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 & 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. doi: 10.1016/S0167-2789(96)00276-X. Google Scholar

[2]

L. Arnold, Random Dynamical Systems,, Springer, (1998). doi: 10.1007/978-3-662-12878-7. Google Scholar

[3]

A. Balanov, N. Janson, D. Postnov and O. Sosnovtseva, Synchronization: From Simple to Complex,, Springer, (2009). Google Scholar

[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. doi: 10.1137/050647281. Google Scholar

[5]

T. Caraballo and P. Kloeden, The persistence of synchronization under environmental noise,, Proc. Roy. Soc. London, 461 (2005), 2257. doi: 10.1098/rspa.2005.1484. Google Scholar

[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, 10 (2003), 491. Google Scholar

[7]

A. Carvalho and M. Primo, Boundary synchronization in parabolic problems with nonlinear boundary conditions,, Dynamics Cont. Discr. Impul. Systems, 7 (2000), 541. Google Scholar

[8]

A. Carvalho, H. Rodrigues and T. Dlotko, Upper semicontinuity of attractors and synchronization,, J. Math. Anal. Appl., 220 (1998), 13. doi: 10.1006/jmaa.1997.5774. Google Scholar

[9]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions,, Lect. Notes Math. 580, (1977). Google Scholar

[10]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta, (1999). Google Scholar

[11]

I. Chueshov, Monotone Random Systems: Theory and Applications,, Lect. Notes Math. 1779, (1779). doi: 10.1007/b83277. Google Scholar

[12]

I. Chueshov, A reduction principle for coupled nonlinear parabolic-hyperbolic PDE,, J. Evol. Eqns., 4 (2004), 591. doi: 10.1007/s00028-004-0175-6. Google Scholar

[13]

I. Chueshov, Invariant manifolds and nonlinear master-slave synchronization in coupled systems,, Appl. Anal., 86 (2007), 269. doi: 10.1080/00036810601097629. Google Scholar

[14]

I. Chueshov, Synchronization in coupled second order in time infinite-dimensional models,, Dyn. Partial Differ. Equ., 13 (2016), 1. doi: 10.4310/DPDE.2016.v13.n1.a1. Google Scholar

[15]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems,, Springer, (2015). doi: 10.1007/978-3-319-22903-4. Google Scholar

[16]

I. Chueshov, Synchronization in Infinite-Dimensional Systems,, book in preparation under the contract with Springer., (). Google Scholar

[17]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,, Memoirs of AMS 912, (2008). doi: 10.1090/memo/0912. Google Scholar

[18]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar

[19]

I. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations,, J. Dyn. Diff. Eqns., 13 (2001), 355. doi: 10.1023/A:1016684108862. Google Scholar

[20]

I. Chueshov and B. Schmalfuss, Master-slave synchronization and invariant manifolds for coupled stochastic systems,, J. Math. Physics, 51 (2010). doi: 10.1063/1.3493646. Google Scholar

[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. doi: 10.3934/dcdsb.2015.20.833. Google Scholar

[22]

X. M. Fan, Random attractor for a damped Sine-Gordon equation with white noise,, Pacific J. Math., 216 (2004), 63. doi: 10.2140/pjm.2004.216.63. Google Scholar

[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. doi: 10.1080/07362990600751860. Google Scholar

[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. doi: 10.1080/07362990601139602. Google Scholar

[25]

J. K. Hale, Diffusive coupling, dissipation, and synchronization,, J. Dyn. Dif. Eqs, 9 (1997), 1. doi: 10.1007/BF02219051. Google Scholar

[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. doi: 10.1090/S0025-5718-1988-0917820-X. Google Scholar

[27]

L. V. Kapitansky and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations,, Leningrad Math. J., 2 (1991), 97. Google Scholar

[28]

G. Leonov and V. Smirnova, Mathematical Problems of Phase Synchronization Theory,, St. Petersburg, (2000). Google Scholar

[29]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,, Dunod, (1969). Google Scholar

[30]

E. Mosekilde, Y. Maistrenko and D. Postnov, Chaotic Synchronization,, World Scientific Publishing Co., (2002). doi: 10.1142/9789812778260. Google Scholar

[31]

O. Naboka, Synchronization of nonlinear oscillations of two coupling Berger plates,, Nonlin. Anal., 67 (2007), 1015. doi: 10.1016/j.na.2006.06.034. Google Scholar

[32]

O. Naboka, Synchronization phenomena in the system consisting of m coupled Berger plates,, J. Math. Anal. Appl., 341 (2008), 1107. doi: 10.1016/j.jmaa.2007.10.068. Google Scholar

[33]

O. Naboka, On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping,, Commun. Pure Appl. Anal., 8 (2009), 1933. doi: 10.3934/cpaa.2009.8.1933. Google Scholar

[34]

G. Osipov, J. Kurths and C. Zhou, Synchronization in Oscillatory Networks,, Springer, (2007). doi: 10.1007/978-3-540-71269-5. Google Scholar

[35]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[36]

G. Prato and G. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar

[37]

J. Robinson, Stability of random attractors under perturbation and approximation,, J. Diff. Eqns., 186 (2002), 652. doi: 10.1016/S0022-0396(02)00038-4. Google Scholar

[38]

H. Rodrigues, Abstract methods for synchronization and applications,, Appl. Anal., 62 (1996), 263. doi: 10.1080/00036819608840483. Google Scholar

[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. doi: 10.1016/j.jde.2009.10.007. Google Scholar

[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). doi: 10.1142/S0219493713500020. Google Scholar

[41]

S. Strogatz, Sync: How Order Emerges From Chaos in the Universe, Nature, and Daily Life,, Hyperion Books, (2003). Google Scholar

[42]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar

[43]

C. W. Wu, Synchronization in Coupled Chaotic Circuits and Systems,, World Scientific Publishing Co., (2002). doi: 10.1142/9789812778420. Google Scholar

show all references

References:
[1]

V. Afraimovich, S. Chow and J. Hale, Synchronization in lattices of coupled oscillators,, Physica D, 103 (1997), 442. doi: 10.1016/S0167-2789(96)00276-X. Google Scholar

[2]

L. Arnold, Random Dynamical Systems,, Springer, (1998). doi: 10.1007/978-3-662-12878-7. Google Scholar

[3]

A. Balanov, N. Janson, D. Postnov and O. Sosnovtseva, Synchronization: From Simple to Complex,, Springer, (2009). Google Scholar

[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. doi: 10.1137/050647281. Google Scholar

[5]

T. Caraballo and P. Kloeden, The persistence of synchronization under environmental noise,, Proc. Roy. Soc. London, 461 (2005), 2257. doi: 10.1098/rspa.2005.1484. Google Scholar

[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, 10 (2003), 491. Google Scholar

[7]

A. Carvalho and M. Primo, Boundary synchronization in parabolic problems with nonlinear boundary conditions,, Dynamics Cont. Discr. Impul. Systems, 7 (2000), 541. Google Scholar

[8]

A. Carvalho, H. Rodrigues and T. Dlotko, Upper semicontinuity of attractors and synchronization,, J. Math. Anal. Appl., 220 (1998), 13. doi: 10.1006/jmaa.1997.5774. Google Scholar

[9]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions,, Lect. Notes Math. 580, (1977). Google Scholar

[10]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta, (1999). Google Scholar

[11]

I. Chueshov, Monotone Random Systems: Theory and Applications,, Lect. Notes Math. 1779, (1779). doi: 10.1007/b83277. Google Scholar

[12]

I. Chueshov, A reduction principle for coupled nonlinear parabolic-hyperbolic PDE,, J. Evol. Eqns., 4 (2004), 591. doi: 10.1007/s00028-004-0175-6. Google Scholar

[13]

I. Chueshov, Invariant manifolds and nonlinear master-slave synchronization in coupled systems,, Appl. Anal., 86 (2007), 269. doi: 10.1080/00036810601097629. Google Scholar

[14]

I. Chueshov, Synchronization in coupled second order in time infinite-dimensional models,, Dyn. Partial Differ. Equ., 13 (2016), 1. doi: 10.4310/DPDE.2016.v13.n1.a1. Google Scholar

[15]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems,, Springer, (2015). doi: 10.1007/978-3-319-22903-4. Google Scholar

[16]

I. Chueshov, Synchronization in Infinite-Dimensional Systems,, book in preparation under the contract with Springer., (). Google Scholar

[17]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,, Memoirs of AMS 912, (2008). doi: 10.1090/memo/0912. Google Scholar

[18]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar

[19]

I. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations,, J. Dyn. Diff. Eqns., 13 (2001), 355. doi: 10.1023/A:1016684108862. Google Scholar

[20]

I. Chueshov and B. Schmalfuss, Master-slave synchronization and invariant manifolds for coupled stochastic systems,, J. Math. Physics, 51 (2010). doi: 10.1063/1.3493646. Google Scholar

[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. doi: 10.3934/dcdsb.2015.20.833. Google Scholar

[22]

X. M. Fan, Random attractor for a damped Sine-Gordon equation with white noise,, Pacific J. Math., 216 (2004), 63. doi: 10.2140/pjm.2004.216.63. Google Scholar

[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. doi: 10.1080/07362990600751860. Google Scholar

[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. doi: 10.1080/07362990601139602. Google Scholar

[25]

J. K. Hale, Diffusive coupling, dissipation, and synchronization,, J. Dyn. Dif. Eqs, 9 (1997), 1. doi: 10.1007/BF02219051. Google Scholar

[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. doi: 10.1090/S0025-5718-1988-0917820-X. Google Scholar

[27]

L. V. Kapitansky and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations,, Leningrad Math. J., 2 (1991), 97. Google Scholar

[28]

G. Leonov and V. Smirnova, Mathematical Problems of Phase Synchronization Theory,, St. Petersburg, (2000). Google Scholar

[29]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,, Dunod, (1969). Google Scholar

[30]

E. Mosekilde, Y. Maistrenko and D. Postnov, Chaotic Synchronization,, World Scientific Publishing Co., (2002). doi: 10.1142/9789812778260. Google Scholar

[31]

O. Naboka, Synchronization of nonlinear oscillations of two coupling Berger plates,, Nonlin. Anal., 67 (2007), 1015. doi: 10.1016/j.na.2006.06.034. Google Scholar

[32]

O. Naboka, Synchronization phenomena in the system consisting of m coupled Berger plates,, J. Math. Anal. Appl., 341 (2008), 1107. doi: 10.1016/j.jmaa.2007.10.068. Google Scholar

[33]

O. Naboka, On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping,, Commun. Pure Appl. Anal., 8 (2009), 1933. doi: 10.3934/cpaa.2009.8.1933. Google Scholar

[34]

G. Osipov, J. Kurths and C. Zhou, Synchronization in Oscillatory Networks,, Springer, (2007). doi: 10.1007/978-3-540-71269-5. Google Scholar

[35]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[36]

G. Prato and G. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar

[37]

J. Robinson, Stability of random attractors under perturbation and approximation,, J. Diff. Eqns., 186 (2002), 652. doi: 10.1016/S0022-0396(02)00038-4. Google Scholar

[38]

H. Rodrigues, Abstract methods for synchronization and applications,, Appl. Anal., 62 (1996), 263. doi: 10.1080/00036819608840483. Google Scholar

[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. doi: 10.1016/j.jde.2009.10.007. Google Scholar

[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). doi: 10.1142/S0219493713500020. Google Scholar

[41]

S. Strogatz, Sync: How Order Emerges From Chaos in the Universe, Nature, and Daily Life,, Hyperion Books, (2003). Google Scholar

[42]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar

[43]

C. W. Wu, Synchronization in Coupled Chaotic Circuits and Systems,, World Scientific Publishing Co., (2002). doi: 10.1142/9789812778420. Google Scholar

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