Journal of Applied Mathematics

The Positive Role of Multiplicative Noise in Complete Synchronization of Unidirectionally Coupled Ring with Three Nodes

Yuzhu Xiao, Sufang Tang, and Zhongkui Sun

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The role of multiplicative noise in the synchronization of unidirectionally coupled ring with three nodes is studied. Based on the theory of stochastic differential equations, we demonstrate that noise plays a positive role in complete synchronization. In numerical simulations, the Lorenz system, Rössler like system, and Hindmarsh-Rose neuron model are employed to demonstrate the correctness of our theoretical result.

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J. Appl. Math., Volume 2014 (2014), Article ID 741961, 7 pages.

First available in Project Euclid: 2 March 2015

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Xiao, Yuzhu; Tang, Sufang; Sun, Zhongkui. The Positive Role of Multiplicative Noise in Complete Synchronization of Unidirectionally Coupled Ring with Three Nodes. J. Appl. Math. 2014 (2014), Article ID 741961, 7 pages. doi:10.1155/2014/741961.

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  • A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge unverisity press, Cambridge, Mass, USA, 2001.
  • S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou, “The synchronization of chaotic systems,” Physics Report, vol. 366, no. 1-2, pp. 1–101, 2002.
  • H. Fujisaka and T. Yamada, “Stability theory of synchronized motion in coupled-oscillator systems,” Progress of Theoretical Physics, vol. 69, no. 1, pp. 32–47, 1983.
  • L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990.
  • L. M. Pecora and T. L. Carroll, “Driving systems with chaotic signals,” Physical Review A, vol. 44, no. 4, pp. 2374–2383, 1991.
  • M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Physical Review Letters, vol. 76, no. 11, pp. 1804–1807, 1996.
  • M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “From phase to lag synchronization in coupled chaotic oscillators,” Physical Review Letters, vol. 78, no. 22, pp. 4193–4196, 1997.
  • Y. Sun and J. Cao, “Adaptive lag synchronization of unknown chaotic delayed neural networks with noise perturbation,” Physics Letters A, vol. 364, pp. 277–285, 2007.
  • L. Kocarev and U. Parlitz, “Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems,” Physical Review Letters, vol. 76, no. 11, pp. 1816–1819, 1996.
  • X. Huang and J. Cao, “Generalized synchronization for delayed chaotic neural networks: a novel coupling scheme,” Nonlinearity, vol. 19, no. 12, pp. 2797–2811, 2006.
  • C. Li, X. Liao, and K. Wong, “Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication,” Physica D: Nonlinear Phenomena, vol. 194, no. 3-4, pp. 187–202, 2004.
  • L. Zhang, X. An, and J. Zhang, “A new chaos synchronization scheme and its application to secure communications,” Nonlinear Dynamics, vol. 73, no. 1-2, pp. 705–722, 2013.
  • M. A. Matías, V. Pérez-Muñuzuri, M. N. Lorenzo, I. P. Mariño, and V. Pérez-Villar, “Observation of a fast rotating wave in rings of coupled chaotic oscillators,” Physical Review Letters, vol. 78, no. 2, p. 219, 1997.
  • M. A. Matías and J. Güémez, “Transient periodic rotating waves and fast propagation of synchronization in linear arrays of chaotic systems,” Physical Review Letters, vol. 81, p. 4124, 1998.
  • Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, Berlin, Germany, 1984.
  • L. O. Chua, M. Itah, L. Kosarev, and K. Eckert, “Chaos synchronization in Chua's circuit,” Journal of Circuits, Systems and Computers, vol. 3, no. 1, p. 93, 1993.
  • H. N. Agiza and M. T. Yassen, “Synchronization of Rossler and Chen chaotic dynamical systems using active control,” Physics Letters A, vol. 278, no. 4, pp. 191–197, 2001.
  • D. Huang, “Simple adaptive-feedback controller for identical chaos synchronization,” Physical Review E, vol. 71, Article ID 037203, 2005.
  • B. Zhang and H. Li, “A new four-dimensional autonomous hyperchaotic system and the synchronization of different chaotic systems by using fast terminal sliding mode control,” Mathematical Problems in Engineering, vol. 2013, Article ID 179428, 8 pages, 2013.
  • X. Li, J. Gu, and W. Xu, “Adaptive integral observer-based synchronization for chaotic systems with unknown parameters and disturbances,” Journal of Applied Mathematics, vol. 2013, Article ID 501421, 8 pages, 2013.
  • A. S. Pikovsky, “Synchronization and stochastization of the ensemble of autogenrators by external noise,” Radiophysics and Quantum Electronics, vol. 27, p. 576, 1984.
  • R. V. Jensen, “Synchronization of randomly driven nonlinear oscillators,” Physical Review E, vol. 58, Article ID R6907, 1998.
  • A. Maritan and J. R. Banavar, “Chaos, noise, and synchronization,” Physical Review Letters, vol. 72, no. 10, article 1451, 1994.
  • A. S. Pikovsky, “Comment on “Chaos, noise, and synchronization”,” Physical Review Letters, vol. 73, no. 21, p. 2931, 1994.
  • H. Herzel and J. Freund, “Chaos, noise, and synchronization reconsidered,” Physical Review E, vol. 52, no. 3, pp. 3238–3241, 1995.
  • G. Malescio, “Noise and synchronization in chaotic systems,” Physical Review E, vol. 53, p. 6551, 1996.
  • E. Sánchez, M. A. Matías, and V. Pérez-Muñuzuri, “Analysis of synchronization of chaotic systems by noise: an experimental study,” Physical Review E, vol. 56, pp. 4068–4071, 1997.
  • C. Lai and C. Zhou, “Synchronization of chaotic maps by symmetric common noise,” Europhysics Letters, vol. 43, no. 4, article 376, 1998.
  • R. Toral, C. R. Mirasso, E. Hernández-García, and O. Piro, “Analytical and numerical studies of noise-induced synchronization of chaotic systems,” Chaos, vol. 11, no. 3, pp. 665–673, 2001.
  • Y. Wu, J. Xu, D. He, and D. J. D. Earn, “Generalized synchronization induced by noise and parameter mismatching in Hindmarsh-Rose neurons,” Chaos, Solitons and Fractals, vol. 23, no. 5, pp. 1605–1611, 2005.
  • C. Zhou and J. Kurths, “Noise-induced synchronization and coherence resonance of a Hodgkin-Huxley model of thermally sensitive neurons,” Chaos, vol. 13, no. 1, pp. 401–409, 2003.
  • C. Zhou, J. Kurths, E. Allaria, S. Boccaletti, R. Meucci, and F. T. Arecchi, “Noise-enhanced synchronization of homoclinic chaos in a CO$_{2}$ laser,” Physical Review E, vol. 67, Article ID 015205(R), 2003.
  • P. Tass, M. G. Rosenblum, J. Weule et al., “Detection of n:m phase locking from noisy data: application to magnetoencephalography,” Physical Review Letters, vol. 81, no. 15, pp. 3291–3294, 1998.
  • Z. Sun and X. Yang, “Generating and enhancing lag synchronization of chaotic systems by white noise,” Chaos, vol. 21, Article ID 033114, 2011.
  • W. Lin and Y. He, “Complete synchronization of the noise-perturbed Chua\textquotesingle s circuits,” Chaos, vol. 15, no. 2, Article ID 023705, 2005.
  • W. Lin, “Realization of synchronization in time-delayed systems with stochastic perturbation,” Journal of Physics. A. Mathematical and Theoretical, vol. 41, no. 23, 235101, 17 pages, 2008.
  • A. Hu and Z. Xu, “Stochastic linear generalized synchronization of chaotic systems via robust control,” Physics Letters A, vol. 372, no. 21, pp. 3814–3818, 2008.
  • J. Cao, Z. Wang, and Y. Sun, “Synchronization in an array of linearly stochastically coupled networks with time delays,” Physica A: Statistical Mechanics and its Applications, vol. 385, no. 2, pp. 718–728, 2007.
  • W. Lin and G. Chen, “Using white noise to enhance synchronization of coupled chaotic systems,” Chaos, vol. 16, no. 1, Article ID 013134, 2006.
  • Y. Xiao, W. Xu, X. Li, and S. Tang, “The effect of noise on the complete synchronization of two bidirectionally coupled piecewise linear chaotic systems,” Chaos, vol. 19, no. 1, Article ID 013131, 2009.
  • Y. M. Vega, M. Vázquez-Prada, and A. F. Pacheco, “Fitness for synchronization of network motifs,” Physica A, vol. 343, no. 1-4, pp. 279–287, 2004.
  • W. Horsthemke and R. Lefever, Noise-Induced Transitions: Theory and Application in Physics, Chemistry and Biology, Springer, Berlin, Germany, 1989.
  • G. Pesce, A. McDaniel, S. Hottovy, J. Wehr, and G. Volpe, “Stratonovich-to-Itô transition in noisy systems with multiplicative feedback,” Nature Communications, vol. 4, Article ID 2733, 2013.
  • W. Lu and T. Chen, “New approach to synchronization analysis of linearly coupled ordinary differential systems,” Physica D. Nonlinear Phenomena, vol. 213, no. 2, pp. 214–230, 2006.
  • L. Arnold, Stochastic Differential Equation and Application, Academic, New York, NY, USA, 1972.
  • R. C. Hilborn, Chaos and Nonlinear Dynamics, Oxford university press, Oxford, UK, 1994.
  • J. L. Hindmarsh and R. M. Rose, “A model of neuronal bursting using three coupled first order differential equations,” Proceedings of the Royal Society of London B, vol. 221, no. 1222, pp. 87–102, 1984.
  • I. A. Heisler, T. Braun, Y. Zhang, G. Hu, and H. A. Cerdeira, “Experimental investigation of partial synchronization in coupled chaotic oscillators,” Chaos, vol. 13, no. 1, pp. 185–194, 2003.
  • P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, Germany, 1995.
  • B. Nana and P. Woafo, “Synchronization in a ring of four mutually coupled van der Pol oscillators: theory and experiment,” Physical Review E, vol. 74, no. 4, Article ID 046213, 2006. \endinput