Synchronization of periodic self-oscillators interacting via memristor-based coupling. (English) Zbl 1448.34074
Summary: A model of two self-sustained oscillators interacting through memristive coupling is studied. The memristive coupling is realized by using a cubic memristor model. Numerical simulation is combined with theoretical analysis by means of quasi-harmonic reduction. It is shown that the specifics of the memristor nonlinearity results in the appearance of infinitely many equilibrium points which form a line of equilibria in the phase space of the system under study. It is established that the possibility to observe the effect of phase locking in the considered system depends on both parameter values and initial conditions. Consequently, the boundaries of the synchronization region are determined by the initial conditions. It is demonstrated that introducing or adding a small term into the memristor state equation gives rise to the disappearance of the line of equilibria and eliminates the dependence of synchronization on the initial conditions.
MSC:
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 34D06 | Synchronization of solutions to ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 94C60 | Circuits in qualitative investigation and simulation of models |
Keywords:
synchronization; periodic self-oscillator; adaptive coupling; line of equilibria; memristorReferences:
| [1] | Adamatzky, A. & Chua, L. (eds.) [2014] Memristor Networks (Springer International Publishing). |
| [2] | Botta, V., Néspoli, C. & Messias, M. [2011] “ Mathematical analysis of a third-order memristor-based Chua’s oscillator,” TEMA Tend. Mat. Apl. Comput.12, 91-99. · Zbl 1263.34076 |
| [3] | Buscarino, A., Fortuna, L., Frasca, M. & Gambuzza, L. [2012] “ A chaotic circuit based on Hewlett-Packard memristor,” Chaos22, 023136. · Zbl 1331.34074 |
| [4] | Buscarino, A., Fortuna, L., Frasca, M. & Gambuzza, L. [2013] “ A gallery of chaotic oscillators based on HP memristor,” Int. J. Bifurcation and Chaos23, 1330015-1-14. · Zbl 1270.34097 |
| [5] | Chen, L., Li, C., Huang, T., Chen, Y., Wen, S. & Qi, J. [2013] “ A synapse memristor model with forgetting effect,” Phys. Lett. A377, 3260-3265. |
| [6] | Chen, M., Li, M., Yu, Q., Bao, B., Xu, Q. & Wang, J. [2015a] “ Dynamics of self-excited attractors and hidden attractors in generalized memristor-based Chua’s circuit,” Nonlin. Dyn.81, 215-226. · Zbl 1430.94107 |
| [7] | Chen, M., Yu, J. & Bao, B. [2015b] “ Finding hidden attractors in imroved memristor-based Chua’s circuit,” Electron. Lett.51, 462-464. |
| [8] | Chua, L. [1971] “ Memristor-the missing circuit element,” IEEE Trans. Circuit Th.CT-18, 507-519. |
| [9] | Chua, L. & Kang, S. [1976] “ Memristive devices and systems,” Proc. IEEE64, 209-223. |
| [10] | Corinto, F., Ascoli, A. & Gilli, M. [2011] “Nonlinear dynamics of memristor oscillators,” IEEE Trans. Circuits Syst.-I: Reg. Papers58, 1323-1336. · Zbl 1468.94570 |
| [11] | Corinto, F. & Forti, M. [2016] “Memristor circuits: Flux-charge analysis method,” IEEE Trans. Circuits Syst.-I: Reg. Papers63, 1997-2009. |
| [12] | Di Ventra, M. & Pershin, Y. [2013] “ The parallel approach,” Nat. Phys.9, 200-202. |
| [13] | Frasca, M., Gambuzza, L., Buscarino, A. & Fortuna, L. [2015] “ Memristor based adaptive coupling for synchronization of two Rössler systems,” Advance in Neural Networks: Computational and Theoretical Issues, , Vol. 37 (Springer International Publishing), pp. 295-300. |
| [14] | Gambuzza, L., Buscarino, A., Fortuna, L. & Frasca, M. [2015a] “ Memristor-based adaptive coupling for consensus and synchronization,” IEEE Trans. Circuits Syst.62, 1175-1184. · Zbl 1468.94603 |
| [15] | Gambuzza, L., Fortuna, L., Frasca, M. & Gale, E. [2015b] “ Experimental evidence of chaos from memristors,” Int. J. Bifurcation and Chaos25, 1550101-1-9. |
| [16] | Ignatov, M., Hansen, M., Ziegler, M. & Kohlstedt, H. [2016] “ Synchronization of two memristively coupled van der Pol oscillators,” Appl. Phys. Lett.108, 084105. |
| [17] | Itoh, M. & Chua, L. [2008] “ Memristor oscillators,” Int. J. Bifurcation and Chaos18, 3183-3206. · Zbl 1165.94300 |
| [18] | Itoh, M. & Chua, L. [2011] “ Memristor Hamiltonian circuits,” Int. J. Bifurcation and Chaos21, 2395-2425. · Zbl 1248.34064 |
| [19] | Itoh, M. & Chua, L. [2017] “ Dynamics of Hamiltonian systems and memristor circuits,” Int. J. Bifurcation and Chaos27, 1730005-1-66. · Zbl 1362.37105 |
| [20] | Jo, S., Chang, T., Ebong, I., Bhadviya, B., Mazumder, P. & Lu, W. [2010] “ Nanoscale memristor device as synapse in neuromorphic systems,” Nano Lett.10, 1297-1301. |
| [21] | Korneev, I. & Semenov, V. [2017] “ Andronov-Hopf bifurcation with and without parameter in a cubic memristor oscillator with a line of equilibria,” Chaos27, 081104. · Zbl 1390.34138 |
| [22] | Korneev, I., Vadivasova, T. & Semenov, V. [2017] “ Hard and soft excitation of oscillations in memristor-based oscillators with a line of equilibria,” Nonlin. Dyn.89, 2829-2843. |
| [23] | Kozma, R., Pino, R. & Pazienza, G. (eds.) [2012] Advances in Neuromorphic Memristor Science and Applications, , Vol. 4 (Springer, Netherlands). |
| [24] | Li, Y., Zhong, Y., Xu, L., Zhang, J., Xu, H., Sun, X. & Miao, X. [2013] “ Ultrafast synaptic events in a chalcogenide memristor,” Scient. Rep.3, 1619. |
| [25] | Messias, M., Nespoli, C. & Botta, V. [2010] “ Hopf bifurcation from lines of equilibria without parameters in memristor oscillators,” Int. J. Bifurcation and Chaos20, 437-450. · Zbl 1188.34060 |
| [26] | Pershin, Y. & Di Ventra, M. [2010] “ Experimental demonstration of associative memory with memristive neural networks,” Neural Netw.23, 881-886. |
| [27] | Pham, V.-T., Buscarino, A., Fortuna, L. & Frasca, M. [2013] “ Simple memristive time-delay chaotic systems,” Int. J. Bifurcation and Chaos23, 1350073-1-9. · Zbl 1270.34191 |
| [28] | Pham, V.-T., Volos, C., Vaidyanathan, S., Le, T. & Vu, V. [2015] “ A memristor-based hyperchaotic system with hidden attractors: Dynamics, synchronization and circuital emulating,” J. Engin. Sci. Technol. Rev.8, 205-214. |
| [29] | Pham, V.-T., Jafari, S. & Kapitaniak, T. [2016a] “ Constructing a chaotic system with an infinite number of equilibrium points,” Int. J. Bifurcation and Chaos26, 1650225-1-7. · Zbl 1354.34075 |
| [30] | Pham, V.-T., Jafari, S., Volos, C. & Kapitaniak, T. [2016b] “ A gallery of chaotic systems with an infinite number of equilibrium points,” Chaos Solit. Fract.93, 58-63. · Zbl 1372.37071 |
| [31] | Pikovsky, A., Rosenblum, M. & Kurths, J. [2001] Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press). · Zbl 0993.37002 |
| [32] | Radwan, A. & Fouda, M. [2015] On the Mathematical Modeling of Memristor, Memcapacitor, and Meminductor, , Vol. 26 (Springer International Publishing). · Zbl 1334.94003 |
| [33] | Riaza, R. [2012] “ Manifolds of equilibria and bifurcations without parameters in memristive circuits,” SIAM J. Appl. Math.72, 877-896. · Zbl 1258.34022 |
| [34] | Semenov, V., Korneev, I., Arinushkin, P., Strelkova, G., Vadivasova, T. & Anishchenko, V. [2015] “ Numerical and experimental studies of attractors in memristor-based Chua’s oscillator with a line of equilibria. Noise-induced effects,” Eur. Phys. J. Special Topics224, 1553-1561. |
| [35] | Serb, A., Bill, J., Khiat, A., Berdan, R., Legenstein, R. & Prodromakis, T. [2016] “ Unsupervised learning in probabilistic neural networks with multi-state metal-oxide memristive synapses,” Nat. Commun.7, 12611. |
| [36] | Tetzlaff, R. (ed.) [2014] Memristor and Memristive Systems (Springer-Verlag, NY). |
| [37] | Vaidyanathan, S. & Volos, C. (eds.) [2017] Advances in Memristors, Memristive Devices and Systems, , Vol. 701 (Springer International Publishing). |
| [38] | Volos, C., Kyprianidis, I., Stouboulos, I., Muñoz-Pacheco, J. & Pham, V.-T. [2015] “ Synchronization of chaotic nonlinear circuits via memristor,” J. Engin. Sci. Technol. Rev.8, 44-51. |
| [39] | Vourkas, I. & Sirakoulis, G. [2016] Memristor-Based Nanoelectronic Computing Circuit and Architectures, , Vol. 19 (Springer International Publishing). |
| [40] | Williamson, A., Schurmann, L., Hiller, L., Klefenz, F., Hoerselmann, I., Husar, P. & Schober, A. [2013] “ Synaptic behavior and STDP of asymmetric nanoscale memristors in biohybrid systems,” Nanoscale5, 7297-7303. |
| [41] | Zhang, J. & Liao, X. [2017] “ Synchronization and chaos in coupled memristor-based Fitzhugh-Nagumo circuits with memristor synapse,” Int. J. Electron. Commun.75, 82-90. |
| [42] | Zhao, Q., Wang, C. & Zhang, X. [2019] “ A universal emulator for memristor, memcapacitor, and meminductor and its chaotic circuit,” Chaos29, 013141. · Zbl 1406.94097 |
| [43] | Zhou, E., Fang, L. & Yang, B. [2018] “ A general method to describe forgetting effect of memristors,” Phys. Lett. A383, 942-948. |
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.
