October  2014, 34(10): 4019-4037. doi: 10.3934/dcds.2014.34.4019

Asymptotic behaviour of a logistic lattice system

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

3. 

Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avda. de la Universidad, s/n, 03202 Elche

Received  December 2012 Revised  February 2013 Published  April 2014

In this paper we study the asymptotic behaviour of solutions of a lattice dynamical system of a logistic type. Namely, we study a system of infinite ordinary differential equations which can be obtained after the spatial discretization of a logistic equation with diffusion. We prove that a global attractor exists in suitable weighted spaces of sequences.
Citation: Tomás Caraballo, Francisco Morillas, José Valero. Asymptotic behaviour of a logistic lattice system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4019-4037. doi: 10.3934/dcds.2014.34.4019
References:
[1]

V. S. Afraimovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps,, Internat. J. Bifur. Chaos, 4 (1994), 631.  doi: 10.1142/S0218127494000459.  Google Scholar

[2]

J. M. Amigó, A. Giménez, F. Morillas and J. Valero, Attractors for a lattice dynamical system generated by non-newtonian fluids modelling suspensions,, Internat. J. Bifur. Chaos, 20 (2010), 2681.  doi: 10.1142/S0218127410027295.  Google Scholar

[3]

P. W. Bates and A. Chmaj, On a discrete convolution model for phase transitions,, Arch. Ration. Mech. Anal., 150 (1999), 281.  doi: 10.1007/s002050050189.  Google Scholar

[4]

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

[5]

P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems,, Internat. J. Bifur. Chaos, 11 (2001), 143.  doi: 10.1142/S0218127401002031.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

S.-N. Chow and J. Mallet-Paret, Pattern formulation and spatial chaos in lattice dynamical systems: I,, IEEE Trans. Circuits Syst., 42 (1995), 746.  doi: 10.1109/81.473583.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

L. O. Chua and T. Roska, The CNN paradigm,, IEEE Trans. Circuits Syst., 40 (1993), 147.  doi: 10.1109/81.222795.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

X. Han, W. Shen and Sh. 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

[24]

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

[25]

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

[26]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium,, J. Theor. Biol., 148 (1991), 49.  doi: 10.1016/S0022-5193(05)80465-5.  Google Scholar

[27]

O. A. Ladyzhenskaya, Some comments to my papers on the theory of attractors for abstract semigroups (in russian),, Zap. Nauchn. Sem. LOMI, 182 (1990), 102.  doi: 10.1007/BF01671002.  Google Scholar

[28]

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

[29]

J. P. Laplante and T. Erneux, Propagating failure in arrays of coupled bistable chemical reactors,, J. Phys. Chem., 96 (1992), 4931.  doi: 10.1021/j100191a038.  Google Scholar

[30]

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

[31]

F. Morillas and J. Valero, A Peano's theorem and attractors for lattice dynamical systems,, Internat. J. Bifur. Chaos, 19 (2009), 557.  doi: 10.1142/S0218127409023196.  Google Scholar

[32]

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

[33]

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

[34]

N. Rashevsky, Mathematical Biophysics,, 3rd revised edition, (1960).   Google Scholar

[35]

A. C. Scott, Analysis of a myelinated nerve model,, Bull. Math. Biophys., 26 (1964), 247.  doi: 10.1007/BF02479046.  Google Scholar

[36]

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

[37]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).   Google Scholar

[38]

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

[39]

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

[40]

E. Zeidler, Nonlinear Functional Analysis and Its Applciations,, Springer, (1986).  doi: 10.1007/978-1-4612-4838-5.  Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

show all references

References:
[1]

V. S. Afraimovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps,, Internat. J. Bifur. Chaos, 4 (1994), 631.  doi: 10.1142/S0218127494000459.  Google Scholar

[2]

J. M. Amigó, A. Giménez, F. Morillas and J. Valero, Attractors for a lattice dynamical system generated by non-newtonian fluids modelling suspensions,, Internat. J. Bifur. Chaos, 20 (2010), 2681.  doi: 10.1142/S0218127410027295.  Google Scholar

[3]

P. W. Bates and A. Chmaj, On a discrete convolution model for phase transitions,, Arch. Ration. Mech. Anal., 150 (1999), 281.  doi: 10.1007/s002050050189.  Google Scholar

[4]

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

[5]

P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems,, Internat. J. Bifur. Chaos, 11 (2001), 143.  doi: 10.1142/S0218127401002031.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

S.-N. Chow and J. Mallet-Paret, Pattern formulation and spatial chaos in lattice dynamical systems: I,, IEEE Trans. Circuits Syst., 42 (1995), 746.  doi: 10.1109/81.473583.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

L. O. Chua and T. Roska, The CNN paradigm,, IEEE Trans. Circuits Syst., 40 (1993), 147.  doi: 10.1109/81.222795.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

X. Han, W. Shen and Sh. 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

[24]

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

[25]

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

[26]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium,, J. Theor. Biol., 148 (1991), 49.  doi: 10.1016/S0022-5193(05)80465-5.  Google Scholar

[27]

O. A. Ladyzhenskaya, Some comments to my papers on the theory of attractors for abstract semigroups (in russian),, Zap. Nauchn. Sem. LOMI, 182 (1990), 102.  doi: 10.1007/BF01671002.  Google Scholar

[28]

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

[29]

J. P. Laplante and T. Erneux, Propagating failure in arrays of coupled bistable chemical reactors,, J. Phys. Chem., 96 (1992), 4931.  doi: 10.1021/j100191a038.  Google Scholar

[30]

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

[31]

F. Morillas and J. Valero, A Peano's theorem and attractors for lattice dynamical systems,, Internat. J. Bifur. Chaos, 19 (2009), 557.  doi: 10.1142/S0218127409023196.  Google Scholar

[32]

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

[33]

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

[34]

N. Rashevsky, Mathematical Biophysics,, 3rd revised edition, (1960).   Google Scholar

[35]

A. C. Scott, Analysis of a myelinated nerve model,, Bull. Math. Biophys., 26 (1964), 247.  doi: 10.1007/BF02479046.  Google Scholar

[36]

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

[37]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).   Google Scholar

[38]

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

[39]

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

[40]

E. Zeidler, Nonlinear Functional Analysis and Its Applciations,, Springer, (1986).  doi: 10.1007/978-1-4612-4838-5.  Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

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