# American Institute of Mathematical Sciences

November  2009, 8(6): 1933-1956. doi: 10.3934/cpaa.2009.8.1933

## On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping

 1 Department of Mechanics and Mathematics, Kharkov National University, Svobody sq. 4, 61077 Kharkov, Ukraine

Received  October 2008 Revised  March 2009 Published  August 2009

The dynamical system generated by a system describing nonlinear oscillations of two coupled Berger plates with nonlinear interior damping and clamped boundary is considered. The dependence of the long-time behavior of the system trajectories on the coupling parameter $\gamma$ is studied in the case of (i) same equations for both plates of the system and damping possibly degenerate at zero; and (ii) different equations and damping non-degenerate at any point. Ultimate synchronization at the level of attractors is proved for both cases, which means that the global attractor of the system approaches the diagonal of the phase space of the system as $\gamma\to\infty$. In case (ii) the structure of the upper limit of the attractor is studied. It coincides with the diagonal of the product of two samples of the attractor to the dynamical system generated by a single plate equation. If both the equations describing the plate dynamics are the same and the damping functions are non-degenerate at any point we prove the synchronization phenomenon for finite large $\gamma$. System synchronization rate is exponential in this case.
Citation: Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933
 [1] Rogério Martins. One-dimensional attractor for a dissipative system with a cylindrical phase space. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 533-547. doi: 10.3934/dcds.2006.14.533 [2] Long Hu, Tatsien Li, Bopeng Rao. Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type. Communications on Pure & Applied Analysis, 2014, 13 (2) : 881-901. doi: 10.3934/cpaa.2014.13.881 [3] Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899 [4] Jin Zhang, Yonghai Wang, Chengkui Zhong. Robustness of exponentially κ-dissipative dynamical systems with perturbations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3875-3890. doi: 10.3934/dcdsb.2017198 [5] Mostafa Abounouh, H. Al Moatassime, J. P. Chehab, S. Dumont, Olivier Goubet. Discrete Schrödinger equations and dissipative dynamical systems. Communications on Pure & Applied Analysis, 2008, 7 (2) : 211-227. doi: 10.3934/cpaa.2008.7.211 [6] Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. Totally dissipative dynamical processes and their uniform global attractors. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1989-2004. doi: 10.3934/cpaa.2014.13.1989 [7] Boling Guo, Zhengde Dai. Attractor for the dissipative Hamiltonian amplitude equation governing modulated wave instabilities. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 783-793. doi: 10.3934/dcds.1998.4.783 [8] Giulia Cavagnari, Antonio Marigonda, Benedetto Piccoli. Optimal synchronization problem for a multi-agent system. Networks & Heterogeneous Media, 2017, 12 (2) : 277-295. doi: 10.3934/nhm.2017012 [9] Lijuan Wang, Qishu Yan. Optimal control problem for exact synchronization of parabolic system. Mathematical Control & Related Fields, 2019, 9 (3) : 411-424. doi: 10.3934/mcrf.2019019 [10] Jin-Liang Wang, Zhi-Chun Yang, Tingwen Huang, Mingqing Xiao. Local and global exponential synchronization of complex delayed dynamical networks with general topology. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 393-408. doi: 10.3934/dcdsb.2011.16.393 [11] Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087 [12] Xin Li, Wenxian Shen, Chunyou Sun. Invariant measures for complex-valued dissipative dynamical systems and applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2427-2446. doi: 10.3934/dcdsb.2017124 [13] Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211 [14] Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative lattice dynamical systems with delays. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 643-663. doi: 10.3934/dcds.2008.21.643 [15] Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120 [16] Xiaojun Chang, Yong Li. Rotating periodic solutions of second order dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 643-652. doi: 10.3934/dcds.2016.36.643 [17] Xiaoming Wang. Numerical algorithms for stationary statistical properties of dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4599-4618. doi: 10.3934/dcds.2016.36.4599 [18] Tatsien Li, Bopeng Rao, Yimin Wei. Generalized exact boundary synchronization for a coupled system of wave equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2893-2905. doi: 10.3934/dcds.2014.34.2893 [19] P.K. Newton. The dipole dynamical system. Conference Publications, 2005, 2005 (Special) : 692-699. doi: 10.3934/proc.2005.2005.692 [20] Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757

2018 Impact Factor: 0.925