Synchronization and waves in a ring of diffusively coupled memristor-based Chua’s circuits. (English) Zbl 1305.37042
Summary: In the present paper we report numerical observations of the spontaneous dynamics of \(N\) identical memristor-based Chua’s circuits bidirectionally coupled in a ring geometry. Two different initial configurations are studied by varying \(N\) and the coupling strength: in the first configuration we consider only one circuit with non-zero initial conditions, in the second one all circuits have uniform random initial conditions. We observed both chaotic and non-chaotic synchronization. In the chaotic synchronized regime we identified emerging chaotic steady waves, characterized by an almost constant frequency and chaotic amplitude. Depending on the initial conditions, in the pseudo-sinusoidal oscillations regime both macroscopic quasi-periodic steady and traveling waves were observed.
MSC:
| 37M05 | Simulation of dynamical systems |
Keywords:
coupled nonlinear oscillators; chaotic synchronization; pattern formation; Chua’s circuit; memristorReferences:
| [1] | Arena, P., Fortuna, L., Frasca, M.: Multi-template approach to realize central pattern generators for artificial locomotion control. Int. J. Circuit Theory Appl. 30, 441-458 (2002) · Zbl 1007.93051 · doi:10.1002/cta.186 |
| [2] | Bilotta, E., Pantano, P., Stranges, F.: Computer graphics meets chaos and hyperchaos. Some key problems. Comput. Graphics (Pergamon) 30, 359-367 (2006) · doi:10.1016/j.cag.2006.02.003 |
| [3] | Bilotta, E., Pantano, P.: The language of chaos. Int. J. Bifurc. Chaos 16, 523-557 (2006) · Zbl 1141.37313 · doi:10.1142/S0218127406014988 |
| [4] | Bilotta, E., Pantano, P., Stranges, F.: A gallery of Chua attractors: part II. Int. J. Bifurc. Chaos 17, 293-380 (2007) · Zbl 1139.37020 · doi:10.1142/S0218127407017343 |
| [5] | Bilotta, E., Stranges, F., Pantano, P.: A gallery of Chua attractors: part III. Int. J. Bifurc. Chaos 17, 657-734 (2007) · Zbl 1139.37021 · doi:10.1142/S0218127407017537 |
| [6] | Bilotta, E., Di Blasi, G., Stranges, F., Pantano, P.: A gallery of Chua attractors: part V. Int. J. Bifurc. Chaos 17, 1383-1511 (2007) · Zbl 1144.37014 · doi:10.1142/S0218127407018099 |
| [7] | Bilotta, E., Di Blasi, G., Stranges, F., Pantano, P.: A gallery of Chua attractors. Part VI. Int. J. Bifurc. Chaos 17, 1801-1910 (2007) · Zbl 1141.37312 · doi:10.1142/S0218127407018105 |
| [8] | Bilotta, E., Pantano, P.: Structural and functional growth in self-reproducers. Complexity 11, 12-29 (2006) · doi:10.1002/cplx.20131 |
| [9] | Bilotta, E., Pantano, P., Vena, S.: Artificial micro-worlds part I: a new approach for studying life-like phenomena. Int. J. Bifurc. Chaos 21, 373-398 (2011) · Zbl 1210.68071 · doi:10.1142/S0218127411028659 |
| [10] | Bilotta, E., Pantano, P., Vena, S.: Artificial micro-worlds part II: cellular automata growth dynamics. Int. J. Bifurc. Chaos 21, 619-645 (2011) · Zbl 1215.68139 · doi:10.1142/S0218127411028672 |
| [11] | Bilotta, E., Pantano, P., Vena, S.: Artificial micro-worlds. Part III: A taxonomy of self-reproducing 2D CA species. Int. J. Bifurc. Chaos 21, 1233-1263 (2011) · Zbl 1248.68341 · doi:10.1142/S0218127411029070 |
| [12] | Bilotta, E., Pantano, P., Vena, S.: Artificial micro-worlds. Part IV: models of complex self-reproducers. Int. J. Bifurc. Chaos 21, 1501-1521 (2011) · Zbl 1248.68342 · doi:10.1142/S0218127411029252 |
| [13] | Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U.: Complex network: structure and dynamics. Phys. Rep. 424, 175-308 (2006) · Zbl 1371.82002 · doi:10.1016/j.physrep.2005.10.009 |
| [14] | Boccaletti, S., Kurths, J., Osipov, G., Valladares, D.L., Zhou, C.S.: The synchronization of chaotic systems. Phys. Rep. 336, 1-101 (2002) · Zbl 0995.37022 · doi:10.1016/S0370-1573(02)00137-0 |
| [15] | Buzsàki, G., Draguhn, A.: Neuronal oscillations in cortical networks. Science 304, 1926-1929 (2004) · doi:10.1126/science.1099745 |
| [16] | Chua, L.: Memristor-the missing circuit element. IEEE Trans. Circuit Theory 18, 507-519 (1971) · doi:10.1109/TCT.1971.1083337 |
| [17] | Collins, J.J., Stewart, I.: A group-theoretic approach to rings of coupled biological oscillators. Biol. Cybern. 71, 95-103 (1994) · Zbl 0804.92009 · doi:10.1007/BF00197312 |
| [18] | Corinto, F., Ascoli, A., Gilli, M.: Nonlinear dynamics of memristor oscillators. IEEE Trans. Circuits Syst. I Regul. Pap. 58, 1323-1336 (2011) · Zbl 1468.94570 · doi:10.1109/TCSI.2010.2097731 |
| [19] | Dormand, J.R., Prince, P.J.: A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6, 19-26 (1980) · Zbl 0448.65045 · doi:10.1016/0771-050X(80)90013-3 |
| [20] | Dos Santos, A.M., Lopes, S.R., Viana, R.L.: Rhythm synchronization and chaotic modulation of coupled van der Pol oscillators in a model for the heartbeat. Physica A 338, 335-355 (2004) · doi:10.1016/j.physa.2004.02.058 |
| [21] | Ermentrout, G.B., Kopell, N.: Inhibition-production patterning in chains of coupled nonlinear oscillators. SIAM J. Appl. Math. 54, 478-507 (1994) · Zbl 0811.92004 · doi:10.1137/S0036139992231964 |
| [22] | Ermentrout, G.B.: The behavior of rings of coupled oscillators. J. Math. Biol. 1985(23), 55-74 (1985) · Zbl 0583.92002 · doi:10.1007/BF00276558 |
| [23] | Frigon, A.: Central pattern generators of the mammalian spinal cord. Neurosci. 18, 56-69 (2012) |
| [24] | Gambino, G., Lombardo, M.C., Sammartino, M.: Global linear feedback control for the generalized Lorenz system. Chaos Solitons Fractals 29, 829-837 (2006) · Zbl 1142.93409 · doi:10.1016/j.chaos.2005.08.072 |
| [25] | Gambino, G., Lombardo, M.C., Sammartino, M.: Turing instability and traveling fronts for a nonlinear reaction-diffusion system with crossdiffusion. Math. Comput. Simul. 82, 1112-1132 (2012) · Zbl 1320.35170 · doi:10.1016/j.matcom.2011.11.004 |
| [26] | Gambino, G., Lombardo, M.C., Sammartino, M.: Turing pattern formation in the Brusselator system with nonlinear diffusion. Phys. Rev. E 88, 042925 (2013) · Zbl 1270.35088 · doi:10.1103/PhysRevE.88.042925 |
| [27] | Gibbons, J.D.: Nonparametric Statistical Inference, 2nd edn. Dekker, New York (1985) · Zbl 0573.62033 |
| [28] | Horikawa, Y.: Metastable and chaotic transient rotating waves in a ring of unidirectionally coupled bistable Lorenz systems. Physica D 261, 8-18 (2013) · Zbl 1417.37126 · doi:10.1016/j.physd.2013.06.003 |
| [29] | Ijspeert, A.J.: Central pattern generators for locomotion control in animals and robots: a review. Neural Netw. 21, 642-653 (2008) · doi:10.1016/j.neunet.2008.03.014 |
| [30] | Itoh, M., Chua, L.O.: Memristor oscillators. Int. J. Bifurc. Chaos 18, 3183-3206 (2008) · Zbl 1165.94300 · doi:10.1142/S0218127408022354 |
| [31] | Jo, S.H., Chang, T., Ebong, I., Bhadviya, B.B., Mazumder, P., Lu, W.: Nanoscale memristor device as synapse in neuromorphic systems. Nano Lett. 10, 1297-1301 (2010) · doi:10.1021/nl904092h |
| [32] | Kleinfeld, D., Sompolinsky, H.: Associative neural network model for the generation of temporal patterns. Theory and application to central pattern generators. Biophys. J. 54, 1039-1051 (1988) · doi:10.1016/S0006-3495(88)83041-8 |
| [33] | Kocarev, L., Parlitz, U.: Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Phys. Rev. Lett. 76, 1816 (1996) · doi:10.1103/PhysRevLett.76.1816 |
| [34] | Kopell, N., Ermentrout, G.B.: Coupled oscillators and the design of central pattern generators. Math. Biosci. 89, 14-23 (1988) · Zbl 0649.92009 |
| [35] | Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Courier Dover Publications, New York (2003) · Zbl 0558.76051 |
| [36] | Lading, B., Mosekilde, E., Yanchuk, S., Yu, M.: Chaotic synchronization between pancreatic beta-cells. Prog. Theor. Phys. Suppl. 139, 164-177 (2000) · doi:10.1143/PTPS.139.164 |
| [37] | Lorenzo, M.N., Marino, I.P., Perez-Munuzuri, V., Matias, M.A., Perez-Villar, V.: Synchronization waves in arrays of driven chaotic systems. Phys. Rev. E 54, R3094 (1996) · doi:10.1103/PhysRevE.54.R3094 |
| [38] | Mansubia, S.C., Mikhailov, A.S., Zanette, D.H.: Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems. World Scientific, Singapore (2004) · Zbl 1119.34001 |
| [39] | Marino, I.P., Perez-Munzuri, V., Matias, M.A.: Desynchronization transitions in rings of coupled chaotic oscillators. Int. J. Bifurc. Chaos 8, 1733-1738 (1998) · Zbl 0941.34026 · doi:10.1142/S021812749800142X |
| [40] | Matias, M.A., Gumez, J., Perez-Munuzuri, V., Marino, I.P., Lorenzo, M.N., Perez-Villar, V.: Observation of a fast rotating wave in rings of coupled chaotic oscillators. Phys. Rev. Lett. 78, 219-222 (1997) · doi:10.1103/PhysRevLett.78.219 |
| [41] | Matias, M.A., Perez-Munuzuri, V., Marino, I.P., Lorenzo, M.N., Perez-Villar, V.: Size instabilities in ring of chaotic synchronized systems. Europhys. Lett. 37, 379-384 (1997) · doi:10.1209/epl/i1997-00159-8 |
| [42] | Mosekilde, E., Maistrenko, Y., Postnov, D.: Chaotic Synchronization: Applications to Living Systems. World Scientific, Singapore (2002) · Zbl 0999.37022 |
| [43] | Muthuswamy, B.: Implementing memristor based chaotic circuits. Int. J. Bifurc. Chaos 20, 1335-1350 (2010) · Zbl 1193.94082 · doi:10.1142/S0218127410026514 |
| [44] | Nana, B., Woafo, P.: Synchronization in a ring of four mutually coupled Van der Pol oscillators: theory and experiment. Phys. Rev. E 74, 1-8 (2006) · doi:10.1103/PhysRevE.74.046213 |
| [45] | Nana, B., Woafo, P.: Synchronized states in a ring of four mutually coupled oscillators and experimental application to secure communications. Commun. Nonlinear Sci. Numer. Simul. 6, 725-1733 (2011) · Zbl 1221.94056 |
| [46] | Nowotny, T., Huerta, R., Rabinovich, M.I.: Neuronal synchrony: peculiarity and generality. Chaos 18, 037119 (2008) · doi:10.1063/1.2949925 |
| [47] | Paoletti, P.; Innocenti, G., Traveling waves in one-dimensional networks of dynamical systems (2011) |
| [48] | Pecora, L.M., Carrol, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821-824 (1990) · Zbl 0938.37019 · doi:10.1103/PhysRevLett.64.821 |
| [49] | Perlikowski, P., Yanchuk, S., Wolfrum, M., Stefanski, A., Mosiolek, P., Kapitaniak, T.: Routes to complex dynamics in a ring of unidirectionally coupled systems. Chaos 20, 013111 (2010) · doi:10.1063/1.3293176 |
| [50] | Pershin, Y.V., Di Ventra, M.: Experimental demonstration of associative memory with memristive neural networks. Neural Netw. 23, 881-886 (2010) · doi:10.1016/j.neunet.2010.05.001 |
| [51] | Pikovsky, A.S., Rosenblum, M.G., Kurths, J.: Phase synchronization in regular and chaotic systems. Int. J. Bifurc. Chaos 10, 2291-2305 (2000) · Zbl 1090.37508 |
| [52] | Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: Phase synchronization of chaotic oscillators. J. Phys. Rev. Lett. 76, 1804-1807 (1996), 78, 4193 (1997) · Zbl 0871.93043 · doi:10.1103/PhysRevLett.76.1804 |
| [53] | Rosenblum, M., Pikovsky, A.: Synchronization: from pendulum clocks to chaotic lasers and chemical oscillators. Contemp. Phys. 44, 401-416 (2003) · doi:10.1080/00107510310001603129 |
| [54] | Shabunin, A., Astakhov, V., Anishchenko, V.: Developing chaos on base of traveling waves in a chain of coupled oscillators with period-doubling. Synchronization and hierarchy of multistability formation. Int. J. Bifurc. Chaos 12, 1895-1907 (2002) · Zbl 1044.37027 · doi:10.1142/S021812740200556X |
| [55] | Wang, L., Duan, M., Duan, S.: Memristive perceptron for combinational logic classification. Math. Probl. Eng. (2013). doi:10.1155/2009/572724 · Zbl 1181.37047 · doi:10.1155/2009/572724 |
| [56] | Wang, Y.W., Guan, Z.H.: Generalized synchronization of continuous chaotic system. Chaos Solitons Fractals 27, 97-101 (2006) · Zbl 1083.37515 · doi:10.1016/j.chaos.2004.12.038 |
| [57] | Zhang, X., Fu, M., Xiao, J., Hu, G.: Self-organization of chaos synchronization and pattern formation in coupled chaotic oscillators. Phys. Rev. E 74, 015202 (2006) · doi:10.1103/PhysRevE.74.015202 |
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