Dynamics, circuit realization, control and synchronization of a hyperchaotic hyperjerk system with coexisting attractors. (English) Zbl 1375.94180
Summary: Although different hyperjerk systems have been discovered, a few hyperjerk systems can exhibit hyperchaotic behavior. In this work, we introduce a new hyperjerk system with hyperchaotic attractors. By investigating dynamics of the system, we have observed the different coexisting attractors such as coexistence of period-2 attractors, or coexistence of period-2 attractor and quasiperiodic attractor. It is worth noting that this striking phenomenon is rarely reported in a hyperjerk system. The proposed system has been realized with electronic components. The agreement between the simulation and experimental results indicates the feasibility of the hyperjerk system. Moreover, chaos control and synchronization of such hyperjerk system have been also reported.
MSC:
| 94C05 | Analytic circuit theory |
| 93C40 | Adaptive control/observation systems |
| 93B52 | Feedback control |
| 93D30 | Lyapunov and storage functions |
Keywords:
hyperchaos; hyperjerk system; quasiperiodic dynamics; coexisting attractor; circuit; control; synchronizationReferences:
| [1] | Schot, S.: Jerk: the time rate of change of acceleration. Am. J. Phys. 46, 1090-1094 (1978) · doi:10.1119/1.11504 |
| [2] | Coullet, P., Tresser, C., Arneodo, A.: A transition to stochasticity for a class of forced oscillators. Phys. Lett. A 72, 268-270 (1979) · Zbl 0985.37501 · doi:10.1016/0375-9601(79)90464-X |
| [3] | Linz, S.J.: Nonlinear dynamical models and jerky motion. Am. J. Phys. 65, 523-526 (1997) · doi:10.1119/1.18594 |
| [4] | Sprott, J.C.: Some simple chaotic jerk functions. Am. J. Phys. 65, 537-543 (1997) · doi:10.1119/1.18585 |
| [5] | Eichhorn, R., Linz, S.J., Hanggi, P.: Simple polynomial classes of chaotic jerky dynamics. Chaos Solitons Fractals 13, 1-15 (2002) · Zbl 0993.37019 · doi:10.1016/S0960-0779(00)00237-X |
| [6] | Malasoma, J.M.: What is the simplest dissipative chaotic jerk equation which is parity invariant. Phys. Lett. A 264, 383-389 (2000) · Zbl 0947.34031 · doi:10.1016/S0375-9601(99)00819-1 |
| [7] | Lainscsek, C., Lettellier, C., Gorodnitsky, I.: Global modeling of the rössler system from the \[z\] z-variable. Phys. Lett. A 314, 409-427 (2003) · Zbl 1038.37063 · doi:10.1016/S0375-9601(03)00912-5 |
| [8] | Ma, J., Wu, X., Chu, R., Zhang, L.: Selection of multi-scroll attractors in Jerk circuits and their verification using Pspice. Nonlinear Dyn. 76, 1951-1962 (2014) · doi:10.1007/s11071-014-1260-1 |
| [9] | Louodop, P., Kountchou, M., Fotsin, H., Bowong, S.: Practical finite-time synchronization of jerk systems: theory and experiment. Nonlinear Dyn. 78, 597-607 (2014) · Zbl 1278.34046 · doi:10.1007/s11071-014-1463-5 |
| [10] | Kengne, J., Njitacke, Z.T., Fotsin, H.: Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dyn. 83, 751-765 (2016) · Zbl 1343.34115 · doi:10.1007/s11071-015-2364-y |
| [11] | Ma, J., Wu, F., Ren, G., Tang, J.: A class of initials-dependent dynamical systems. Appl. Math. Comput. 298, 65-76 (2017) · Zbl 1411.37035 |
| [12] | Wang, C., Chu, R., Ma, J.: Controlling a chaotic resonator by means of dynamics track control. Complexity 21, 370-378 (2015) · doi:10.1002/cplx.21572 |
| [13] | Elhadj, Z., Sprott, J.C.: Transformation of 4-D dynamical systems to hyperjerk form. Palest. J. Math. 2, 38-45 (2013) · Zbl 1343.37023 |
| [14] | Chlouverakis, K.E., Sprott, J.C.: Chaotic hyperjerk systems. Chaos Solitons Fractals 28, 739-746 (2006) · Zbl 1106.37024 · doi:10.1016/j.chaos.2005.08.019 |
| [15] | Dalkiran, F.Y., Sprott, J.C.: Simple chaotic hyperjerk system. Int. J. Bifurcat. Chaos 26, 1650,189 (2016) · doi:10.1142/S0218127416501893 |
| [16] | Linz, S.J.: On hyperjerky systems. Chaos Solitons Fractals 37, 741-747 (2008) · Zbl 1148.37026 · doi:10.1016/j.chaos.2006.09.059 |
| [17] | Munmuangsaen, B., Srisuchinwong, B.: Elementary chaotic snap flows. Chaos Solitons Fractals 44, 995-1003 (2011) · doi:10.1016/j.chaos.2011.08.008 |
| [18] | Bao, B., Zou, X., Liu, Z., Hu, F.: Generalized memory element and chaotic memory system. Int. J. Bifurcat. Chaos 23, 1350,135-1350,412 (2013) · Zbl 1275.34064 · doi:10.1142/S0218127413501356 |
| [19] | Vaidyanathan, S., Volos, C., Pham, V.T., Madhavan, K.: Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its SPICE implementation. Arch. Control Sci. 25, 135-158 (2015) · Zbl 1446.93045 |
| [20] | Vaidyanathan, S.: Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system via backsteping control method. Arch. Control Sci. 26, 311-338 (2016) · Zbl 1446.93042 |
| [21] | Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining lyapunov exponents from a time series. Physica D 16, 285-317 (1985) · Zbl 0585.58037 · doi:10.1016/0167-2789(85)90011-9 |
| [22] | Hens, C., Dana, S.K., Feudel, U.: Extreme multistability: attractors manipulation and robustness. Chaos 25, 053,112 (2015) · Zbl 1374.34219 · doi:10.1063/1.4921351 |
| [23] | Li, C., Sprott, J.C.: Finding coexisting attractors using amplitude control. Nonlinear Dyn. 78, 2059-2064 (2014) · doi:10.1007/s11071-014-1568-x |
| [24] | Zeng, Z., Huang, T., Zheng, W.: Multistability of recurrent networks with time-varying delays and the piecewise linear activation function. IEEE Trans. Neural Netw. 21, 1371-1377 (2010) · Zbl 1343.37023 |
| [25] | Pisarchik, A.N., Feudel, U.: Control of multistability. Phys. Rep. 540, 167-218 (2014) · Zbl 1357.34105 · doi:10.1016/j.physrep.2014.02.007 |
| [26] | Guan, Z.H., Lai, Q., Chi, M., Cheng, X.M., Liu, F.: Analysis of a new three-dimensional system with multiple chaotic attractors. Nonlinear Dyn. 75, 331-343 (2014) · Zbl 1281.34065 · doi:10.1007/s11071-013-1069-3 |
| [27] | Lai, Q., Chen, S.: Research on a new 3d autonomous chaotic system with coexisting attractors. Optik 127, 3000-3004 (2016) · doi:10.1016/j.ijleo.2015.12.089 |
| [28] | Leipnik, R.B., Newton, T.A.: Double strange attractors in rigid body motion with linear feedback control. Phys. Lett. A 86, 63-87 (1981) · doi:10.1016/0375-9601(81)90165-1 |
| [29] | Ngonghala, C., Feudel, U., Showalter, K.: Extreme multistability in a chemical model system. Phys. Rev. E 83, 056,206 (2011) · doi:10.1103/PhysRevE.83.056206 |
| [30] | Vaithianathan, V., Veijun, J.: Coexistence of four different attractors in a fundamental power system model. IEEE Trans. Circuits Syst. I(46), 405-409 (1999) |
| [31] | Zeng, Z., Zheng, W.: Multistability of neural networks with time-varying delays and concave-convex characteristic. IEEE Trans. Neural Netw. Learn. Syst. 23, 293-305 (2012) · doi:10.1109/TNNLS.2011.2179311 |
| [32] | Kengne, J., Chedjou, J.C., Kom, M., Kyamakya, K., Tamba, V.K.: Regular oscillations, chaos, and multistability in a system of two coupled van der pol oscillators: numerical and experimental studies. Nonlinear Dyn. 76, 1119-1132 (2014) · doi:10.1007/s11071-013-1195-y |
| [33] | Kengne, J., Chedjou, J.C., Fozin, T.F., Kyamakya, K., Kenne, G.: On the analysis of semiconductor diode based chaotic and hyperchaotic chaotic generators—a case study. Nonlinear Dyn. 77, 373-386 (2014) · doi:10.1007/s11071-014-1301-9 |
| [34] | Kengne, J.: Coexistence of chaos with hyperchaos, period-3 doubling bifurcation, and transient chaos in the hyperchaotic oscillator with gyrators. Int. J. Bifurcat. Chaos 25, 1550,052 (2015) · doi:10.1142/S0218127415500522 |
| [35] | Lai, Q., Hu, B., Guan, Z.H., Li, T., Zheng, D.F., Wu, Y.H.: Multistability and bifurcation in a delayed neural network. Neurocomputing 127, 785-792 (2016) · doi:10.1016/j.neucom.2016.05.064 |
| [36] | Kengne, J., Njitacke, Z.T., Fotsin, H.B.: Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dyn. 83, 751-766 (2016) · Zbl 1343.34115 · doi:10.1007/s11071-015-2364-y |
| [37] | Buscarino, A., Fortuna, L., Frasca, M., Gambuzza, L.V.: A chaotic circuit based on Hewlett-Packard memristor. Chaos 22, 023,136 (2012) · Zbl 1331.34074 · doi:10.1063/1.4729135 |
| [38] | Fortuna, L., Frasca, M., Xibilia, M.G.: Chua’s Circuit Implementation: Yesterday, Today and Tomorrow. World Scientific, Singapore (2009) · doi:10.1142/7200 |
| [39] | Buscarino, A., Fortuna, L., Frasca, M., Sciuto, G.: A Concise Guide to Chaotic Electronic Circuits. Springer, Berlin (2014) · doi:10.1007/978-3-319-05900-6 |
| [40] | Sprott, J.C.: A proposed standard for the publication of new chaotic systems. Int. J. Bifurcat. Chaos 21, 2391-2394 (2011) · doi:10.1142/S021812741103009X |
| [41] | Kaneko, K.: Clustering, coding, switching, hierarchical ordering, and control in network of chaotic elements. Physica D 41, 137-172 (1990) · Zbl 0709.58520 · doi:10.1016/0167-2789(90)90119-A |
| [42] | Wang, Z., Sun, W., Wei, Z., Zhang, S.: Dynamical and delayed feedback control for a 3D jerk system with hidden attractor. Nonlinear Dyn. 82, 577-588 (2015) · Zbl 1348.34113 · doi:10.1007/s11071-015-2177-z |
| [43] | Kingni, S.T., Jafari, S., Simo, H., Woafo, P.: 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, 76 (2014) · doi:10.1140/epjp/i2014-14076-4 |
| [44] | Li, C., Sprott, J.C.: Amplitude control approach for chaotic signals. Nonlinear Dyn. 73, 1335-1341 (2013) · Zbl 1281.34070 · doi:10.1007/s11071-013-0866-z |
| [45] | Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, Upper Saddle River (2002) · Zbl 1003.34002 |
| [46] | Pecora, L., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821-824 (1990) · Zbl 0938.37019 · doi:10.1103/PhysRevLett.64.821 |
| [47] | Sundarapandian, V., Pehlivan, I.: Analysis, control, synchronization, and circuit design of a novel chaotic system. Math. Comput. Model. 55, 1904-1915 (2012) · Zbl 1255.93076 · doi:10.1016/j.mcm.2011.11.048 |
| [48] | Volos, C.K., Kyprianidis, I.M., Stouboulos, I.N.: Image encryption process based on chaotic synchronization phenomena. Signal Process. 93, 1328-1340 (2013) · doi:10.1016/j.sigpro.2012.11.008 |
| [49] | Banerjee, S.: Chaos Synchronization and Cryptography for Secure Communication. IGI Global, USA (2010) |
| [50] | Wu, X., He, Y., Yu, W., Yin, B.: A new chaotic attractor and its synchronization implementation. Circuits Syst. Signal Process. 34, 1747-1768 (2015) · doi:10.1007/s00034-014-9946-7 |
| [51] | Tayebi, A., Berber, S., Swain, A.: Performance analysis of chaotic DSSS-CDMA synchronization under jamming attack. Circuits Syst. Signal Process. 35, 4350-4371 (2016) · Zbl 1366.94161 · doi:10.1007/s00034-016-0266-y |
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.
