• Previous Article
    Extinction and coexistence of species for a diffusive intraguild predation model with B-D functional response
  • DCDS-B Home
  • This Issue
  • Next Article
    Coexistence and extinction in Time-Periodic Volterra-Lotka type systems with nonlocal dispersal
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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[20]

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

[21]

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

[22]

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

[23]

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

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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[1]

Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101

[2]

Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106

[3]

Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems. Evolution Equations & Control Theory, 2017, 6 (4) : 559-586. doi: 10.3934/eect.2017028

[4]

Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072

[5]

Boris Andreianov, Halima Labani. Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2179-2199. doi: 10.3934/cpaa.2012.11.2179

[6]

Abiti Adili, Bixiang Wang. Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 643-666. doi: 10.3934/dcdsb.2013.18.643

[7]

Abiti Adili, Bixiang Wang. Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise. Conference Publications, 2013, 2013 (special) : 1-10. doi: 10.3934/proc.2013.2013.1

[8]

Klemens Fellner, Wolfang Prager, Bao Q. Tang. The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinetic & Related Models, 2017, 10 (4) : 1055-1087. doi: 10.3934/krm.2017042

[9]

Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216

[10]

Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203

[11]

Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515

[12]

Laurent Desvillettes, Klemens Fellner. Entropy methods for reaction-diffusion systems. Conference Publications, 2007, 2007 (Special) : 304-312. doi: 10.3934/proc.2007.2007.304

[13]

A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65

[14]

Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49

[15]

John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851

[16]

Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150

[17]

Yiqiu Mao. Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3935-3947. doi: 10.3934/dcdsb.2018118

[18]

Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. Evolution Equations & Control Theory, 2018, 7 (4) : 571-585. doi: 10.3934/eect.2018027

[19]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[20]

C. van der Mee, Stella Vernier Piro. Travelling waves for solid-gas reaction-diffusion systems. Conference Publications, 2003, 2003 (Special) : 872-879. doi: 10.3934/proc.2003.2003.872

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (73)
  • HTML views (355)
  • Cited by (0)

Other articles
by authors

[Back to Top]