ISSN 1817-3020 (Print)
ISSN 2542-193X (Online)

Korneev I. A., Slepnev A. V., Semenov V. V., Vadivasova T. E. Mutual Synchronization of Dissipatively Coupled Memristive Self-Oscillators. //Izvestiya of Saratov University. New series. Series: Physics. , 2020, vol. 20, iss. 3, pp. 210-221. DOI: https://doi.org/10.18500/1817-3020-2020-20-3-210-221

Published online:
31.08.2020
Language:
Russian
UDC:
530.182

# Mutual Synchronization of Dissipatively Coupled Memristive Self-Oscillators

Autors:
Korneev Ivan Alexandrovich, Saratov State University
Slepnev Andrei Viacheslavovich, Saratov State University
Semenov Vladimir Viktorovich, Saratov State University
Vadivasova Tatyana Evgen'evna, Saratov State University
Abstract:

Background and Objectives: Dynamical systems containing memristive elements, i.e. elements with “memory”, represent a special class of dynamical systems that can be named memristive systems. In memristive systems, the synchronization of oscillations has some features. However, a complete description of these features is still lacking. For the most part, this concerns the synchronization of memristive self-oscillators, both identical and with frequency detuning. In this paper, we study two resistively coupled self-oscillators containing memristive conductivities. We consider both the complete synchronization of identical selfoscillators and the frequency synchronization of self-oscillators with detuning. The features of synchronization effects associated with the memristive nature of partial systems are revealed. The influence of “non-ideality” of memristive elements and the duration of the establishment processes in this case are analyzed. Materials and Methods: Using numerical integration methods for various parameter values, approximate solutions of a system of ordinary differential equations that describe the dynamics of two coupled memristive self-oscillators are obtained. Projections of phase trajectories are plotted on various planes, as well as regions of synchronization of oscillations of the system under study. Results: It is shown that the memristive systems under consideration demonstrate both the effect of complete synchronization (with full identity of partial systems) and the effect of frequency synchronization (in the presence of frequency detuning). The complete synchronization of oscillations is characterized by the presence of a threshold for the coupling, the value of which continuously depends on the initial conditions, in particular, on the initial values of variables that specify the states of memristive elements. With a constant value of the coupling coefficient, there is a continuous dependence of the boundaries of the frequency synchronization region on the initial conditions. The introduction of a parameter characterizing the rate of memristors “forgetting” their initial state (“non-ideality”) leads to the disappearance of the dependence of the type of the steady state on the initial conditions. Conclusion: The fundamental phenomenon of synchronization is inherent in memristive self-oscillators, which allows them to be attributed to the class of self-oscillating systems. The influence of the initial conditions on the effects of synchronization can be considered as a general property of “ideal” memristive systems.

Key words:
DOI:
10.18500/1817-3020-2020-20-3-210-221
References:
1. Chua L. O. Memristor – The missing circuit element. IEEE Transactions on Electron Devices, 1971, vol. 18, pp. 507–519. DOI: https://doi.org/10.1109/TCT.1971.1083337
2. Strukov D. B., Snider G. S., Stewart D. R., Williams R. S. The missing memristor found. Nature, 2008, vol. 453, pp. 80–83. DOI: https://doi.org/10.1038/nature06932
3. Berzina T., Smerieri A., Bernabó M., Pucci A., Ruggeri G., Erokhin V., Fontana M. Optimization of an organic memristor as an adaptive memory element. Journal of Applied Physics, 2009, vol. 105, no. 12, pp. 124515. DOI: https://doi.org/10.1063/1.3153944
4. Jeong H. Y., Kim J. Y., Kim J. W., Hwang J. O., Kim J. E., Lee J. Y., Choi S. Y. Graphene oxide thin fi lms for fl exible nonvolatile memory applications. Nano Letters, 2010, vol. 10, no. 11, pp. 4381–4386. DOI: https://doi.org/10.1021/nl101902k
5. Chang T., Jo S.-H., Kim K.-H., Sheridan P., Gaba S., Lu W. Synaptic behaviors and modeling of a metal oxide memristive device. Applied Physics A, 2011, vol. 102, pp. 857–863. DOI: https://doi.org/10.1007/s00339-011-6296-1
6. Yang Y., Sheridan P., Lu W. Complementary resistive switching in tantalum oxide-based resistive memory devices. Applied Physics Letters, 2012, vol. 100, no. 20, pp. 203–112. DOI: https://doi.org/10.1063/1.4719198
7. Strachan J., Torrezan A., Miao F., Pickett M., Yang J., Yi W., Medeiros-Ribeiro G., Williams R. State dynamics and modeling of tantalum oxide memristors. IEEE Transactions on Electron Devices, 2013, vol. 60, no. 7, pp. 2194–2202. DOI: https://doi.org/10.1109/TED.2013.2264476
8. Liu G., Chen Y., Wang C., Zhang W., Li R.-W., Wang L. Polymer memristor for information storage and neuromorphic applications. Materials Horizons, 2014, vol. 1, no. 5, pp. 489–506. DOI: https://doi.org/10.1039/C4MH00067F
9. Erokhina S., Sorokin V., Erokhin V. Polyaniline-based organic memristive device fabricated bylayed-by-layed deposition technique. Electronic Materials Letters. 2015, vol. 11, no. 5, pp. 801–805. DOI: https://doi.org/10.1007/s13391-015-4329-1
10. Chua L. O., Kang S. M. Memristive devices and systems. Proceedings of the IEEE. 1976, vol. 64, iss. 2, pp. 209–223. DOI: https://doi.org/10.1109/PROC.1976.10092
11. Volos C. K., Kyprianidis I. M., Stouboulos I. N., MunozPacheko J. M., Pham V. T. Synchronization of chaotic nonlinear circuits via a memristor. Journal of Engineering Science & Technology Review, 2015, vol. 8, iss. 2, pp. 44–51.
12. Anishchenko V. S., Astakhov V. V., Neiman A. B., Vadivasova T. E., Schimansky-Geier L. Nonlinear dynamics of chaotic and stochastic systems: tutorial and modern developments. Springer Science & Business Media, 2007. 455 p.
13. Messias M., Nespoli C., Botta V. A. Hopf bifurcation from lines of equilibria without parameters in memristor oscillators. International Journal of Bifurcation and Chaos, 2010, vol. 20, no. 2, pp. 437–450. DOI: https://doi.org/10.1142/S0218127410025521
14. Pirani V. A. B., Néspoli C., Messias M. Mathematical Analisys of a Third-order Memristor-based Chua’s Oscillator. Trends in Applied and Computational Mathematics, 2011, vol. 12, no. 2, pp. 91–99. DOI: https://doi.org/10.5540/tema.2011.012.02.0091
15. Riaza R. Manifolds of equilibria and bifurcations without parameters in memristive circuits. SIAM Journal on Applied Mathematics, 2012, vol. 72, iss. 3, pp. 877–896. DOI: https://doi.org/10.1137/100816559
16. Fitch A. L., Yu D., Iu H. H. C., Sreeram V. Hyperchaos on memristor-based modified canonical Chuas circuit. International Journal of Bifurcation and Chaos, 2012, vol. 22, no. 6, pp. 1250133–1250138. DOI: https://doi.org/10.1142/S0218127412501337
17. Li Q., Hu S., Tang S., Zeng G. Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria and its implementation. International Journal of Circuit Theory and Applications, 2014, vol. 42, iss. 11, pp. 1172–1188. DOI: https://doi.org/10.1002/cta.1912
18. Pham V. T., Volos C. K., Vaidyanathan S., Le T. P. , Vu V. Y. A memristor-based hyperchaotic system with hidden attractors: dynamics, synchronization and circuital emulating. Journal of Engineering Science & Technology Review, 2015, vol. 8, iss. 2, pp. 205–214.
19. Kengne J., Tabekoung Z. N., Namba V. K., Negou A. N. Periodicity, chaos and multiple attractors in a memristorbased Shinrikis circuit. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015, vol. 25, pp. 103126. DOI: https://doi.org/10.1063/1.4934653
20. Semenov V., Korneev I., Arinushkin P., Strelkova G., Vadivasova T., Anishchenko V. Numerical and experimental studies of attractors in memristor-based Chua’s oscillator with a line of equilibria. Noise-induced effects. The European Physical Journal Special Topics, 2015, vol. 224, iss. 8, pp.1553–1561. DOI: https://doi.org/10.1140/epjst/e2015-02479-6
21. Korneev I. A., Vadivasova T. E., Semenov V. V. Hard and soft excitation of oscillations in memristor-based oscillators with a line of equilibria. Nonlinear Dynamics, 2017, vol. 89, iss. 4, pp. 2829–2843. DOI: https://doi.org/10.1007/s11071-017-3628-5
22. Korneev I. A., Semenov V. V. Andronov-Hopf bifurcation with and without parameter in a cubic memristor oscillator with a line of equilibria. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2017, vol. 27, iss. 8, pp. 081104(1–5). DOI: https://doi.org/10.1063/1.4996401
23. Frasca M., Gambuzza L., Buscarino A., Fortuna L. Implementation of adaptive coupling through memristor. Physica Status Solidi C, 2014, vol. 12, iss. 1–2, pp. 206210. DOI: https://doi.org/10.1002/pssc.201400097
24. Gambuzza L., Buscarino A., Fortuna L., Frasca M. Memristor-based adaptive coupling for consensus and synchronization. IEEE Transactions on Circuits and Systems I: Regular Papers, 2015, vol. 62, iss. 4, pp.1175–1184. DOI: https://doi.org/10.1109/TCSI.2015.2395631
25. Volos C. K., Pham V.-T., Vaidyanathan S., Kyprianidis I. M., Stouboulos I. N. The case of bidirectionally coupled nonlinear circuits via a memristor. Advances and Applications in Nonlinear Control Systems, 2016, vol. 635, pp. 317–350. DOI: https://doi.org/10.1007/978-3-319-30169-3_15
26. Ignatov M., Hansen M., Ziegler M., Kohlstedt H. Synchronization of two memristively coupled van der Pol oscillators. Applied Physics Letters, 2016, vol. 108, iss. 8, pp. 84–105. DOI: https://doi.org/10.1063/1.4942832
27. Korneev I. A., Shabalina O. G., Semenov V. V., Vadivasova T. E. Synchronization self-sustained oscillators interacting through the memristor. Izvestiya VUZ, Applied Nonlinear Dynamics, 2018, vol. 26, no. 2, pp. 24–40. DOI: https://doi.org/10.18500/0869-6632-2018-26-2-24-40
28. Xu F., Zhang J., Fang T., Huang Sh., Wang M. Synchronous dynamics in neural system coupled with memristive synapse. Nonlinear Dynamics, 2018, vol. 92, no. 3, pp. 1395–1402. DOI: https://doi.org/10.1007/s11071-018-4134-0
29. Yang X., Cao J., Yu W. Exponential synchronization of memristive Cohen–Grossberg neural networks with mixed delays. Cognitive Neurodynamics, 2014, vol. 8, no. 3, pp. 239–249. DOI: https://doi.org/10.1007/s11571-013-9277-6
30. Hu X., Duan Sh. Adaptive synchronization of memristorbased chaotic neural systems. Journal of Engineering Science and Technology Review, 2015, vol. 8, iss. 2, pp. 17–23.
31. Yangand X., Ho D. W. C. Synchronization of delayed memristive neural networks: Robust analysis approach. IEEE Transactions on Cybernetics, 2016, vol. 46, iss. 12, no. 2, pp. 3377–3387. DOI: https://doi.org/10.1109/TCYB.2015.2505903
32. Zhao H., Li L., Peng H., Kurths J., Xiao J., Yang Y. Anti-synchronization for stochastic memristor-based neural networks with non-modeled dynamics via adaptive control approach. The European Physical Journal B, 2015, vol. 88, iss. 5, pp. 1–10. DOI: https://doi.org/10.1140/epjb/e2015-50798-9
33. Wang C., Lv M., Alsaedi A., Ma J. Synchronization stability and pattern selection in a memristive neuronal network. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2017, vol. 27, iss. 11, pp. 113108(1–8). DOI: https://doi.org/10.1063/1.5004234
34. Zhang L., Yang Y., Wang F. Lag synchronization for fractional-order memristive neural networks via period intermittent control. Nonlinear Dynamics, 2017, vol. 89, pp. 367–381. DOI: https://doi.org/10.1007/s11071-017-3459-4
35. Chen C., Li L., Peng H., Yang Y., Li T. Synchronization control of coupled memristor-based neural networks with mixed delays and stochastic perturbations. Neural Processing Letters, 2018, vol. 47, no. 2, pp. 679–696. DOI: https://doi.org/10.1007/s11063-017-9675-6
36. Fiedler B., Liebscher S., Alexander J. Generic Hopf bifurcation from lines of equilibria without parameters: I. theory. Journal of Differential Equations, 2000, vol. 167, iss. 1, pp. 16–35. DOI: https://doi.org/10.1006/jdeq.2000.3779
37. Chen L., Li Ch., Huang T., Chen Y., Wen Sh., Qi J. A synapse memristor model with forgetting effect. Physics Letters A, 2013, vol. 377, iss. 45–48, pp. 3260–3265. DOI: https://doi.org/10.1016/j.physleta.2013.10.024
38. Zhou E., Fang L., Yang B. A general method to describe forgetting effect of memristor. Physics Letters A, 2019, vol. 383, iss. 10, pp. 942–948. DOI: https://doi.org/10.1016/j.physleta.2018.12.028
Full text: