February  2021, 26(2): 775-794. doi: 10.3934/dcdsb.2020134

Stabilities and dynamic transitions of the Fitzhugh-Nagumo system

1. 

College of Mathematics, Sichuan University, Chengdu, Sichuan, 610065, China

2. 

School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, PR China

3. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 610054, China

* Corresponding author: Quan Wang

Received  January 2020 Published  April 2020

Fund Project: This research was supported by the NSFC, Grant No.11901408 and 11711306

The article aims to examine the dynamic transition of the reaction-diffusion Fitzhugh-Nagumo system defined on a thin spherical shell and a 2D-rectangular domain. The mathematical tool employed is the theory of phase transition dynamics established for dissipative dynamical systems. The main results in this paper include two parts. First, for the system on a thin spherical shell, we only focus on the transition from a real simple eigenvalue. More precisely, if the first eigenspace is three–dimensional, the system undergoes either a continuous transition or a jump transition. Besides, a mix transition is also allowed if the first eigenspace is one–dimensional. Second, for the system on a rectangular domain, both the transitions from a simple real eigenvalue and a pair of simple complex eigenvalues are considered. Our results imply that two steady-state solutions bifurcate, which are either attractors or saddle points, and a Hopf bifurcation is also possible in the system on the rectangular domain.

Citation: Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134
References:
[1]

P. Carter and B. Sandstede, Fast pulses with oscillatory tails in the FitzHugh-Nagumo system, SIAM Journal Mathematic Analysis, 47 (2015), 3393-3441.  doi: 10.1137/140999177.  Google Scholar

[2]

C.-N. ChenS.-Y. Kung and Y. Morita, Planar standing wavefronts in the Fitzhugh-Nagumo equations, SIAM Journal Mathematic Analysis, 46 (2014), 657-690.  doi: 10.1137/130907793.  Google Scholar

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C.-N. ChenC.-C. Chen and C.-C. Huang, Traveling waves for the Fitzhugh-Nagumo system on an infinite channel, Journal of Differential Equations, 261 (2016), 3010-3041.  doi: 10.1016/j.jde.2016.05.014.  Google Scholar

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P. Cornwell and C. K. R. T. Jones, On the existence and stability of fast traveling waves in a doubly diffusive Fitzhugh-Nagumo system, SIAM Journal Applied Dynamical Systems, 17 (2018), 754-787.  doi: 10.1137/17M1149432.  Google Scholar

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R. 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

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M. O. Gani and T. Ogawa, Instability of periodic traveling wave solutions in a modified Fitzhugh-Nagumo model for excitable media, Applied Mathematics and Computation, 256 (2015), 968-984.  doi: 10.1016/j.amc.2015.01.109.  Google Scholar

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F. Sánchez-GarduñoA. L. KrauseJ. A. Castillo and P. Padilla, Turing-Hopf patterns on growing domains: The torus and the sphere, Journal of Theoretical Biology, 481 (2019), 136-150.  doi: 10.1016/j.jtbi.2018.09.028.  Google Scholar

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D. HanM. Hernandez and Q. Wang, Dynamical transitions of a low-dimensional model for Rayleigh-Bénard convection under a vertical magnetic field, Chaos, Solitons and Fractals, 114 (2018), 370-380.  doi: 10.1016/j.chaos.2018.06.027.  Google Scholar

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P. van Heijster and B. Sandstede, Bifurcations to travelling planar spots in a three-component Fitzhugh-Nagumo system, Physica D Nonlinear Phenomena, 275 (2014), 19-34.  doi: 10.1016/j.physd.2014.02.001.  Google Scholar

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T. Ma and S. Wang, Dynamic transition and pattern formation for chemotaction system, Discrete and Continuous Dynamical System Series B, 19 (2014), 2089-2835.  doi: 10.1088/0951-7715/24/4/012.  Google Scholar

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[18]

Y. MaoD. Yan and C. Lu, Dynamic transitions and stability for the acetabularia whorl formation, Discrete and Continuous Dynamical System Series B, 24 (2019), 5989-6004.  doi: 10.3934/dcdsb.2019117.  Google Scholar

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[20]

S. ÖzerT. Şengül and Q. Wang, Multiple equilibria and transitions in spherical MHD equations, Communications in Mathematical Sciences, 17 (2019), 1531-1555.  doi: 10.4310/CMS.2019.v17.n6.a3.  Google Scholar

[21]

S. Özer and T. Sengül, Transitions of spherical thermohaline circulation to multiple equilibria, Journal of Mathematical Fluid Mechanics, 20 (2018), 499-515.  doi: 10.1007/s00021-017-0331-8.  Google Scholar

[22]

J. Pedlosky, Geophysical Fluid Dynamics, Springer Science & Business Media, 2013. Google Scholar

[23]

M. Schonbek, Boundary value problem for the Fitzhugh-Nagumo equations, Journal of Differential Equations, 30 (1978), 119-147.  doi: 10.1016/0022-0396(78)90027-X.  Google Scholar

[24]

T. Sengul and S. Wang, Pattern formation and dynamic transition for magnetohydrodynamic convection, Communications on Pure and Applied Analysis, 13 (2014), 2609-2639.  doi: 10.3934/cpaa.2014.13.2609.  Google Scholar

[25]

E. P. Zemskov and I. R. Epstein, Wave propagation in a Fitzhugh-Nagumo-type model with modified excitability, Physical Review E, 82 (2010), 026207, 6 pp. doi: 10.1103/PhysRevE.82.026207.  Google Scholar

[26]

E. P. Zemskov, M. A. Tsyganov and W. Horsthemke, Multifront regime of a piecewise-linear Fitzhugh-Nagumo model with cross diffusion, Physical Review E, 99 (2019), 062214, 9 pp. doi: 10.1103/PhysRevE.99.062214.  Google Scholar

[27]

Q. Zheng and J. Shen, Pattern formation in the Fitzhugh-Nagumo model, Computers & Mathematics with Applications, 70 (2015), 1082-1097.  doi: 10.1016/j.camwa.2015.06.031.  Google Scholar

show all references

References:
[1]

P. Carter and B. Sandstede, Fast pulses with oscillatory tails in the FitzHugh-Nagumo system, SIAM Journal Mathematic Analysis, 47 (2015), 3393-3441.  doi: 10.1137/140999177.  Google Scholar

[2]

C.-N. ChenS.-Y. Kung and Y. Morita, Planar standing wavefronts in the Fitzhugh-Nagumo equations, SIAM Journal Mathematic Analysis, 46 (2014), 657-690.  doi: 10.1137/130907793.  Google Scholar

[3]

C.-N. ChenC.-C. Chen and C.-C. Huang, Traveling waves for the Fitzhugh-Nagumo system on an infinite channel, Journal of Differential Equations, 261 (2016), 3010-3041.  doi: 10.1016/j.jde.2016.05.014.  Google Scholar

[4]

P. Cornwell and C. K. R. T. Jones, On the existence and stability of fast traveling waves in a doubly diffusive Fitzhugh-Nagumo system, SIAM Journal Applied Dynamical Systems, 17 (2018), 754-787.  doi: 10.1137/17M1149432.  Google Scholar

[5]

H. DijkstraT. SengulJ. Shen and S. Wang, Dynamic transitions of quasi-geostrophic channel flow, SIAM Journal on Applied Mathematics, 75 (2015), 2361-2378.  doi: 10.1137/15M1008166.  Google Scholar

[6]

R. 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

[7]

M. O. Gani and T. Ogawa, Instability of periodic traveling wave solutions in a modified Fitzhugh-Nagumo model for excitable media, Applied Mathematics and Computation, 256 (2015), 968-984.  doi: 10.1016/j.amc.2015.01.109.  Google Scholar

[8]

F. Sánchez-GarduñoA. L. KrauseJ. A. Castillo and P. Padilla, Turing-Hopf patterns on growing domains: The torus and the sphere, Journal of Theoretical Biology, 481 (2019), 136-150.  doi: 10.1016/j.jtbi.2018.09.028.  Google Scholar

[9]

D. HanM. Hernandez and Q. Wang, Dynamical transitions of a low-dimensional model for Rayleigh-Bénard convection under a vertical magnetic field, Chaos, Solitons and Fractals, 114 (2018), 370-380.  doi: 10.1016/j.chaos.2018.06.027.  Google Scholar

[10]

P. van Heijster and B. Sandstede, Bifurcations to travelling planar spots in a three-component Fitzhugh-Nagumo system, Physica D Nonlinear Phenomena, 275 (2014), 19-34.  doi: 10.1016/j.physd.2014.02.001.  Google Scholar

[11]

C. KieuT. SengulQ. Wang and D. Yan, On the Hopf (double Hopf) bifurcations and transitions of two layer western boundary currents, Communications in Nonlinear Science and Numerical Simulation, 65 (2018), 196-215.  doi: 10.1016/j.cnsns.2018.05.010.  Google Scholar

[12]

M. Kuznetsov, A. Kolobov and A. Polezhaev, Pattern formation in a reaction-diffusion system of Fitzhugh-Nagumo type before the onset of subcritical Turing bifurcation, Physical Review E, 95 (2017), 052208, 7 pp. doi: 10.1103/PhysRevE.95.052208.  Google Scholar

[13]

J.-L. LionsR. Teman and S. Wang, New formation of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288.  doi: 10.1088/0951-7715/5/2/001.  Google Scholar

[14]

T. Ma and S. Wang, Phase Transition Dynamics, Springer, New York, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[15]

T. Ma and S. Wang, Dynamic transition and pattern formation for chemotaction system, Discrete and Continuous Dynamical System Series B, 19 (2014), 2089-2835.  doi: 10.1088/0951-7715/24/4/012.  Google Scholar

[16]

T. Ma and S. Wang, Bifurcation Theory and Application, Series A: Monographs and Treatises, 53. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi: 10.1142/9789812701152.  Google Scholar

[17]

Y. Mao, Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain, Discrete and Continuous Dynamical System Series B, 23 (2018), 3935-3947.  doi: 10.3934/dcdsb.2018118.  Google Scholar

[18]

Y. MaoD. Yan and C. Lu, Dynamic transitions and stability for the acetabularia whorl formation, Discrete and Continuous Dynamical System Series B, 24 (2019), 5989-6004.  doi: 10.3934/dcdsb.2019117.  Google Scholar

[19]

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

[20]

S. ÖzerT. Şengül and Q. Wang, Multiple equilibria and transitions in spherical MHD equations, Communications in Mathematical Sciences, 17 (2019), 1531-1555.  doi: 10.4310/CMS.2019.v17.n6.a3.  Google Scholar

[21]

S. Özer and T. Sengül, Transitions of spherical thermohaline circulation to multiple equilibria, Journal of Mathematical Fluid Mechanics, 20 (2018), 499-515.  doi: 10.1007/s00021-017-0331-8.  Google Scholar

[22]

J. Pedlosky, Geophysical Fluid Dynamics, Springer Science & Business Media, 2013. Google Scholar

[23]

M. Schonbek, Boundary value problem for the Fitzhugh-Nagumo equations, Journal of Differential Equations, 30 (1978), 119-147.  doi: 10.1016/0022-0396(78)90027-X.  Google Scholar

[24]

T. Sengul and S. Wang, Pattern formation and dynamic transition for magnetohydrodynamic convection, Communications on Pure and Applied Analysis, 13 (2014), 2609-2639.  doi: 10.3934/cpaa.2014.13.2609.  Google Scholar

[25]

E. P. Zemskov and I. R. Epstein, Wave propagation in a Fitzhugh-Nagumo-type model with modified excitability, Physical Review E, 82 (2010), 026207, 6 pp. doi: 10.1103/PhysRevE.82.026207.  Google Scholar

[26]

E. P. Zemskov, M. A. Tsyganov and W. Horsthemke, Multifront regime of a piecewise-linear Fitzhugh-Nagumo model with cross diffusion, Physical Review E, 99 (2019), 062214, 9 pp. doi: 10.1103/PhysRevE.99.062214.  Google Scholar

[27]

Q. Zheng and J. Shen, Pattern formation in the Fitzhugh-Nagumo model, Computers & Mathematics with Applications, 70 (2015), 1082-1097.  doi: 10.1016/j.camwa.2015.06.031.  Google Scholar

Figure 1.  Neutral stability surface $ \tilde{\alpha}_{c}(D, \epsilon) $
Figure 2.  The regions separating two types of transitions. Region A, jump transitions from a simple real eigenvalue; Region B, jump transitions from a pair of simple complex eigenvalues
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