Advertisement

Differential coupling contributes to synchronization via a capacitor connection between chaotic circuits

  • Yu-Meng Xu
  • Zhao Yao
  • Aatef Hobiny
  • Jun MaEmail author
Article

Abstract

Nonlinear oscillators and circuits can be coupled to reach synchronization and consensus. The occurrence of complete synchronization means that all oscillators can maintain the same amplitude and phase, and it is often detected between identical oscillators. However, phase synchronization means that the coupled oscillators just keep pace in oscillation even though the amplitude of each node could be different. For dimensionless dynamical systems and oscillators, the synchronization approach depends a great deal on the selection of coupling variable and type. For nonlinear circuits, a resistor is often used to bridge the connection between two or more circuits, so voltage coupling can be activated to generate feedback on the coupled circuits. In this paper, capacitor coupling is applied between two Pikovsk-Rabinovich (PR) circuits, and electric field coupling explains the potential mechanism for differential coupling. Then symmetric coupling and cross coupling are activated to detect synchronization stability, separately. It is found that resistor-based voltage coupling via a single variable can stabilize the synchronization, and the energy flow of the controller is decreased when synchronization is realized. Furthermore, by applying appropriate intensity for the coupling capacitor, synchronization is also reached and the energy flow across the coupling capacitor is helpful in regulating the dynamical behaviors of coupled circuits, which are supported by a continuous energy exchange between capacitors and the inductor. It is also confirmed that the realization of synchronization is dependent on the selection of a coupling channel. The approach and stability of complete synchronization depend on symmetric coupling, which is activated between the same variables. Cross coupling between different variables just triggers phase synchronization. The capacitor coupling can avoid energy consumption for the case with resistor coupling, and it can also enhance the energy exchange between two coupled circuits.

Key words

Synchronization Voltage coupling Chaotic circuit Capacitor coupling 

CLC number

O59 TN710 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adesnik H, Bruns W, Taniguchi H, et al., 2012. A neural circuit for spatial summation in visual cortex. Nature, 490(7419): 226–231. https://doi.org/10.1038/nature11526Google Scholar
  2. Andrievskii BR, Fradkov AL, 2004. Control of chaos: methods and applications. II. Applications. Autom Remote Contr, 65(4):505–533. https://doi.org/10.1023/B:AURC.0000023528.59389.09MathSciNetzbMATHGoogle Scholar
  3. Balenzuela P, García-Ojalvo J, 2005. Role of chemical synapses in coupled neurons with noise. Phys Rev E, 72: 021901. https://doi.org/10.1103/PhysRevE.72.021901Google Scholar
  4. Bao BC, Jiang T, Xu Q, et al., 2016. Coexisting infinitely many attractors in active band-pass filter-based memristive circuit. Nonl Dynam, 86(3):1711–1723. https://doi.org/10.1007/s11071-016-2988-6Google Scholar
  5. Bao BC, Jiang T, Wang GY, et al., 2017. Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability. Nonl Dynam, 89(2):1157–1171. https://doi.org/10.1007/s11071-017-3507-0Google Scholar
  6. Bennett MVL, 1997. Gap junctions as electrical synapses. J Neurocytol, 26(6):349–366. https://doi.org/10.1023/A:1018560803261Google Scholar
  7. Bennett MVL, 2000. Electrical synapses, a personal perspective (or history). Brain Res Rev, 32(1):16–28. https://doi.org/10.1016/S0165-0173(99)00065-XMathSciNetGoogle Scholar
  8. Boccaletti S, Farini A, Arecchi FT, 1997. Adaptive synchronization of chaos for secure communication. Phys Rev E, 55(5):4979–4981. https://doi.org/10.1103/PhysRevE.55.4979Google Scholar
  9. Budhathoki RK, Sah MP, Adhikari SP, et al., 2013. Composite behavior of multiple memristor circuits. IEEE Trans Circ Syst I, 60(10):2688–2700. https://doi.org/10.1109/TCSI.2013.2244320MathSciNetGoogle Scholar
  10. Burić N, Todorović K, Vasović N, 2008. Synchronization of bursting neurons with delayed chemical synapses. Phys Rev E, 78:036211. https://doi.org/10.1103/PhysRevE.78.036211Google Scholar
  11. Buscarino A, Fortuna L, Frasca M, 2009. Experimental robust synchronization of hyperchaotic circuits. Phys D, 238(18): 1917–1922. https://doi.org/10.1016/j.physd.2009.06.021zbMATHGoogle Scholar
  12. Buscarino A, Fortuna L, Frasca M, et al., 2012. A chaotic circuit based on Hewlett-Packard memristor. Chaos, 22(2):023136. https://doi.org/10.1063/1.4729135MathSciNetzbMATHGoogle Scholar
  13. Davison IG, Ehlers MD, 2011. Neural circuit mechanisms for pattern detection and feature combination in olfactory cortex. Neuron, 70(1):82–94. https://doi.org/10.1016/j.neuron.2011.02.047Google Scholar
  14. Eccles JC, 1982. The synapse: from electrical to chemical transmission. Ann Rev Neurosci, 5(1):325–339. https://doi.org/10.1146/annurev.ne.05.030182.001545Google Scholar
  15. Fell J, Axmacher N, 2011. The role of phase synchronization in memory processes. Nat Rev Neurosci, 12(2):105–118. https://doi.org/10.1038/nrn2979Google Scholar
  16. Guo SL, Xu Y, Wang CN, et al., 2017. Collective response, synapse coupling and field coupling in neuronal network. Chaos Sol Fract, 105:120–127. https://doi.org/10.1016/j.chaos.2017.10.019MathSciNetGoogle Scholar
  17. Guo SL, Ma J, Alsaedi A, 2018. Suppression of chaos via control of energy flow. Pramana, 90(3):39. https://doi.org/10.1007/s12043-018-1534-0Google Scholar
  18. Hahn SL, 1996. Hilbert Transforms in Signal Processing. Artech House, Boston, USA.zbMATHGoogle Scholar
  19. Hanias MP, Giannaris G, Spyridakis A, et al., 2006. Time series analysis in chaotic diode resonator circuit. Chaos Sol Fract, 27(2):569–573. https://doi.org/10.1016/j.chaos.2005.03.051zbMATHGoogle Scholar
  20. He DH, Shi PL, Stone L, 2003. Noise-induced synchronization in realistic models. Phys Rev E, 67(2):027201. https://doi.org/10.1103/PhysRevE.67.027201Google Scholar
  21. Ikezi H, de Grassie JS, Jensen TH, 1983. Observation of multiple-valued attractors and crises in a driven nonlinear circuit. Phys Rev A, 28(2):1207–1209. https://doi.org/10.1103/PhysRevA.28.1207MathSciNetGoogle Scholar
  22. Kiliç R, Alçi M, Çam U, et al., 2002. Improved realization of mixed-mode chaotic circuit. Int J Bifurc Chaos, 12(6): 1429–1435. https://doi.org/10.1142/S0218127402005236Google Scholar
  23. Kopell N, Ermentrout B, 2004. Chemical and electrical synapses perform complementary roles in the synchronization of interneuronal networks. PNAS, 101(43):15482–15487. https://doi.org/10.1073/pnas.0406343101Google Scholar
  24. Li XM, Wang J, Hu WH, 2007. Effects of chemical synapses on the enhancement of signal propagation in coupled neurons near the canard regime. Phys Rev E, 76:041902. https://doi.org/10.1103/PhysRevE.76.041902MathSciNetGoogle Scholar
  25. Louodop P, Fotsin H, Kountchou M, et al., 2014. Finite-time synchronization of tunnel-diode-based chaotic oscillators. Phys Rev E, 89:032921. https://doi.org/10.1103/PhysRevE.89.032921Google Scholar
  26. Lv M, Ma J, Yao YG, et al., 2018. Synchronization and wave propagation in neuronal network under field coupling. Sci China Technol Sci, in press. https://doi.org/10.1007/s11431-018-9268-2Google Scholar
  27. Ma J, Song XL, Tang J, et al., 2015. Wave emitting and propagation induced by autapse in a forward feedback neuronal network. Neurocomputing, 167:378–389. https://doi.org/10.1016/j.neucom.2015.04.056Google Scholar
  28. Ma J, Mi L, Zhou P, et al., 2017. Phase synchronization between two neurons induced by coupling of electromagnetic field. Appl Math Comput, 307:321–328. https://doi.org/10.1016/j.amc.2017.03.002MathSciNetGoogle Scholar
  29. Muthuswamy B, Chua LO, 2010. Simplest chaotic circuit. Int J Bifurc Chaos, 20(5):1567–1580. https://doi.org/10.1142/S0218127410027076Google Scholar
  30. Muthuswamy B, Kokate PP, 2009. Memristor-based chaotic circuits. IETE Tech Rev, 26(6):417–429. https://doi.org/10.4103/0256-4602.57827Google Scholar
  31. Nazzal JM, Natsheh AN, 2007. Chaos control using slidingmode theory. Chaos Sol Fract, 33(2):695–702. https://doi.org/10.1016/j.chaos.2006.01.071Google Scholar
  32. Neiman A, Schimansky-Geier L, Cornell-Bell A, et al., 1999. Noise-enhanced phase synchronization in excitable media. Phys Rev Lett, 83(23):4896–4899. https://doi.org/10.1103/PhysRevLett.83.4896Google Scholar
  33. Parlitz U, Junge L, Lauterborn W, et al., 1996. Experimental observation of phase synchronization. Phys Rev E, 54(2): 2115–2117. https://doi.org/10.1103/PhysRevE.54.2115Google Scholar
  34. Pikovsky AS, 1981. A dynamical model for periodic and chaotic oscillations in the Belousov-Zhabotinsky reaction. Phys Lett A, 85(1):13–16. https://doi.org/10.1016/0375-9601(81)90626-5Google Scholar
  35. Pikovsky AS, Rabinovich MI, 1978. A simple autogenerator with stochastic behaviour. Sov Phys Dokl, 23:183–185.Google Scholar
  36. Pikovsky A, Rosenblum M, Kurths J, 2000. Phase synchronization in regular and chaotic systems. Int J Bifurc Chaos, 10(10):2291–2305. https://doi.org/10.1142/S0218127400001481MathSciNetzbMATHGoogle Scholar
  37. Qin HX, Ma J, Jin WY, et al., 2014. Dynamics of electric activities in neuron and neurons of network induced by autapses. Sci China Technol Sci, 57(5):936–946. https://doi.org/10.1007/s11431-014-5534-0Google Scholar
  38. Ren GD, Zhou P, Ma J, et al., 2017. Dynamical response of electrical activities in digital neuron circuit driven by autapse. Int J Bifurc Chaos, 27(12):1750187. https://doi.org/10.1142/S0218127417501875MathSciNetGoogle Scholar
  39. Ren GD, Xue YX, Li YW, et al., 2019. Field coupling benefits signal exchange between Colpitts systems. Appl Math Comput, 342:45–54. https://doi.org/10.1016/j.amc.2018.09.017MathSciNetGoogle Scholar
  40. Schöll E, Schuster HG, 2008. Handbook of Chaos Control. John Wiley & Sons, Weinheim, Germany.Google Scholar
  41. Song XL, Wang CN, Ma J, et al., 2015. Transition of electric activity of neurons induced by chemical and electric autapses. Sci China Technol Sci, 58(6):1007–1014. https://doi.org/10.1007/s11431-015-5826-zGoogle Scholar
  42. Tang J, Zhang J, Ma J, et al., 2017. Astrocyte calcium wave induces seizure-like behavior in neuron network. Sci China Technol Sci, 60(7):1011–1018. https://doi.org/10.1007/s11431-016-0293-9Google Scholar
  43. Timmer J, Rust H, Horbelt W, et al., 2000. Parametric, nonparametric and parametric modelling of a chaotic circuit time series. Phys Lett A, 274(3–4): 123–134. https://doi.org/10.1016/S0375-9601(00)00548-XMathSciNetzbMATHGoogle Scholar
  44. Wang CN, Ma J, Liu Y, et al., 2012. Chaos control, spiral wave formation, and the emergence of spatiotemporal chaos in networked Chua circuits. Nonl Dynam, 67(1):139–146. https://doi.org/10.1007/s11071-011-9965-xzbMATHGoogle Scholar
  45. Wang CN, Chu RT, Ma J, 2015. Controlling a chaotic resonator by means of dynamic track control. Complexity, 21(1): 370–378. https://doi.org/10.1002/cplx.21572MathSciNetGoogle Scholar
  46. Wu CW, Chua LO, 1993. A simple way to synchronize chaotic systems with applications to secure communication systems. Int J Bifurc Chaos, 3(6):1619–1627. https://doi.org/10.1142/S0218127493001288zbMATHGoogle Scholar
  47. Xu Y, Jia Y, Ma J, et al., 2018. Collective responses in electrical activities of neurons under field coupling. Sci Rep, 8(1):1349. https://doi.org/10.1038/s41598-018-19858-1Google Scholar
  48. Yu WT, Zhang J, Tang J, 2017. Effects of dynamic synapses, neuronal coupling, and time delay on firing of neuron. Acta Phys Sin, 66:200201.Google Scholar
  49. Zaher AA, Abu-Rezq A, 2011. On the design of chaos-based secure communication systems. Commun Nonl Sci Numer Simul, 16(9):3721–3737. https://doi.org/10.1016/j.cnsns.2010.12.032Google Scholar
  50. Zhang G, Ma J, Alsaedi A, et al., 2018a. Dynamical behavior and application in Josephson junction coupled by memristor. Appl Math Comput, 321:290–299. https://doi.org/10.1016/j.amc.2017.10.054MathSciNetGoogle Scholar
  51. Zhang G, Wu FQ, Hayat T, et al., 2018b. Selection of spatial pattern on resonant network of coupled memristor and Josephson junction. Commun Nonl Sci Numer Simul, 65:79–90. https://doi.org/10.1016/j.cnsns.2018.05.018MathSciNetGoogle Scholar

Copyright information

© Zhejiang University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of PhysicsLanzhou University of TechnologyLanzhouChina
  2. 2.NAAM-Research Group, Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations