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  February 2021 Early access  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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

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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