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November  2018, 23(9): 3787-3797. doi: 10.3934/dcdsb.2018077

Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type

1. 

Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France

2. 

Normandie Univ, France

3. 

An Giang University, Long Xuyen City, Vietnam

* Corresponding author

Received  May 2016 Revised  November 2017 Published  March 2018

Fund Project: This research was funded by Region Normandie France and the ERDF (European Regional Development Fund) project XTERM.

We focus on the long time behavior of complex networks of reaction-diffusion systems. We prove the existence of the global attractor and the $L^{∞}$-bound for networks of $n$ reaction-diffusion systems that belong to a class that generalizes the FitzHugh-Nagumo reaction-diffusion equations.

Citation: B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077
References:
[1]

B. Ambrosio and J.-P. Françcoise, Propagation of Bursting Oscillations, Phil. Trans. R. Soc. A., 367 (2009), 4863-4875.  doi: 10.1098/rsta.2009.0143.  Google Scholar

[2]

B. Ambrosio and M. A. Aziz-Alaoui, Synchronization and control of coupled reaction-diffusion systems of the FitzHugh-Nagumo-type, Comput. Math. Appl., 64 (2012), 934-943.  doi: 10.1016/j.camwa.2012.01.056.  Google Scholar

[3]

B. Ambrosio and M. A. Aziz-Alaoui, Basin of Attraction of Solutions with Pattern Formation in Slow-Fast Reaction-Diffusion Systems, Acta Biotheoretica, 64 (2016), 311-325.  doi: 10.1007/s10441-016-9294-z.  Google Scholar

[4]

B. Ambrosio, M. A. Aziz-Alaoui and V. L. E. Phan, Large time behavior and synchronization for a complex network system of reaction-diffusion systems, preprint, arXiv: 1504.07763. Google Scholar

[5]

A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., 62 (1983), 441-491.   Google Scholar

[6]

E. ConwayD. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16.  doi: 10.1137/0135001.  Google Scholar

[7]

G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neurosciences Springer, 2010.  Google Scholar

[8]

R. A. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[9]

C. Foias and G. Prodi, Sur le comportement global des solutions non stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.   Google Scholar

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations Springer, 1981.  Google Scholar

[11]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544.   Google Scholar

[12]

E. M. Izhikevich Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting MIT Press, Cambridge, MA, 2007.  Google Scholar

[13]

C. K. R. T. Jones, Stability of the travelling wave solution of the FitzHugh-Nagumo system, Transactions of the AMS, 286 (1984), 431-469.  doi: 10.1090/S0002-9947-1984-0760971-6.  Google Scholar

[14]

N. Kopell and D. Ruelle, Bounds on complexity in reaction-diffusion systems, SIAM J. Appl. Math, 46 (1986), 68-80.  doi: 10.1137/0146007.  Google Scholar

[15]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type Providence, Rhode Island, Transl. of Math. Monographs 23,1968.  Google Scholar

[16]

J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires Dunod, Paris, 1969.  Google Scholar

[17]

M. Marion, Finite Dimensionnal Attractors associated with Partly Dissipative Reaction-Diffusion Systems, SIAM, J. Math. Anal., 20 (1989), 816-844.  doi: 10.1137/0520057.  Google Scholar

[18]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE., 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.  Google Scholar

[19]

J. Rauch and J. Smoller, Qualitative theory of the fitzhugh nagumo equations, Advances in Mathematics, 27 (1978), 12-44.  doi: 10.1016/0001-8708(78)90075-0.  Google Scholar

[20]

J. Robinson, Infinite-Dimensional Dynamical Systems Cambridge University Press, 2001.  Google Scholar

[21]

F. Rothe, Global Solutions of Reaction-Diffusion Systems Springer-Verlag, 1984.  Google Scholar

[22]

J. Smoller, Shock Waves and Reaction-Diffusion Equations Springer, 1994.  Google Scholar

[23]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer, 1988.  Google Scholar

show all references

References:
[1]

B. Ambrosio and J.-P. Françcoise, Propagation of Bursting Oscillations, Phil. Trans. R. Soc. A., 367 (2009), 4863-4875.  doi: 10.1098/rsta.2009.0143.  Google Scholar

[2]

B. Ambrosio and M. A. Aziz-Alaoui, Synchronization and control of coupled reaction-diffusion systems of the FitzHugh-Nagumo-type, Comput. Math. Appl., 64 (2012), 934-943.  doi: 10.1016/j.camwa.2012.01.056.  Google Scholar

[3]

B. Ambrosio and M. A. Aziz-Alaoui, Basin of Attraction of Solutions with Pattern Formation in Slow-Fast Reaction-Diffusion Systems, Acta Biotheoretica, 64 (2016), 311-325.  doi: 10.1007/s10441-016-9294-z.  Google Scholar

[4]

B. Ambrosio, M. A. Aziz-Alaoui and V. L. E. Phan, Large time behavior and synchronization for a complex network system of reaction-diffusion systems, preprint, arXiv: 1504.07763. Google Scholar

[5]

A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., 62 (1983), 441-491.   Google Scholar

[6]

E. ConwayD. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16.  doi: 10.1137/0135001.  Google Scholar

[7]

G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neurosciences Springer, 2010.  Google Scholar

[8]

R. A. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[9]

C. Foias and G. Prodi, Sur le comportement global des solutions non stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.   Google Scholar

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations Springer, 1981.  Google Scholar

[11]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544.   Google Scholar

[12]

E. M. Izhikevich Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting MIT Press, Cambridge, MA, 2007.  Google Scholar

[13]

C. K. R. T. Jones, Stability of the travelling wave solution of the FitzHugh-Nagumo system, Transactions of the AMS, 286 (1984), 431-469.  doi: 10.1090/S0002-9947-1984-0760971-6.  Google Scholar

[14]

N. Kopell and D. Ruelle, Bounds on complexity in reaction-diffusion systems, SIAM J. Appl. Math, 46 (1986), 68-80.  doi: 10.1137/0146007.  Google Scholar

[15]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type Providence, Rhode Island, Transl. of Math. Monographs 23,1968.  Google Scholar

[16]

J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires Dunod, Paris, 1969.  Google Scholar

[17]

M. Marion, Finite Dimensionnal Attractors associated with Partly Dissipative Reaction-Diffusion Systems, SIAM, J. Math. Anal., 20 (1989), 816-844.  doi: 10.1137/0520057.  Google Scholar

[18]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE., 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.  Google Scholar

[19]

J. Rauch and J. Smoller, Qualitative theory of the fitzhugh nagumo equations, Advances in Mathematics, 27 (1978), 12-44.  doi: 10.1016/0001-8708(78)90075-0.  Google Scholar

[20]

J. Robinson, Infinite-Dimensional Dynamical Systems Cambridge University Press, 2001.  Google Scholar

[21]

F. Rothe, Global Solutions of Reaction-Diffusion Systems Springer-Verlag, 1984.  Google Scholar

[22]

J. Smoller, Shock Waves and Reaction-Diffusion Equations Springer, 1994.  Google Scholar

[23]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer, 1988.  Google Scholar

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