Advertisement

Journal of Zhejiang University-SCIENCE A

, Volume 19, Issue 12, pp 889–903 | Cite as

Synchronization stability between initial-dependent oscillators with periodical and chaotic oscillation

  • Fu-qiang Wu
  • Jun Ma
  • Guo-dong Ren
Article

Abstract

The selection of periodical or chaotic attractors becomes initial-dependent in that setting different initial values can trigger a different profile of attractors in a dynamical system with memory by adding a nonlinear term such as z2y in the Rössler system. The memory effect means that the outputs are very dependent on the initial value for variable z, e.g. magnetic flux for a memristor. In this study, standard nonlinear analyses, including phase portrait, bifurcation analysis, and Lyapunov exponent analysis were carried out. Synchronization between two coupled oscillators and a network was investigated by resetting initial states. A statistical synchronization factor was calculated to find the dependence of synchronization on the coupling intensity when different initial values were selected. Our results show that the dynamics of the attractor depends on the selection of the initial value for one variable z. In the case of coupling between two oscillators, appropriate initial values are selected to trigger two different nonlinear oscillators (periodical and chaotic). Results show that complete synchronization between periodical oscillators, chaotic oscillators, and periodical and chaotic oscillators can be realized by applying an appropriate unidirectional coupling intensity. In particular, two periodical oscillators can be coupled bidirectionally to reach chaotic synchronization so that periodical oscillation is modulated to become chaotic. When the memory effect is considered on some nodes of a chain network, enhancement of memory function can decrease the synchronization, while a small region for intensity of memory function can contribute to the synchronization of the network. Finally, dependence of attractor formation on the initial setting was verified on the field programmable gate array (FPGA) circuit in digital signal processing (DSP) builder block under Matlab/Simulink.

Key words

Synchronization Bifurcation Synchronization factor Field programmable gate array (FPGA) 

初始值敏感的周期和混沌振荡模态系统同步稳定性

摘要

目的

具有记忆特性的振子系统的模态选择对初始值具有敏感性。本文旨在探讨初始值控制的振子的耦合同步稳定一致性问题。

创新点

1. 两个周期振子耦合后达到混沌同步;2. 周期振子和混沌振子耦合后达到周期性振荡同步。

方法

1. 通过分岔分析,研究振荡模态和初始值选择之间的关系(图2、6 和8);2. 通过数值计算,研究两个周期振子在耦合下的混沌同步关系(图 7);3. 通过计算同步因子和斑图,分析同步一致 性对耦合强度与记忆函数增益的依赖程度(图9 和10);4. 通过现场可编程门阵列验证动力系统 模态对初始值的依赖程度(图11 和12)。

结论

1. 具有记忆函数的非线性振子的动力学行为(如吸引子)在参数固定的情况下与初始值选取有 关。2. 不同类型振子的耦合可以达到多样同步行为;周期振子耦合达到混沌同步;周期振子耦合 混沌振子可以抑制混沌。3. 包含记忆函数的振子 网络耦合同步非常困难。

关键词

同步 分岔 同步因子 现场可编程门阵列 

CLC number

O59 TN710 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anshan H, 1988. A study of the chaotic phenomena in Chua’s circuit. IEEE International Symposium on Circuits and Systems, p.273–276. https://doi.org/10.1109/ISCAS.1988.14919 Google Scholar
  2. Carroll TL, Pecora LM, 1993. Synchronizing nonautonomous chaotic circuits. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 40(10):646–650. https://doi.org/10.1109/82.246166 CrossRefGoogle Scholar
  3. Chen S, Wang D, Li C, et al., 2004. Synchronizing strictfeedback chaotic system via a scalar driving signal. Chaos, 14(3):539–544. https://doi.org/10.1063/1.1749233 MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cheng AL, Chen YY, 2017. Analyzing the synchronization of Rössler systems—when trigger-and-reinject is equally important as the spiral motio. Physics Letters A, 381(42):3641–3651. https://doi.org/10.1016/j.physleta.2017.09.042 MathSciNetCrossRefGoogle Scholar
  5. de la Fraga LG, Tlelo-Cuautle E, 2014. Optimizing the maximum Lyapunov exponent and phase space portraits in multi-scroll chaotic oscillators. Nonlinear Dynamics, 76(2):1503–1515. https://doi.org/10.1007/s11071-013-1224-x CrossRefGoogle Scholar
  6. Fixman M, 1984. Absorption by static traps: initial-value and steady-state problems. Journal of Chemical Physics, 81(8):3666–3677. https://doi.org/10.1063/1.448116 CrossRefGoogle Scholar
  7. Friedman D, Strowbridge BW, 2003. Both electrical and chemical synapses mediate fast network oscillations in the olfactory bulb. Neurophysiology, 89(5):2601–2610. https://doi.org/10.1152/jn.00887.2002 CrossRefGoogle Scholar
  8. Giordano R, Aloisio A, 2011. Fixed-latency, multi-gigabit serial links with Xilinx FPGAs. IEEE Transactions on Nuclear Science, 58(1):194–201. https://doi.org/10.1109/TNS.2010.2101083 CrossRefGoogle Scholar
  9. Graubard K, Hartline DK, 1987. Full-wave rectification from a mixed electrical-chemical synapse. Science, 237(4814): 535–537. https://doi.org/10.1126/science.2885921 CrossRefGoogle Scholar
  10. Györgyi L, Field RJ, 1992. A three-variable model of deterministic chaos in the Belousov-Zhabotinsky reaction. Nature, 355(6363):808–810. https://doi.org/10.1038/355808a0 CrossRefGoogle Scholar
  11. Heagy JF, Carroll TL, Pecora LM, 1995. Desynchronization by periodic orbits. Physical Review E, 52(2):R1253–R1256. https://doi.org/10.1103/PhysRevE.52.R1253 CrossRefGoogle Scholar
  12. Honein T, Chien N, Herrmann G, 1991. On conservation laws for dissipative systems. Physics Letters A, 155(4-5):223–224. https://doi.org/10.1016/0375-9601(91)90472-K MathSciNetCrossRefGoogle Scholar
  13. Hu G, Xiao J, Yang J, et al., 1997. Synchronization of spatiotemporal chaos and its applications. Physical Review E, 56(3):2738–2746. https://doi.org/10.1103/PhysRevE.56.2738 CrossRefGoogle Scholar
  14. Hunt BR, Ott E, Yorke JA, 1997. Differentiable generalized synchronization of chaos. Physical Review E, 55(44): 4029–4034. https://doi.org/10.1103/PhysRevE.55.4029 MathSciNetCrossRefGoogle Scholar
  15. Jin W, Lin Q, Wang A, et al., 2017. Computer simulation of noise effects of the neighborhood of stimulus threshold for a mathematical model of homeostatic regulation of sleep-wake cycles. Complexity, Article No. 4797545. https://doi.org/10.1155/2017/4797545 CrossRefzbMATHGoogle Scholar
  16. Lau FCM, Tse CK, 2003. Approximate-optimal detector for chaos communication systems. International Journal of Bifurcation and Chaos, 13(5):1329–1335. https://doi.org/10.1142/S0218127403007266 CrossRefzbMATHGoogle Scholar
  17. Li C, Chen L, Aihara K, 2008. Impulsive control of stochastic systems with applications in chaos control, chaos synchronization, and neural networks. Chaos, 18(2):23132. https://doi.org/10.1063/1.2939483 MathSciNetCrossRefzbMATHGoogle Scholar
  18. Lin W, He Y, 2005. Complete synchronization of the noiseperturbed Chua’s circuits. Chaos, 15(2):23705. https://doi.org/10.1063/1.1938627 CrossRefGoogle Scholar
  19. Liu X, Ma G, Jiang X, et al., 2016. H8 stochastic synchronization for master-slave semi-Markovian switching system via sliding mode control. Complexity, 21(6):430–441. https://doi.org/10.1002/cplx.21702 MathSciNetCrossRefGoogle Scholar
  20. Luo ACJ, Han RPS, 2000. The dynamics of stochastic and resonant layers in a periodically driven pendulum. Chaos Solitons & Fractals, 11(14):2349–2359. https://doi.org/10.1016/S0960-0779(99)00162-9 MathSciNetCrossRefzbMATHGoogle Scholar
  21. Luo ACJ, Min F, 2011. Synchronization dynamics of two different dynamical systems. Chaos Solitons & Fractals, 44(6):362–380. https://doi.org/10.1016/j.chaos.2010.12.011 MathSciNetCrossRefzbMATHGoogle Scholar
  22. Lv M, Wang C, Ren G, et al., 2016. Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dynamics, 85(3):1479–1490. https://doi.org/10.1007/s11071-016-2773-6 CrossRefGoogle Scholar
  23. Ma J, Li F, Huang L, et al., 2011. Complete synchronization, phase synchronization and parameters estimation in a realistic chaotic system. Communications in Nonlinear Science and Numerical Simulation, 16(9):3770–3785. https://doi.org/10.1016/j.cnsns.2010.12.030 CrossRefzbMATHGoogle Scholar
  24. Ma J, Wu F, Ren G, et al., 2017a. A class of initials-dependent dynamical systems. Applied Mathematics and Computation, 298:65–76. https://doi.org/10.1016/j.amc.2016.11.004 MathSciNetCrossRefGoogle Scholar
  25. Ma J, Mi L, Zhou P, et al., 2017b. Phase synchronization between two neurons induced by coupling of electromagnetic field. Applied Mathematics and Computation, 307:321–328. https://doi.org/10.1016/j.amc.2017.03.002 MathSciNetCrossRefGoogle Scholar
  26. Ma J, Wu F, Wang C, 2017c. Synchronization behaviors of coupled neurons under electromagnetic radiation. International Journal of Modern Physics B, 31(2):1650251.MathSciNetCrossRefGoogle Scholar
  27. Mahmoud GM, Mahmoud EE, 2010. Complete synchronization of chaotic complex nonlinear systems with uncertain parameters. Nonlinear Dynamics, 62(4):875–882. https://doi.org/10.1007/s11071-010-9770-y MathSciNetCrossRefzbMATHGoogle Scholar
  28. Mankin R, Laas K, Laas T, et al., 2018. Memory effects for a stochastic fractional oscillator in a magnetic field. Physical Review E, 97(1):12145. https://doi.org/10.1103/PhysRevE.97.012145 CrossRefzbMATHGoogle Scholar
  29. Martinet B, Adrian RJ, 1988. Rayleigh-Benard convection: experimental study of time-dependent instabilities. Experiments in Fluids, 6(5):316–322. https://doi.org/10.1007/BF00538822 CrossRefGoogle Scholar
  30. Osipov GV, Pikovsky AS, Kurths J, 2002. Phase synchronization of chaotic rotators. Physical Review Letters, 88(4): 54102. https://doi.org/10.1103/PhysRevLett.88.054102 CrossRefGoogle Scholar
  31. Pano-Azucena AD, Rangel-Magdaleno JJ, Tlelo-Cuautle E, et al., 2017. Arduino-based chaotic secure communication system using multi-directional multi-scroll chaotic oscillators. Nonlinear Dynamics, 87(4):2203–2217. https://doi.org/10.1007/s11071-016-3184-4 CrossRefGoogle Scholar
  32. Pecora LM, Carroll TL, 2015. Synchronization of chaotic systems. Chaos, 25(9):97611. https://doi.org/10.1063/1.4917383 CrossRefGoogle Scholar
  33. Pena Ramirez J, Fey RHB, Nijmeijer H, 2013. Synchronization of weakly nonlinear oscillators with Huygens’ coupling. Chaos, 23(3):33118. https://doi.org/10.1063/1.4816360 MathSciNetCrossRefzbMATHGoogle Scholar
  34. Peng JH, Ding EJ, Ding M, et al., 1996. Synchronizing hyperchaos with a scalar transmitted signal. Physical Review Letters, 76(6):904–907. https://doi.org/10.1103/PhysRevLett.76.904 CrossRefGoogle Scholar
  35. Pereda AE, 2014. Electrical synapses and their functional interactions with chemical synapses. Nature Reviews Neuroscience, 15:250–263. https://doi.org/10.1038/nrn3708 CrossRefGoogle Scholar
  36. Podvigin NF, Bagaeva TV, Podvigina DN, et al., 2008. Selective self-synchronization of impulse flows in neuronal networks of the visual system. Biophysics, 53(2):177–181. https://doi.org/10.1134/S0006350908020097 CrossRefGoogle Scholar
  37. Rössler OE, 1976. An equation for continuous chaos. Physics Letters A, 57(5):397–398. https://doi.org/10.1016/0375-9601(76)90101-8 CrossRefzbMATHGoogle Scholar
  38. Ruggieri M, Speciale MP, 2017. On the construction of conservation laws: a mixed approach. Journal of Mathematical Physics, 58(2):23510. https://doi.org/10.1063/1.4976189 MathSciNetCrossRefzbMATHGoogle Scholar
  39. Sieberer LM, Huber SD, Altman E, et al., 2013. Dynamical critical phenomena in driven-dissipative systems. Physical Review Letters, 110(19):195301. https://doi.org/10.1103/PhysRevLett.110.195301 CrossRefGoogle Scholar
  40. Szot P, 2012. Common factors among Alzheimer’s disease, Parkinson’s disease, and epilepsy: possible role of the noradrenergic nervous system. Epilepsia, 53(S1):61–66. https://doi.org/10.1111/j.1528-1167.2012.03476.x CrossRefGoogle Scholar
  41. Tang YX, Khalaf AJM, Rajagopal K, et al., 2018. A new nonlinear oscillator with infinite number of coexisting hidden and self-excited attractors. Chinese Physics B, 27(4):40502. https://doi.org/10.1088/1674-1056/27/4/040502 CrossRefGoogle Scholar
  42. Tlelo-Cuautle E, Carbajal-Gomez VH, Obeso-Rodelo PJ, et al., 2015a. FPGA realization of a chaotic communication system applied to image processing. Nonlinear Dynamics, 82(4):1879–1892. https://doi.org/10.1007/s11071-015-2284-x MathSciNetCrossRefGoogle Scholar
  43. Tlelo-Cuautle E, Rangel-Magdaleno JJ, Pano-Azucena AD, et al., 2015b. FPGA realization of multi-scroll chaotic oscillators. Communications in Nonlinear Science and Numerical Simulation, 27(1-3):66–80. https://doi.org/10.1016/j.cnsns.2015.03.003 MathSciNetCrossRefGoogle Scholar
  44. Tlelo-Cuautle E, Rangel-Magdaleno J, de la Fraga LG, 2016a. Engineering Applications of FPGAs: Chaotic Systems, Artificial Neural Networks, Random Number Generators, and Secure Communication Systems. Springer, Cham, Switzerland. https://doi.org/10.1007/978-3-319-34115-6 CrossRefGoogle Scholar
  45. Tlelo-Cuautle E, Pano-Azucena AD, Rangel-Magdaleno JJ, et al., 2016b. Generating a 50-scroll chaotic attractor at 66 MHz by using FPGAs. Nonlinear Dynamics, 85(4): 2143–2157. https://doi.org/10.1007/s11071-016-2820-3 CrossRefGoogle Scholar
  46. Tlelo-Cuautle E, de Jesus Quintas-Valles A, de la Fraga LG, et al., 2016c. VHDL descriptions for the FPGA implementation of PWL-function-based multi-scroll chaotic oscillators. PLoS ONE, 11(2):e0168300. https://doi.org/10.1371/journal.pone.0168300 CrossRefGoogle Scholar
  47. Tlelo-Cuautle E, de la Fraga LG, Pham VT, et al., 2017. Dynamics, FPGA realization and application of a chaotic system with an infinite number of equilibrium points. Nonlinear Dynamics, 89(2):1129–1139. https://doi.org/10.1007/s11071-017-3505-2 CrossRefGoogle Scholar
  48. Trejo-Guerra R, Tlelo-Cuautle E, Jiménez-Fuentes JM, et al., 2012. Integrated circuit generating 3-and 5-scroll attractors. Communications in Nonlinear Science and Numerical Simulation, 17(11):4328–4335. https://doi.org/10.1016/j.cnsns.2012.01.029 MathSciNetCrossRefGoogle Scholar
  49. Trejo-Guerra R, Tlelo-Cuautle E, Carbajal-Gomez VH, et al., 2013. A survey on the integrated design of chaotic oscillators. Applied Mathematics and Computation, 219(10): 5113–5122. https://doi.org/10.1016/j.amc.2012.11.021 MathSciNetCrossRefzbMATHGoogle Scholar
  50. Ueda Y, Akamatsu N, 1981. Chaotically transitional phenomena in the forced negative-resistance oscillator. IEEE Transactions on Circuits and Systems, 28(3):217–224. https://doi.org/10.1109/TCS.1981.1084975 MathSciNetCrossRefGoogle Scholar
  51. Vaidyanathan S, Volos C, 2016. Advances and Applications in Chaotic Systems. Springer, Cham, Switzerland. https://doi.org/10.1007/978-3-319-30279-9 CrossRefzbMATHGoogle Scholar
  52. Vyas S, Huang H, Gale JT, et al., 2016. Neuronal complexity in subthalamic nucleus is reduced in Parkinson’s disease. IEEE Transactions Neurology System Rehabilitation Engineering, 24(1):36–45. https://doi.org/10.1109/TNSRE.2015.2453254 CrossRefGoogle Scholar
  53. Wang C, Ma J, 2018. A review and guidance for pattern selection in spatiotemporal system. International Journal of Modern Physics B, 32(6):1830003.MathSciNetCrossRefGoogle Scholar
  54. Wang G, Jin W, Liu H, et al., 2018. The synchronization of asymmetric-structured electric coupling neuronal system. International Journal of Modern Physics B, 32(4): 1850040.CrossRefGoogle Scholar
  55. Wang H, Wang Q, Lu Q, 2011. Bursting oscillations, bifurcation and synchronization in neuronal systems. Chaos Solitons & Fractals, 44(8):667–675. https://doi.org/10.1016/j.chaos.2011.06.003 CrossRefzbMATHGoogle Scholar
  56. Wang Y, Ma J, Xu Y, et al., 2017. The electrical activity of neurons subject to electromagnetic induction and Gaussian white noise. International Journal of Bifurcation and Chaos, 27(2):1750030. https://doi.org/10.1142/S0218127417500304 MathSciNetCrossRefzbMATHGoogle Scholar
  57. Wu F, Wang C, Xu Y, et al., 2016. Model of electrical activity in cardiac tissue under electromagnetic induction. Science Reports, 6(1):28. https://doi.org/10.1038/s41598-016-0031-2 CrossRefGoogle Scholar
  58. Wu F, Wang C, Jin W, et al., 2017. Dynamical responses in a new neuron model subjected to electromagnetic induction and phase noise. Physica A: Statistical Mechanics and Its Applications, 469:81–88. https://doi.org/10.1016/j.physa.2016.11.056 MathSciNetCrossRefzbMATHGoogle Scholar
  59. Xu Y, Ying H, Jia Y, et al., 2017. Autaptic regulation of electrical activities in neuron under electromagnetic induction. Science Reports, 7:43452. https://doi.org/10.1038/srep43452 CrossRefGoogle Scholar
  60. Yamada T, Fujisaka H, 1986. Intermittency caused by chaotic modulation. I: Analysis with a multiplicative noise model. Progress of Theoretical Physics, 76(3):582–591. https://doi.org/10.1143/PTP.76.582 Google Scholar
  61. Yang SS, Duan CK, 1998. Generalized synchronization in chaotic systems. Chaos Solitons & Fractals, 9(10):1703–1707. https://doi.org/10.1016/S0960-0779(97)00149-5 MathSciNetCrossRefzbMATHGoogle Scholar
  62. Yang T, Yang LB, Yang CM, 1997. Impulsive control of Lorenz system. Physica D: Nonlinear Phenomena, 110(1-2):18–24. https://doi.org/10.1016/S0167-2789(97)00116-4 MathSciNetCrossRefzbMATHGoogle Scholar
  63. Zhang XH, Liu SQ, 2018. Stochastic resonance and synchronization behaviors of excitatory-inhibitory small-world network subjected to electromagnetic induction. Chinese Physics B, 27(4):40501. https://doi.org/10.1088/1674-1056/27/4/040501 CrossRefGoogle Scholar

Copyright information

© Zhejiang University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsLanzhou University of TechnologyLanzhouChina
  2. 2.School of ScienceChongqing University of Posts and TelecommunicationsChongqingChina
  3. 3.NAAM-Research Group, Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations