• 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Editura Academiei Republicii Socialiste România, (1976). Google Scholar

[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. Google Scholar

[6]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems,, Stochastics & Dynamics, 6 (2006), 1. doi: 10.1142/S0219493706001621. Google Scholar

[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. Google Scholar

[8]

J. Bell, Some threshhold results for models of myelinated nerves,, Mathematical Biosciences, 54 (1981), 181. doi: 10.1016/0025-5564(81)90085-7. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

L. O. Chua and T. Roska, The CNN paradigm,, IEEE Trans. Circuits Syst., 40 (1993), 147. Google Scholar

[19]

L. O. Chua and L. Yang, Cellular neural networks: Theory,, IEEE Trans. Circuits Syst., 35 (1988), 1257. doi: 10.1109/31.7600. Google Scholar

[20]

L. O. Chua and L. Yang, Cellular neural neetworks: Applications,, IEEE Trans. Circuits Syst., 35 (1988), 1273. doi: 10.1109/31.7601. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

R. Kapral, Discrete models for chemically reacting systems,, J. Math. Chem., 6 (1991), 113. doi: 10.1007/BF01192578. Google Scholar

[29]

S. Kato, On existence and uniqueness conditions for nonlinear ordinary differential equations in Banach spaces,, Funkcialaj Ekvacioj., 19 (1976), 239. Google Scholar

[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. Google Scholar

[31]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium,, J. Theor. Biol., 148 (1991), 49. Google Scholar

[32]

O. A. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,", Cambridge University Press, (1991). doi: 10.1017/CBO9780511569418. Google Scholar

[33]

J. P. Laplante and T. Erneux, Propagating failure in arrays of coupled bistable chemical reactors,, J. Phys. Chem., 96 (1992), 4931. Google Scholar

[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. Google Scholar

[35]

Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems,, Chaos, 27 (2006), 1080. doi: 10.1016/j.chaos.2005.04.089. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[40]

N. Rashevsky, "Mathematical Biophysics: Physico-Mathematical Foundations of Biology,", Third revised edition, (1960). Google Scholar

[41]

A. C. Scott, Analysis of a myelinated nerve model,, Bull. Math. Biophys., 26 (1964), 247. Google Scholar

[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. Google Scholar

[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. Google Scholar

[44]

B. Wang, Dynamics of systems of infinite lattices,, J. Differential Equations, 221 (2006), 224. doi: 10.1016/j.jde.2005.01.003. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[51]

S. Zhou, Attractors for first order dissipative lattice dynamical systems,, Physica D, 178 (2003), 51. doi: 10.1016/S0167-2789(02)00807-2. Google Scholar

[52]

S. Zhou, Attractors and approximations for lattice dynamical systems,, J. Differential Equations, 200 (2004), 342. doi: 10.1016/j.jde.2004.02.005. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Editura Academiei Republicii Socialiste România, (1976). Google Scholar

[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. Google Scholar

[6]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems,, Stochastics & Dynamics, 6 (2006), 1. doi: 10.1142/S0219493706001621. Google Scholar

[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. Google Scholar

[8]

J. Bell, Some threshhold results for models of myelinated nerves,, Mathematical Biosciences, 54 (1981), 181. doi: 10.1016/0025-5564(81)90085-7. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

L. O. Chua and T. Roska, The CNN paradigm,, IEEE Trans. Circuits Syst., 40 (1993), 147. Google Scholar

[19]

L. O. Chua and L. Yang, Cellular neural networks: Theory,, IEEE Trans. Circuits Syst., 35 (1988), 1257. doi: 10.1109/31.7600. Google Scholar

[20]

L. O. Chua and L. Yang, Cellular neural neetworks: Applications,, IEEE Trans. Circuits Syst., 35 (1988), 1273. doi: 10.1109/31.7601. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

R. Kapral, Discrete models for chemically reacting systems,, J. Math. Chem., 6 (1991), 113. doi: 10.1007/BF01192578. Google Scholar

[29]

S. Kato, On existence and uniqueness conditions for nonlinear ordinary differential equations in Banach spaces,, Funkcialaj Ekvacioj., 19 (1976), 239. Google Scholar

[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. Google Scholar

[31]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium,, J. Theor. Biol., 148 (1991), 49. Google Scholar

[32]

O. A. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,", Cambridge University Press, (1991). doi: 10.1017/CBO9780511569418. Google Scholar

[33]

J. P. Laplante and T. Erneux, Propagating failure in arrays of coupled bistable chemical reactors,, J. Phys. Chem., 96 (1992), 4931. Google Scholar

[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. Google Scholar

[35]

Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems,, Chaos, 27 (2006), 1080. doi: 10.1016/j.chaos.2005.04.089. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[40]

N. Rashevsky, "Mathematical Biophysics: Physico-Mathematical Foundations of Biology,", Third revised edition, (1960). Google Scholar

[41]

A. C. Scott, Analysis of a myelinated nerve model,, Bull. Math. Biophys., 26 (1964), 247. Google Scholar

[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. Google Scholar

[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. Google Scholar

[44]

B. Wang, Dynamics of systems of infinite lattices,, J. Differential Equations, 221 (2006), 224. doi: 10.1016/j.jde.2005.01.003. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[51]

S. Zhou, Attractors for first order dissipative lattice dynamical systems,, Physica D, 178 (2003), 51. doi: 10.1016/S0167-2789(02)00807-2. Google Scholar

[52]

S. Zhou, Attractors and approximations for lattice dynamical systems,, J. Differential Equations, 200 (2004), 342. doi: 10.1016/j.jde.2004.02.005. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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]

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

[8]

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

[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]

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

[18]

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

[19]

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

[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

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (8)

[Back to Top]