# American Institute of Mathematical Sciences

• Previous Article
On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes
• DCDS Home
• This Issue
• Next Article
Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems
January  2014, 34(1): 51-77. doi: 10.3934/dcds.2014.34.51

## On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems

 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla 2 Department d'Economia Aplicada, Facultat d'Economia, Universitat de València, Campus del Tarongers s/n, 46022-València, Spain 3 Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avda. de la Universidad, s/n, 03202 Elche

Received  November 2012 Revised  January 2013 Published  June 2013

In this paper we first prove a rather general theorem about existence of solutions for an abstract differential equation in a Banach space by assuming that the nonlinear term is in some sense weakly continuous.
We then apply this result to a lattice dynamical system with delay, proving also the existence of a global compact attractor for such system.
Citation: Tomás Caraballo, Francisco Morillas, José Valero. On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 51-77. doi: 10.3934/dcds.2014.34.51
##### References:
 [1] V. S. Afraĭmovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), 631. doi: 10.1142/S0218127494000459. [2] A. Y. Abdallah, Exponential attractors for first-order lattice dynamical systems,, J. Math. Anal. Appl., 339 (2008), 217. doi: 10.1016/j.jmaa.2007.06.054. [3] J. M. Amigó, Á. Giménez, F. Morillas and J. Valero, Attractors for a lattice dynamical system generated by non-Newtonian fluids modeling suspensions,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2681. doi: 10.1142/S0218127410027295. [4] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Editura Academiei Republicii Socialiste România, (1976). [5] P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions,, Arch. Ration. Mech. Anal., 150 (1999), 281. doi: 10.1007/s002050050189. [6] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems,, Stochastics & Dynamics, 6 (2006), 1. doi: 10.1142/S0219493706001621. [7] P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143. doi: 10.1142/S0218127401002031. [8] J. Bell, Some threshhold results for models of myelinated nerves,, Mathematical Biosciences, 54 (1981), 181. doi: 10.1016/0025-5564(81)90085-7. [9] J. Bell and C. Cosner, Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons,, Quarterly Appl. Math., 42 (1984), 1. [10] W.-J. Beyn and S. Yu. Pilyugin, Attractors of reaction diffusion systems on infinite lattices,, J. Dynam. Differential Equations, 15 (2003), 485. doi: 10.1023/B:JODY.0000009745.41889.30. [11] T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise,, Front. Math. China, 3 (2008), 317. doi: 10.1007/s11464-008-0028-7. [12] T. Caraballo, F. Morillas and J. Valero, Random Attractors for stochastic lattice systems with non-Lipschitz nonlinearity,, J. Diff. Equat. App., 17 (2011), 161. doi: 10.1080/10236198.2010.549010. [13] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities,, J. Differential Equations, 253 (2012), 667. doi: 10.1016/j.jde.2012.03.020. [14] S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems. I,, IEEE Trans. Circuits Syst., 42 (1995), 746. doi: 10.1109/81.473583. [15] S.-N. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems,, J. Differential Equations, 149 (1998), 248. doi: 10.1006/jdeq.1998.3478. [16] S.-N. Chow, J. Mallet-Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations,, Random Computational Dynamics, 4 (1996), 109. [17] S.-N. Chow and W. Shen, Dynamics in a discrete Nagumo equation: Spatial topological chaos,, SIAM J. Appl. Math., 55 (1995), 1764. doi: 10.1137/S0036139994261757. [18] L. O. Chua and T. Roska, The CNN paradigm,, IEEE Trans. Circuits Syst., 40 (1993), 147. [19] L. O. Chua and L. Yang, Cellular neural networks: Theory,, IEEE Trans. Circuits Syst., 35 (1988), 1257. doi: 10.1109/31.7600. [20] L. O. Chua and L. Yang, Cellular neural neetworks: Applications,, IEEE Trans. Circuits Syst., 35 (1988), 1273. doi: 10.1109/31.7601. [21] A. Pérez-Muñuzuri, V. Pérez-Muñuzuri, V. Pérez-Villar and L. O. Chua, Spiral waves on a 2-D array of nonlinear circuits,, IEEE Trans. Circuits Syst., 40 (1993), 872. [22] R. Dogaru and L. O. Chua, Edge of chaos and local activity domain of Fitz-Hugh-Nagumo equation,, Internat. J. Bifur. Chaos, 8 (1988), 211. doi: 10.1142/S0218127498000152. [23] T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems,, Physica D, 67 (1993), 237. doi: 10.1016/0167-2789(93)90208-I. [24] M. Gobbino and M. Sardella, On the connectedness of attractors for dynamical systems,, J. Differential Equations, 133 (1997), 1. doi: 10.1006/jdeq.1996.3166. [25] A. M. Gomaa, On existence of solutions and solution sets of differential equations and differential inclusions with delay in Banach spaces,, J. Egyptian Math. Soc., 20 (2012), 79. doi: 10.1016/j.joems.2012.08.007. [26] X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise,, J. Math. Anal. Appl., 376 (2011), 481. doi: 10.1016/j.jmaa.2010.11.032. [27] X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces,, J. Differential Equations, 250 (2011), 1235. doi: 10.1016/j.jde.2010.10.018. [28] R. Kapral, Discrete models for chemically reacting systems,, J. Math. Chem., 6 (1991), 113. doi: 10.1007/BF01192578. [29] S. Kato, On existence and uniqueness conditions for nonlinear ordinary differential equations in Banach spaces,, Funkcialaj Ekvacioj., 19 (1976), 239. [30] J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells,, SIAM J. Appl. Math., 47 (1987), 556. doi: 10.1137/0147038. [31] J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium,, J. Theor. Biol., 148 (1991), 49. [32] O. A. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,", Cambridge University Press, (1991). doi: 10.1017/CBO9780511569418. [33] J. P. Laplante and T. Erneux, Propagating failure in arrays of coupled bistable chemical reactors,, J. Phys. Chem., 96 (1992), 4931. [34] V. Lakshmikantham, A. R. Mitchell and R. W. Mitchell, On the existence of solutions of differential equations of retarde type in a Banach space,, Annales Polonici Mathematici, 35 (): 253. [35] Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems,, Chaos, 27 (2006), 1080. doi: 10.1016/j.chaos.2005.04.089. [36] J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems,, J. Dynam. Differential Equations, 11 (1999), 49. doi: 10.1023/A:1021841618074. [37] V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions,, Set-Valued Anal., 6 (1998), 83. doi: 10.1023/A:1008608431399. [38] F. Morillas and J. Valero, Peano's theorem and attractors for lattice dynamical systems,, Internat. J. Bifur. Chaos, 19 (2009), 557. doi: 10.1142/S0218127409023196. [39] F. Morillas and J. Valero, On the connectedness of the attainability set for lattice dynamical systems,, J. Diff. Equat. App., 18 (2012), 675. doi: 10.1080/10236198.2011.574621. [40] N. Rashevsky, "Mathematical Biophysics: Physico-Mathematical Foundations of Biology,", Third revised edition, (1960). [41] A. C. Scott, Analysis of a myelinated nerve model,, Bull. Math. Biophys., 26 (1964), 247. [42] W. Shen, Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices,, SIAM J. Appl. Math., 56 (1996), 1379. doi: 10.1137/S0036139995282670. [43] A. Sikorska-Nowak, Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals,, Discuss. Math. Differ. Incl. Control Optim., 27 (2007), 315. doi: 10.7151/dmdico.1087. [44] B. Wang, Dynamics of systems of infinite lattices,, J. Differential Equations, 221 (2006), 224. doi: 10.1016/j.jde.2005.01.003. [45] B. Wang, Asymptotic behavior of non-autonomous lattice systems,, J. Math. Anal. Appl., 331 (2007), 121. doi: 10.1016/j.jmaa.2006.08.070. [46] X. Wang, Sh. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems,, Nonlinear Anal., 72 (2010), 483. doi: 10.1016/j.na.2009.06.094. [47] W. Yan, Y. Li and Sh. Ji, Random attractors for first order stochastic retarded lattice dynamical systems,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3319566. [48] C. Zhao and S. Zhou, Attractors of retarded first order lattice systems,, Nonlinearity, 20 (2007), 1987. doi: 10.1088/0951-7715/20/8/010. [49] C. Zhao and Sh. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications,, J. Math. Anal. Appl., 354 (2009), 78. doi: 10.1016/j.jmaa.2008.12.036. [50] C. Zhao, S. Zhou and W. Wang, Compact kernel sections for lattice systems with delays,, Nonlinear Analysis TMA, 70 (2009), 1330. doi: 10.1016/j.na.2008.02.015. [51] S. Zhou, Attractors for first order dissipative lattice dynamical systems,, Physica D, 178 (2003), 51. doi: 10.1016/S0167-2789(02)00807-2. [52] S. Zhou, Attractors and approximations for lattice dynamical systems,, J. Differential Equations, 200 (2004), 342. doi: 10.1016/j.jde.2004.02.005. [53] S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems,, J. Differential Equations, 224 (2006), 172. doi: 10.1016/j.jde.2005.06.024. [54] B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation,, J. Differential Equations, 96 (1992), 1. doi: 10.1016/0022-0396(92)90142-A.

show all references

##### References:
 [1] V. S. Afraĭmovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4 (1994), 631. doi: 10.1142/S0218127494000459. [2] A. Y. Abdallah, Exponential attractors for first-order lattice dynamical systems,, J. Math. Anal. Appl., 339 (2008), 217. doi: 10.1016/j.jmaa.2007.06.054. [3] J. M. Amigó, Á. Giménez, F. Morillas and J. Valero, Attractors for a lattice dynamical system generated by non-Newtonian fluids modeling suspensions,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2681. doi: 10.1142/S0218127410027295. [4] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Editura Academiei Republicii Socialiste România, (1976). [5] P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions,, Arch. Ration. Mech. Anal., 150 (1999), 281. doi: 10.1007/s002050050189. [6] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems,, Stochastics & Dynamics, 6 (2006), 1. doi: 10.1142/S0219493706001621. [7] P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143. doi: 10.1142/S0218127401002031. [8] J. Bell, Some threshhold results for models of myelinated nerves,, Mathematical Biosciences, 54 (1981), 181. doi: 10.1016/0025-5564(81)90085-7. [9] J. Bell and C. Cosner, Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons,, Quarterly Appl. Math., 42 (1984), 1. [10] W.-J. Beyn and S. Yu. Pilyugin, Attractors of reaction diffusion systems on infinite lattices,, J. Dynam. Differential Equations, 15 (2003), 485. doi: 10.1023/B:JODY.0000009745.41889.30. [11] T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise,, Front. Math. China, 3 (2008), 317. doi: 10.1007/s11464-008-0028-7. [12] T. Caraballo, F. Morillas and J. Valero, Random Attractors for stochastic lattice systems with non-Lipschitz nonlinearity,, J. Diff. Equat. App., 17 (2011), 161. doi: 10.1080/10236198.2010.549010. [13] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities,, J. Differential Equations, 253 (2012), 667. doi: 10.1016/j.jde.2012.03.020. [14] S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems. I,, IEEE Trans. Circuits Syst., 42 (1995), 746. doi: 10.1109/81.473583. [15] S.-N. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems,, J. Differential Equations, 149 (1998), 248. doi: 10.1006/jdeq.1998.3478. [16] S.-N. Chow, J. Mallet-Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations,, Random Computational Dynamics, 4 (1996), 109. [17] S.-N. Chow and W. Shen, Dynamics in a discrete Nagumo equation: Spatial topological chaos,, SIAM J. Appl. Math., 55 (1995), 1764. doi: 10.1137/S0036139994261757. [18] L. O. Chua and T. Roska, The CNN paradigm,, IEEE Trans. Circuits Syst., 40 (1993), 147. [19] L. O. Chua and L. Yang, Cellular neural networks: Theory,, IEEE Trans. Circuits Syst., 35 (1988), 1257. doi: 10.1109/31.7600. [20] L. O. Chua and L. Yang, Cellular neural neetworks: Applications,, IEEE Trans. Circuits Syst., 35 (1988), 1273. doi: 10.1109/31.7601. [21] A. Pérez-Muñuzuri, V. Pérez-Muñuzuri, V. Pérez-Villar and L. O. Chua, Spiral waves on a 2-D array of nonlinear circuits,, IEEE Trans. Circuits Syst., 40 (1993), 872. [22] R. Dogaru and L. O. Chua, Edge of chaos and local activity domain of Fitz-Hugh-Nagumo equation,, Internat. J. Bifur. Chaos, 8 (1988), 211. doi: 10.1142/S0218127498000152. [23] T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems,, Physica D, 67 (1993), 237. doi: 10.1016/0167-2789(93)90208-I. [24] M. Gobbino and M. Sardella, On the connectedness of attractors for dynamical systems,, J. Differential Equations, 133 (1997), 1. doi: 10.1006/jdeq.1996.3166. [25] A. M. Gomaa, On existence of solutions and solution sets of differential equations and differential inclusions with delay in Banach spaces,, J. Egyptian Math. Soc., 20 (2012), 79. doi: 10.1016/j.joems.2012.08.007. [26] X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise,, J. Math. Anal. Appl., 376 (2011), 481. doi: 10.1016/j.jmaa.2010.11.032. [27] X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces,, J. Differential Equations, 250 (2011), 1235. doi: 10.1016/j.jde.2010.10.018. [28] R. Kapral, Discrete models for chemically reacting systems,, J. Math. Chem., 6 (1991), 113. doi: 10.1007/BF01192578. [29] S. Kato, On existence and uniqueness conditions for nonlinear ordinary differential equations in Banach spaces,, Funkcialaj Ekvacioj., 19 (1976), 239. [30] J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells,, SIAM J. Appl. Math., 47 (1987), 556. doi: 10.1137/0147038. [31] J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium,, J. Theor. Biol., 148 (1991), 49. [32] O. A. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,", Cambridge University Press, (1991). doi: 10.1017/CBO9780511569418. [33] J. P. Laplante and T. Erneux, Propagating failure in arrays of coupled bistable chemical reactors,, J. Phys. Chem., 96 (1992), 4931. [34] V. Lakshmikantham, A. R. Mitchell and R. W. Mitchell, On the existence of solutions of differential equations of retarde type in a Banach space,, Annales Polonici Mathematici, 35 (): 253. [35] Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems,, Chaos, 27 (2006), 1080. doi: 10.1016/j.chaos.2005.04.089. [36] J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems,, J. Dynam. Differential Equations, 11 (1999), 49. doi: 10.1023/A:1021841618074. [37] V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions,, Set-Valued Anal., 6 (1998), 83. doi: 10.1023/A:1008608431399. [38] F. Morillas and J. Valero, Peano's theorem and attractors for lattice dynamical systems,, Internat. J. Bifur. Chaos, 19 (2009), 557. doi: 10.1142/S0218127409023196. [39] F. Morillas and J. Valero, On the connectedness of the attainability set for lattice dynamical systems,, J. Diff. Equat. App., 18 (2012), 675. doi: 10.1080/10236198.2011.574621. [40] N. Rashevsky, "Mathematical Biophysics: Physico-Mathematical Foundations of Biology,", Third revised edition, (1960). [41] A. C. Scott, Analysis of a myelinated nerve model,, Bull. Math. Biophys., 26 (1964), 247. [42] W. Shen, Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices,, SIAM J. Appl. Math., 56 (1996), 1379. doi: 10.1137/S0036139995282670. [43] A. Sikorska-Nowak, Retarded functional differential equations in Banach spaces and Henstock-Kurzweil-Pettis integrals,, Discuss. Math. Differ. Incl. Control Optim., 27 (2007), 315. doi: 10.7151/dmdico.1087. [44] B. Wang, Dynamics of systems of infinite lattices,, J. Differential Equations, 221 (2006), 224. doi: 10.1016/j.jde.2005.01.003. [45] B. Wang, Asymptotic behavior of non-autonomous lattice systems,, J. Math. Anal. Appl., 331 (2007), 121. doi: 10.1016/j.jmaa.2006.08.070. [46] X. Wang, Sh. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems,, Nonlinear Anal., 72 (2010), 483. doi: 10.1016/j.na.2009.06.094. [47] W. Yan, Y. Li and Sh. Ji, Random attractors for first order stochastic retarded lattice dynamical systems,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3319566. [48] C. Zhao and S. Zhou, Attractors of retarded first order lattice systems,, Nonlinearity, 20 (2007), 1987. doi: 10.1088/0951-7715/20/8/010. [49] C. Zhao and Sh. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications,, J. Math. Anal. Appl., 354 (2009), 78. doi: 10.1016/j.jmaa.2008.12.036. [50] C. Zhao, S. Zhou and W. Wang, Compact kernel sections for lattice systems with delays,, Nonlinear Analysis TMA, 70 (2009), 1330. doi: 10.1016/j.na.2008.02.015. [51] S. Zhou, Attractors for first order dissipative lattice dynamical systems,, Physica D, 178 (2003), 51. doi: 10.1016/S0167-2789(02)00807-2. [52] S. Zhou, Attractors and approximations for lattice dynamical systems,, J. Differential Equations, 200 (2004), 342. doi: 10.1016/j.jde.2004.02.005. [53] S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems,, J. Differential Equations, 224 (2006), 172. doi: 10.1016/j.jde.2005.06.024. [54] B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation,, J. Differential Equations, 96 (1992), 1. doi: 10.1016/0022-0396(92)90142-A.
 [1] Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809 [2] Yejuan Wang, Lin Yang. Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1961-1987. doi: 10.3934/dcdsb.2018257 [3] Ahmed Y. Abdallah. Exponential attractors for second order lattice dynamical systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 803-813. doi: 10.3934/cpaa.2009.8.803 [4] Xiaoying Han. Exponential attractors for lattice dynamical systems in weighted spaces. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 445-467. doi: 10.3934/dcds.2011.31.445 [5] Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215 [6] Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control & Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012 [7] Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281 [8] Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281 [9] 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 [10] 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 [11] Ahmed Y. Abdallah. Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (1) : 55-69. doi: 10.3934/cpaa.2006.5.55 [12] Anhui Gu. Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-31. doi: 10.3934/dcdsb.2019104 [13] 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 [14] B. Coll, A. Gasull, R. Prohens. On a criterium of global attraction for discrete dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (3) : 537-550. doi: 10.3934/cpaa.2006.5.537 [15] David Cheban. Global attractors of nonautonomous quasihomogeneous dynamical systems. Conference Publications, 2001, 2001 (Special) : 96-101. doi: 10.3934/proc.2001.2001.96 [16] Stefano Galatolo. Global and local complexity in weakly chaotic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1607-1624. doi: 10.3934/dcds.2003.9.1607 [17] Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167 [18] Neville J. Ford, Stewart J. Norton. Predicting changes in dynamical behaviour in solutions to stochastic delay differential equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 367-382. doi: 10.3934/cpaa.2006.5.367 [19] 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 [20] 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

2017 Impact Factor: 1.179