doi:10.1007/s11071-015-2056-7 |
| [15] | M. Inoue, A. Nagayoshi, A chaos neuro-computer. Phys. Lett. A 158(8), 373-376 (1991) · doi:10.1016/0375-9601(91)90677-Z |
| [16] | S. Jafari, J.C. Sprott, S.M.R.H. Golpayegani, Elementary quadratic chaotic flows with no equilibria. Phys. Lett. A 377(9), 699-702 (2013) · Zbl 1428.34059 · doi:10.1016/j.physleta.2013.01.009 |
| [17] | S. Jafari, J.C. Sprott, Simple chaotic flows with a line equilibrium. Chaos Solitons Fractals 57, 79-84 (2013) · Zbl 1355.37056 · doi:10.1016/j.chaos.2013.08.018 |
| [18] | S.T. Kingni, L. Keuninckx, P. Woafo, G. Van der Sande, J. Danckaert, Dissipative chaos, Shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: theory and electronic implementation. Nonlinear Dyn. 73(1-2), 1111-1123 (2013) · Zbl 1281.34069 · doi:10.1007/s11071-013-0856-1 |
| [19] | S. Kingni, S. Jafari, H. Simo, P. Woafo, Three-dimensional chaotic autonomous system with only one stable equilibrium: analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form. Eur. Phys. J. Plus 129(5), 1-16 (2014) · doi:10.1140/epjp/i2014-14076-4 |
| [20] | S.T. Kingni, G.S.M. Ngueuteu, P. Woafo, Bursting generation mechanism in a three-dimensional autonomous system, chaos control, and synchronization in its fractional-order form. Nonlinear Dyn. 76(2), 1169-1183 (2014) · doi:10.1007/s11071-013-1200-5 |
| [21] | M.S. Krishna, S. Das, K. Biswas, B. Goswami, Fabrication of a fractional order capacitor with desired specifications: a study on process identification and characterization. Electron Devices IEEE Trans. 58(11), 4067-4073 (2011) · doi:10.1109/TED.2011.2166763 |
| [22] | N.V. Kuznetsov, G.A. Leonov, S. Seledzhi, Hidden oscillations in nonlinear control systems. IFAC Proc. Vol. 18(1), 2506-2510 (2011) · Zbl 0448.58028 |
| [23] | G.A. Leonov, N.V. Kuznetsov, V. Vagaitsev, Analytical-numerical method for attractor localization of generalized Chua’s system. IFAC Proc. Vol. 4(1), 29-33 (2010) |
| [24] | G.A. Leonov, N.V. Kuznetsov, Algorithms for searching for hidden oscillations in the Aizerman and Kalman problems, in Doklady Mathematics, vol. 1 (Springer, Berlin, 2011), pp. 475-481 · Zbl 1247.34063 |
| [25] | G.A. Leonov, N.V. Kuznetsov, Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems. IFAC Proc. Vol. 18(1), 2494-2505 (2011) |
| [26] | G.A. Leonov, N.V. Kuznetsov, Hidden oscillations in dynamical systems. 16 Hilbert’s problem, Aizerman’s and Kalman’s conjectures, hidden attractors in Chua’s circuits. J. Math. Sci. 201(5), 645-662 (2014) · Zbl 1338.37045 · doi:10.1007/s10958-014-2017-6 |
| [27] | G.A. Leonov, N.V. Kuznetsov, M. Kiseleva, E. Solovyeva, A. Zaretskiy, Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor. Nonlinear Dyn. 77(1-2), 277-288 (2014) · doi:10.1007/s11071-014-1292-6 |
| [28] | G.A. Leonov, N.V. Kuznetsov, V. Vagaitsev, Localization of hidden Chua’s attractors. Phys. Lett. A 375(23), 2230-2233 (2011) · Zbl 1242.34102 · doi:10.1016/j.physleta.2011.04.037 |
| [29] | G.A. Leonov, N.V. Kuznetsov, V. Vagaitsev, Hidden attractor in smooth Chua systems. Phys. D Nonlinear Phenom. 241(18), 1482-1486 (2012) · Zbl 1277.34052 · doi:10.1016/j.physd.2012.05.016 |
| [30] | G.A. Leonov, N.V. Kuznetsov, Analytical-numerical methods for hidden attractors’ localization: the 16th Hilbert problem, Aizerman and Kalman conjectures, and Chua circuits, in Numerical Methods for Differential Equations, Optimization, and Technological Problems (Springer, Berlin, 2013), pp. 41-64 · Zbl 1267.65200 |
| [31] | G.A. Leonov, N.V. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int. J. Bifurc. Chaos 23(01), 1330002 (2013) · Zbl 1270.34003 · doi:10.1142/S0218127413300024 |
| [32] | A.Y.T. Leung, X.-F. Li, Y.-D. Chu, X.-B. Rao, Synchronization of fractional-order chaotic systems using unidirectional adaptive full-state linear error feedback coupling. Nonlinear Dyn. 82(1), 185-199 (2015) · Zbl 1348.93168 · doi:10.1007/s11071-015-2148-4 |
| [33] | C. Li, J.C. Sprott, Chaotic flows with a single nonquadratic term. Phys. Lett. A 378(3), 178-183 (2014) · Zbl 1396.35002 · doi:10.1016/j.physleta.2013.11.004 |
| [34] | E.N. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130-141 (1963) · Zbl 1417.37129 · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 |
| [35] | J. Lü, G. Chen, A new chaotic attractor coined. Int. J. Bifurc. Chaos 12(03), 659-661 (2002) · Zbl 1063.34510 · doi:10.1142/S0218127402004620 |
| [36] | B. Maundy, A. Elwakil, S. Gift, On a multivibrator that employs a fractional capacitor. Analog Integr. Circuits Signal Process. 62(1), 99-103 (2010) · doi:10.1007/s10470-009-9329-3 |
| [37] | M. Molaie, S. Jafari, J.C. Sprott, S.M.R.H. Golpayegani, Simple chaotic flows with one stable equilibrium. Int. J. Bifurc. Chaos 23(11), 1350188 (2013) · Zbl 1284.34064 · doi:10.1142/S0218127413501885 |
| [38] | Y. Nakamura, A. Sekiguchi, The chaotic mobile robot. Robot. Automa. IEEE Trans. 17(6), 898-904 (2001) · doi:10.1109/70.976022 |
| [39] | G.M. Ngueuteu, P. Woafo, Dynamics and synchronization analysis of coupled fractional-order nonlinear electromechanical systems. Mech. Res. Commun. 46, 20-25 (2012) · doi:10.1016/j.mechrescom.2012.08.003 |
| [40] | V. Petrov, V. Gaspar, J. Masere, K. Showalter, Controlling chaos in the Belousov-Zhabotinsky reaction. Nature 361(6409), 240-243 (1993) · doi:10.1038/361240a0 |
| [41] | P. Perlikowski, S. Yanchuk, M. Wolfrum, A. Stefanski, P. Mosiolek, T. Kapitaniak, Routes to complex dynamics in a ring of unidirectionally coupled systems. Chaos Interdiscip. J. Nonlinear Sci. 20(1), 013111-013120 (2010) · doi:10.1063/1.3293176 |
| [42] | V.-T. Pham, C. Volos, S. Jafari, Z. Wei, X. Wang, Constructing a novel no-equilibrium chaotic system. Int. J. Bifurc. Chaos 24(05), 1450073 (2014) · Zbl 1296.34114 · doi:10.1142/S0218127414500734 |
| [43] | A.G. Radwan, Stability analysis of the fractional-order \[\text{ RL }\beta \text{ C }\alpha\] RLβCα circuit. J. Fract. Calc. Appl. 3(1), 1-15 (2012) · Zbl 1488.34288 |
| [44] | A.G. Radwan, K.N. Salama, Passive and active elements using fractional circuit. Circuits Syst. I Regul. Pap. IEEE Trans. 58(10), 2388-2397 (2011) · Zbl 1468.94775 · doi:10.1109/TCSI.2011.2142690 |
| [45] | A.G. Radwan, K.N. Salama, Fractional-order RC and RL circuits. Circuits Syst. Signal Process. 31(6), 1901-1915 (2012) · doi:10.1007/s00034-012-9432-z |
| [46] | O.E. Rössler, An equation for continuous chaos. Phys. Lett. A 57(5), 397-398 (1976) · Zbl 1371.37062 · doi:10.1016/0375-9601(76)90101-8 |
| [47] | A. Silchenko, T. Kapitaniak, V. Anishchenko, Noise-enhanced phase locking in a stochastic bistable system driven by a chaotic signal. Phys. Rev. E 59(2), 1593 (1999) · doi:10.1103/PhysRevE.59.1593 |
| [48] | C.P. Silva, Shil’nikov’s theorem—a tutorial. Circuits Syst. I Fundam. Theory Appl. IEEE Trans. 40(10), 675-682 (1993) · Zbl 0850.93352 · doi:10.1109/81.246142 |
| [49] | L.J. Sheu, A speech encryption using fractional chaotic systems. Nonlinear Dyn. 65(1-2), 103-108 (2011) · Zbl 1251.94013 · doi:10.1007/s11071-010-9877-1 |
| [50] | J.C. Sprott, Some simple chaotic flows. Phys. Rev. E 50(2), R647 (1994) · doi:10.1103/PhysRevE.50.R647 |
| [51] | J.C. Sprott, A new class of chaotic circuit. Phys. Lett. A 266(1), 19-23 (2000) · doi:10.1016/S0375-9601(00)00026-8 |
| [52] | J.C. Sprott, Simplest dissipative chaotic flow. Phys. Lett. A 228(4), 271-274 (1997) · Zbl 1043.37504 · doi:10.1016/S0375-9601(97)00088-1 |
| [53] | M.S. Tavazoei, M. Haeri, A necessary condition for double scroll attractor existence in fractional-order systems. Phys. Lett. A 367(1), 102-113 (2007) · Zbl 1209.37037 · doi:10.1016/j.physleta.2007.05.081 |
| [54] | M.C. Tripathy, K. Biswas, S. Sen, A design example of a fractional-order Kerwin-Huelsman-Newcomb biquad filter with two fractional capacitors of different order. Circuits Syst. Signal Process. 32(4), 1523-1536 (2013) · doi:10.1007/s00034-012-9539-2 |
| [55] | X. Wang, G. Chen, Constructing a chaotic system with any number of equilibria. Nonlinear Dyn. 71(3), 429-436 (2013) · doi:10.1007/s11071-012-0669-7 |
| [56] | J. Wang, G. Feng, Bifurcation and chaos in discrete-time BVP oscillator. Int. J. Non Linear Mech. 45(6), 608-620 (2010) · doi:10.1016/j.ijnonlinmec.2009.04.004 |
| [57] | Y. Wang, X. Liao, T. Xiang, K.-W. Wong, D. Yang, Cryptanalysis and improvement on a block cryptosystem based on iteration a chaotic map. Phys. Lett. A 363(4), 277-281 (2007) · Zbl 1197.94209 · doi:10.1016/j.physleta.2006.11.023 |
| [58] | Z. Wei, Dynamical behaviors of a chaotic system with no equilibria. Phys. Lett. A 376(2), 102-108 (2011) · Zbl 1255.37013 · doi:10.1016/j.physleta.2011.10.040 |
| [59] | S. Westerlund, L. Ekstam, Capacitor theory. Dielectr. Electr. Insul. IEEE Trans. 1(5), 826-839 (1994) · doi:10.1109/94.326654 |
| [60] | T. Zhou, G. Chen, Classification of chaos in 3-D autonomous quadratic systems-I: basic framework and methods. Int. J. Bifurc. Chaos 16(09), 2459-2479 (2006) · Zbl 1185.37092 · doi:10.1142/S0218127406016203 |
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