American Institute of Mathematical Sciences

November  2016, 21(9): 2969-2990. doi: 10.3934/dcdsb.2016082

Synchronization in coupled stochastic sine-Gordon wave model

 1 School of Mathematics & Informatics, Karazin Kharkov National University, Kharkov 61022, Ukraine 2 School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan 430074 3 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074

Received  December 2015 Revised  March 2016 Published  October 2016

The asymptotic synchronization at the level of global random attractors is investigated for a class of coupled stochastic second order in time evolution equations. The main focus is on sine-Gordon type models perturbed by additive white noise. The model describes distributed Josephson junctions. The analysis makes extensive use of the method of quasi-stability.
Citation: Igor Chueshov, Peter E. Kloeden, Meihua Yang. Synchronization in coupled stochastic sine-Gordon wave model. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 2969-2990. doi: 10.3934/dcdsb.2016082
References:
 [1] V. Afraimovich, S. Chow and J. Hale, Synchronization in lattices of coupled oscillators, Physica D, 103 (1997), 442-451. doi: 10.1016/S0167-2789(96)00276-X. [2] L. Arnold, Random Dynamical Systems, Springer, Berlin/Heidelberg/New York, 1998. doi: 10.1007/978-3-662-12878-7. [3] A. Balanov, N. Janson, D. Postnov and O. Sosnovtseva, Synchronization: From Simple to Complex, Springer, Berlin-Heidelberg, 2009. [4] T. Caraballo, I. Chueshov and P. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2007), 1489-1507. doi: 10.1137/050647281. [5] T. Caraballo and P. Kloeden, The persistence of synchronization under environmental noise, Proc. Roy. Soc. London, 461 (2005), 2257-2267. doi: 10.1098/rspa.2005.1484. [6] T. Caraballo and J. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics Cont. Discr. Impul. Systems, Series A, 10 (2003), 491-513. [7] A. Carvalho and M. Primo, Boundary synchronization in parabolic problems with nonlinear boundary conditions, Dynamics Cont. Discr. Impul. Systems, 7 (2000), 541-560. [8] A. Carvalho, H. Rodrigues and T. Dlotko, Upper semicontinuity of attractors and synchronization, J. Math. Anal. Appl., 220 (1998), 13-41. doi: 10.1006/jmaa.1997.5774. [9] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lect. Notes Math. 580, Springer, Berlin, 1977. [10] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999 (in Russian); English translation: Acta, Kharkov, 2002. [11] I. Chueshov, Monotone Random Systems: Theory and Applications, Lect. Notes Math. 1779, Springer, Berlin/Heidelberg/New York, 2002. doi: 10.1007/b83277. [12] I. Chueshov, A reduction principle for coupled nonlinear parabolic-hyperbolic PDE, J. Evol. Eqns., 4 (2004), 591-612. doi: 10.1007/s00028-004-0175-6. [13] I. Chueshov, Invariant manifolds and nonlinear master-slave synchronization in coupled systems, Appl. Anal., 86 (2007), 269-286. doi: 10.1080/00036810601097629. [14] I. Chueshov, Synchronization in coupled second order in time infinite-dimensional models, Dyn. Partial Differ. Equ., 13 (2016), 1-29. doi: 10.4310/DPDE.2016.v13.n1.a1. [15] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, Cham/Heidelberg/New York, 2015. doi: 10.1007/978-3-319-22903-4. [16] I. Chueshov, Synchronization in Infinite-Dimensional Systems,, book in preparation under the contract with Springer., (). [17] I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Memoirs of AMS 912, AMS, Providence, 2008. doi: 10.1090/memo/0912. [18] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9. [19] I. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dyn. Diff. Eqns., 13 (2001), 355-380. doi: 10.1023/A:1016684108862. [20] I. Chueshov and B. Schmalfuss, Master-slave synchronization and invariant manifolds for coupled stochastic systems, J. Math. Physics, 51 (2010), 102702, 23pp. doi: 10.1063/1.3493646. [21] I. Chueshov and B. Schmalfuss, Stochastic dynamics in a fluid-plate interaction model with the only longitudinal deformations of the plate, Disc. Conts. Dyn. Systems - B, 20 (2015), 833-852. doi: 10.3934/dcdsb.2015.20.833. [22] X. M. Fan, Random attractor for a damped Sine-Gordon equation with white noise, Pacific J. Math., 216 (2004), 63-76. doi: 10.2140/pjm.2004.216.63. [23] X. M. Fan, Attractors for a damped stochastic wave equation of Sine-Gordon type with sublinear multiplicative noise, Stochastic Anal. Appl., 24 (2006), 767-793. doi: 10.1080/07362990600751860. [24] X. M. Fan and Y. Wang, Fractal dimension of attractors for a stochastic wave equation with nonlinear damping and white noise, Stochastic Anal. Appl., 25 (2007), 381-396. doi: 10.1080/07362990601139602. [25] J. K. Hale, Diffusive coupling, dissipation, and synchronization, J. Dyn. Dif. Eqs, 9 (1997), 1-52. doi: 10.1007/BF02219051. [26] J. K. Hale, X. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp., 50 (1988), 89-123. doi: 10.1090/S0025-5718-1988-0917820-X. [27] L. V. Kapitansky and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Leningrad Math. J., 2 (1991), 97-117. [28] G. Leonov and V. Smirnova, Mathematical Problems of Phase Synchronization Theory, St. Petersburg, Nauka, 2000 (in Russian). [29] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. [30] E. Mosekilde, Y. Maistrenko and D. Postnov, Chaotic Synchronization, World Scientific Publishing Co., River Edge, NJ, 2002. doi: 10.1142/9789812778260. [31] O. Naboka, Synchronization of nonlinear oscillations of two coupling Berger plates, Nonlin. Anal., TMA, 67 (2007), 1015-1026. doi: 10.1016/j.na.2006.06.034. [32] O. Naboka, Synchronization phenomena in the system consisting of m coupled Berger plates, J. Math. Anal. Appl., 341 (2008), 1107-1124. doi: 10.1016/j.jmaa.2007.10.068. [33] O. Naboka, On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping, Commun. Pure Appl. Anal., 8 (2009), 1933-1956. doi: 10.3934/cpaa.2009.8.1933. [34] G. Osipov, J. Kurths and C. Zhou, Synchronization in Oscillatory Networks, Springer, Berlin/Heidelberg, 2007. doi: 10.1007/978-3-540-71269-5. [35] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [36] G. Prato and G. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. [37] J. Robinson, Stability of random attractors under perturbation and approximation, J. Diff. Eqns., 186 (2002), 652-669. doi: 10.1016/S0022-0396(02)00038-4. [38] H. Rodrigues, Abstract methods for synchronization and applications, Appl. Anal., 62 (1996), 263-296. doi: 10.1080/00036819608840483. [39] Z. W. Shen, S. F. Zhou and W. X. Shen, One-dimensional random attractor and rotation number of the stochastic damped Sine-Gordon equation, J.Diff. Eqns., 248 (2010), 1432-1457. doi: 10.1016/j.jde.2009.10.007. [40] W. X. Shen, Z. W. Shen and S. F. Zhou, Asymptotic dynamics of a class of coupled oscillators driven by white noises, Stoch. and Dyn., 13 (2013), 1350002[23 pages]. doi: 10.1142/S0219493713500020. [41] S. Strogatz, Sync: How Order Emerges From Chaos in the Universe, Nature, and Daily Life, Hyperion Books, New York, 2003. [42] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8. [43] C. W. Wu, Synchronization in Coupled Chaotic Circuits and Systems, World Scientific Publishing Co., River Edge, NJ, 2002. doi: 10.1142/9789812778420.

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References:
 [1] V. Afraimovich, S. Chow and J. Hale, Synchronization in lattices of coupled oscillators, Physica D, 103 (1997), 442-451. doi: 10.1016/S0167-2789(96)00276-X. [2] L. Arnold, Random Dynamical Systems, Springer, Berlin/Heidelberg/New York, 1998. doi: 10.1007/978-3-662-12878-7. [3] A. Balanov, N. Janson, D. Postnov and O. Sosnovtseva, Synchronization: From Simple to Complex, Springer, Berlin-Heidelberg, 2009. [4] T. Caraballo, I. Chueshov and P. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2007), 1489-1507. doi: 10.1137/050647281. [5] T. Caraballo and P. Kloeden, The persistence of synchronization under environmental noise, Proc. Roy. Soc. London, 461 (2005), 2257-2267. doi: 10.1098/rspa.2005.1484. [6] T. Caraballo and J. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics Cont. Discr. Impul. Systems, Series A, 10 (2003), 491-513. [7] A. Carvalho and M. Primo, Boundary synchronization in parabolic problems with nonlinear boundary conditions, Dynamics Cont. Discr. Impul. Systems, 7 (2000), 541-560. [8] A. Carvalho, H. Rodrigues and T. Dlotko, Upper semicontinuity of attractors and synchronization, J. Math. Anal. Appl., 220 (1998), 13-41. doi: 10.1006/jmaa.1997.5774. [9] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lect. Notes Math. 580, Springer, Berlin, 1977. [10] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999 (in Russian); English translation: Acta, Kharkov, 2002. [11] I. Chueshov, Monotone Random Systems: Theory and Applications, Lect. Notes Math. 1779, Springer, Berlin/Heidelberg/New York, 2002. doi: 10.1007/b83277. [12] I. Chueshov, A reduction principle for coupled nonlinear parabolic-hyperbolic PDE, J. Evol. Eqns., 4 (2004), 591-612. doi: 10.1007/s00028-004-0175-6. [13] I. Chueshov, Invariant manifolds and nonlinear master-slave synchronization in coupled systems, Appl. Anal., 86 (2007), 269-286. doi: 10.1080/00036810601097629. [14] I. Chueshov, Synchronization in coupled second order in time infinite-dimensional models, Dyn. Partial Differ. Equ., 13 (2016), 1-29. doi: 10.4310/DPDE.2016.v13.n1.a1. [15] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, Cham/Heidelberg/New York, 2015. doi: 10.1007/978-3-319-22903-4. [16] I. Chueshov, Synchronization in Infinite-Dimensional Systems,, book in preparation under the contract with Springer., (). [17] I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Memoirs of AMS 912, AMS, Providence, 2008. doi: 10.1090/memo/0912. [18] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9. [19] I. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dyn. Diff. Eqns., 13 (2001), 355-380. doi: 10.1023/A:1016684108862. [20] I. Chueshov and B. Schmalfuss, Master-slave synchronization and invariant manifolds for coupled stochastic systems, J. Math. Physics, 51 (2010), 102702, 23pp. doi: 10.1063/1.3493646. [21] I. Chueshov and B. Schmalfuss, Stochastic dynamics in a fluid-plate interaction model with the only longitudinal deformations of the plate, Disc. Conts. Dyn. Systems - B, 20 (2015), 833-852. doi: 10.3934/dcdsb.2015.20.833. [22] X. M. Fan, Random attractor for a damped Sine-Gordon equation with white noise, Pacific J. Math., 216 (2004), 63-76. doi: 10.2140/pjm.2004.216.63. [23] X. M. Fan, Attractors for a damped stochastic wave equation of Sine-Gordon type with sublinear multiplicative noise, Stochastic Anal. Appl., 24 (2006), 767-793. doi: 10.1080/07362990600751860. [24] X. M. Fan and Y. Wang, Fractal dimension of attractors for a stochastic wave equation with nonlinear damping and white noise, Stochastic Anal. Appl., 25 (2007), 381-396. doi: 10.1080/07362990601139602. [25] J. K. Hale, Diffusive coupling, dissipation, and synchronization, J. Dyn. Dif. Eqs, 9 (1997), 1-52. doi: 10.1007/BF02219051. [26] J. K. Hale, X. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp., 50 (1988), 89-123. doi: 10.1090/S0025-5718-1988-0917820-X. [27] L. V. Kapitansky and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Leningrad Math. J., 2 (1991), 97-117. [28] G. Leonov and V. Smirnova, Mathematical Problems of Phase Synchronization Theory, St. Petersburg, Nauka, 2000 (in Russian). [29] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. [30] E. Mosekilde, Y. Maistrenko and D. Postnov, Chaotic Synchronization, World Scientific Publishing Co., River Edge, NJ, 2002. doi: 10.1142/9789812778260. [31] O. Naboka, Synchronization of nonlinear oscillations of two coupling Berger plates, Nonlin. Anal., TMA, 67 (2007), 1015-1026. doi: 10.1016/j.na.2006.06.034. [32] O. Naboka, Synchronization phenomena in the system consisting of m coupled Berger plates, J. Math. Anal. Appl., 341 (2008), 1107-1124. doi: 10.1016/j.jmaa.2007.10.068. [33] O. Naboka, On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping, Commun. Pure Appl. Anal., 8 (2009), 1933-1956. doi: 10.3934/cpaa.2009.8.1933. [34] G. Osipov, J. Kurths and C. Zhou, Synchronization in Oscillatory Networks, Springer, Berlin/Heidelberg, 2007. doi: 10.1007/978-3-540-71269-5. [35] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [36] G. Prato and G. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. [37] J. Robinson, Stability of random attractors under perturbation and approximation, J. Diff. Eqns., 186 (2002), 652-669. doi: 10.1016/S0022-0396(02)00038-4. [38] H. Rodrigues, Abstract methods for synchronization and applications, Appl. Anal., 62 (1996), 263-296. doi: 10.1080/00036819608840483. [39] Z. W. Shen, S. F. Zhou and W. X. Shen, One-dimensional random attractor and rotation number of the stochastic damped Sine-Gordon equation, J.Diff. Eqns., 248 (2010), 1432-1457. doi: 10.1016/j.jde.2009.10.007. [40] W. X. Shen, Z. W. Shen and S. F. Zhou, Asymptotic dynamics of a class of coupled oscillators driven by white noises, Stoch. and Dyn., 13 (2013), 1350002[23 pages]. doi: 10.1142/S0219493713500020. [41] S. Strogatz, Sync: How Order Emerges From Chaos in the Universe, Nature, and Daily Life, Hyperion Books, New York, 2003. [42] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8. [43] C. W. Wu, Synchronization in Coupled Chaotic Circuits and Systems, World Scientific Publishing Co., River Edge, NJ, 2002. doi: 10.1142/9789812778420.
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