ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Korneev I. A., Slepnev A. V., Semenov V. V., Vadivasova T. E. Wave processes in a ring of memristively coupled self-excited oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics, 2020, vol. 28, iss. 3, pp. 324-340. DOI: 10.18500/0869-6632-2020-28-3-324-340

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Wave processes in a ring of memristively coupled self-excited oscillators

Korneev Ivan Aleksandrovich, Saratov State University
Slepnev Andrej Vjacheslavovich, Saratov State University
Vadivasova Tatjana Evgenevna, Saratov State University

The purpose of this work is to reveal intrinsic peculiarities of the dynamics and spatial structure formation in an ensemble of the coupled van der Pol self-oscillators in a case of memristive coupling. Two models of memristive coupling are considered: an idealised memristive model and a real one exhibiting the effect of «forgetting» of an initial state after a long time. Methods. Numerical simulation of the equations describing the system under study by means of the fourthorder Runge–Kutta method is carried out. Further exploration consists in plotting and analysis of space-time diagrams and instantaneous spatial profiles of dynamical regimes obtained by means of varying the initial conditions and parameter values. Results. It is shown that the shape of an instantaneous spatial profile fully depends on initial conditions in a case of ideal memristive coupling. Choosing the initial conditions, one can realize coexistence of different clusters with qualitatively different kinds of the dynamics (for instance, coexisting travelling waves and regimes of full synchronization). Such a phenomenon disappears in a system with the «forgetting» effect. Conclusion. The properties of memristive coupling strongly impact the behaviour of interacting self-oscillators. The system with ideal memristive coupling is very sensitive to initial conditions. It allows to control the dynamics in a broad range and to change the spatial profile form varying the initial conditions. 


1. Chua L.O. Memristor–The missing circuit element // IEEE Transactions on circuit theory. 1971. Vol. 1. P. 507–519.

2. Chua L.O., Kang S.M. Memristive devices and systems // Proceedings of the IEEE. 1976. Vol. 64, no. 2. P. 209–223.

3. Strukov D.B., Snider G.S., Stewart D.R., and Williams R.S. The missing memristor found // Nature. 2008. Vol. 453. P. 80–83.

4. Berzina T., Smerieri A., Bernabo 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, № 12. 124515.

5. Jeong H.Y. et al. Graphene oxide thin films for flexible nonvolatile memory applications // Nano letters. 2010. Vol. 10, no. 11. P. 4381–4386.

6. 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. P. 857–863.

7. Yang Y., Sheridan P., Lu W. Complementary resistive switching in tantalum oxide-based resistive memory devices // Applied Physics Letters. 2012. Vol. 100, no. 20. P. 203–112.

8. 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. P. 2194–2202.

9. Kim S., Choi S., Lu W. Comprehensive physical model of dynamic resistive switching in an oxide memristor // ACS Nano. 2014. Vol. 8, no. 3. P. 2369–2376.

10. Liu G., Chen Y., Wang C., Zhang W., Li R.-W., and Wang L. Polymer memristor for information storage and neuromorphic applications // Materials Horizons. 2014. Vol. 1, no. 5. P. 489–506.

11. Demin V., Erokhin V., Emelyanov A., Battistoni S., Baldi G., Iannotta S., Kashkarov P., and Kovalchuk M. Hardware elementary perceptron based on polyaniline memristive devices // Organic Electronics. 2015. Vol. 25. P. 16–20.

12. Erokhina S., Sorokin V., Erokhin V. Polyaniline-based organic memristive device fabricated by layed-by-layed deposition technique // Electronic Materials Letters. 2015. Vol. 11, no. 5. P. 801–805.

13. Pershin Y.V., Di Ventra M. Practical approach to programmable analog circuits with memristors // IEEE Transactions on Circuits and Systems I. 2010. Vol. 57. P. 1857–1864.

14. Pershin Y.V., Di Ventra M. Memory effects in complex materials and nanoscale systems // Advances in Physics. 2011. Vol. 60. P. 145–227.

15. Chew Z., Li L. Printed circuit board based memristor in adaptive lowpass filter // Electronics Letters. 2012. Vol. 48, no. 25. P. 1610–1611.

16. Di Ventra M., Pershin Y. The parallel approach // Nature Physics. 2013. Vol. 9, no. 4. P. 200–202.

17. Yang J., Strukov D., Stewart D. Memristive devices for computing // Nature Nanotechnology. 2013. Vol. 8, no. 1. P. 13–24.

18. Tetzlaff R. Memristor and Memristive Systems. New York: Springer Science & Business Media, 2014.

19. Vourkas I., Sirakoulis G.C. Memristor-Based Nanoelectronic Computing Circuits and Architectures. Cham: Springer International Publishing, 2016. Vol. 19.

20. Pershin V., Di Ventra M. Experimental demonstration of associative memory with memristive neural networks // Neural networks. 2010. Vol. 23, no. 7. P. 881–886.

21. Jo S.H., Chang T., Ebong I., Bhadviya B.B., Mazumder P., Lu W. Nanoscale memristor device as synapse in neuromorphic systems // Nano letters. 2010. Vol. 10, no. 4. P. 1297–1301.

22. Wu A. and Zeng Z. Dynamic behaviors of memristor-based recurrent neural networks with timevarying delays // Neural networks. 2012. Vol. 36. P. 1–10.

23. Guo Z., Wang J., Yan Z. Global exponential dissipativity and stabilization of memristor-based recurrent neural networks with time-varying delays // Neural Netw. 2013. Vol. 48. P. 158–172.

24. Guo Z., Wang J., Yan Z. Attractivity analysis of memristor-based cellular neural networks with time-varying delays // IEEE transactions on neural networks and learning systems. 2014. Vol. 25, no. 4. P. 704–717.

25. Zhao H., Li L., Peng H., Kurths J., Xiao J., Yang Y. Anti-synchronization for stochastic memristorbased neural networks with non-modeled dynamics via adaptrive control approach // EPJ B. 2015. Vol. 88, no. 5. P. 1–10.

26. Li R., Cao J., Tu Z. Passivity analysis of memristive neural networks with probabilistic timevarying delays // Neurocomputing. 2016. Vol. 191. P. 249–262.

27. Itoh M., Chua L.O. Memristor oscillators // International journal of bifurcation and chaos. 2008. Vol. 18. P. 3183–3206.

28. 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. P. 437–450.

29. Botta V.A., Nespoli C., Messias M. ´ Mathematical analysis of a third-order memristor-based Chua’s oscillator // Trends in Applied and Computational Mathematics. 2011. Vol. 12, no. 2. P. 91–99.

30. Riaza R. Manifolds of equilibria and bifurcations without parameters in memristive circuits // SIAM Journal on Applied Mathematics. 2012. Vol. 72, no. 3. P. 877–896.

31. 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. P. 1172–1188.

32. Semenov V.V., Korneev I.A., Arinushkin P.A., Strelkova G.I., Vadivasova T.E., Anishchenko V.S. Numerical and experimental studies of attractors in memristor-based Chua’s oscillator with a line of equilibria. Noise-induced effects // EPJ Special Topics. 2015. Vol. 224, no. 8. P. 1553–1561.

33. Korneev I.A., Vadivasova T.E., Semenov V.V. Hard and soft excitation of oscillations in memristorbased oscillators with a line of equilibria // Nonlinear dynamics. 2017. Vol. 89, no. 4. P. 2829–2843.

34. 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 and Technology Review. 2015. Vol. 8. P. 205–214.

35. Kengne J., Tabekoung Z.N., Namba V.K., Negou A.N. Periodicity, chaos and multiple attractors in a memristor-based Shinriki‘s circuit // Chaos. 2015. Vol. 25. 103126.

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, no. 1. P. 16–35.

37. Korneev I.A., Semenov V.V. Andronov-Hopf bifurcation with and without parameter in a cubic memristor oscillator with a line of equilibria // Chaos. 2017. Vol. 27, no. 8. 081104.

38. Lu M., Wang C.N., Ren G.D., et al. Model of electrical activity in a neuron under magnetic flow effect // Nonlinear dynamics. 2016. Vol. 85. P. 1479–1490.

39. Ma J., Zhang G., Hayat T., Ren G. Model electrical activity of neuron under electric field // Nonlinear dynamics. 2018. Vol. 92, no. 3. P. 1395–1402.

40. Wu F.Q., Wang C.N., Xu Y., et al. Model of electrical activity in cardiac tissue under electromagnetic induction // Scientific reports. 2016. Vol. 6, P. 28.

41. Frasca M., Gambuzza L., Buscarino A., Fortuna L. Implementation of adaptive coupling through memristor // Physica Status Solidi. 2014. Vol. 12, no. 1–2. P. 206–210.

42. Gambuzza L., Buscarino A., Fortuna L., and Frasca M. Memristor-based adaptive coupling for consensus and synchronization // IEEE Transactions on Circuits and Systems I. 2015. Vol. 62, no. 4. P. 1175–1184.

43. Volos Ch.K., Pham V.T., Vaidyanathan S., Kyprianidis I.M., Stouboulos I.N. The case of bidirectionally coupled nonlinear circuits via a memristor. Cham: Springer, 2016. Vol. 635. P. 317–350.

44. Ignatov M., Hansen M., Ziegler M., Kohlstedt H. Synchronization of two memristively coupled van der Pol oscillators // Applied Physics Letters. 2016. Vol. 108, no. 8. P. 84–105.

45. 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. P. 1395–1402.

46. 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. 

47. Pham V.T., Buscarino A., Fortuna L., Frasca M. Autowaves in memristive cellular neural networks // International journal of bifurcation and chaos. 2012. Vol. 22, no. 8. P. 1230027.

48. Buscarino A., Corradino C., Fortuna L., Frasca M., Chua L. Turing patterns in memristive cellular nonlinear networks // IEEE Transactions on Circuits and Systems I. 2016. Vol. 99. P. 1–9.

49. Ma J., Wu F.Q., Hayat T., et al. Electromagnetic induction and radiation-induced abnormality of wave propagation in excitable media // Physica A. 2017. Vol. 486. P. 508–516.

50. Wang C., Lv M., Alsaedi A., Ma J. Synchronization stability and pattern selection in a memristive neuronal network // Chaos. 2017. Vol. 27. P. 113108(1-8).

51. Xu F., Zhang J., Jin M., Huang Sh., Fang T. Chimera states and synchronization behavior in multilayer memristive neural networks // Nonlinear Dynamics. 2018. Vol. 94, no. 2. P. 775–783.

52. Chen L., Li Ch., Huang T., Chen Y., Wen Sh., Qi J. A synapse memristor model with fogetting effect // Physics Letters A. 2013. Vol. 377. P. 3260–3265.

53. Zhou E., Fang L., Yang B. A general method ti describe fogetting effect of memristor // Physics Letters A. 2019. Vol. 383, no. 10, P. 942–948.

54. Korneev I.A., Semenov V.V., Vadivasova T.E. Synchronization of periodic self-oscillators interacting via memristor-based coupling // International journal of bifurcation and chaos. 2020. P. 1–8.