
-
Previous Article
Limit speed of traveling wave solutions for the perturbed generalized KdV equation
- DCDS-S Home
- This Issue
-
Next Article
The amplitude system for a Simultaneous short-wave Turing and long-wave Hopf instability
Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.
Readers can access Online First articles via the “Online First” tab for the selected journal.
Wiggly canards: Growth of traveling wave trains through a family of fast-subsystem foci
1. | Department of Mathematics, University of California, Irvine, Irvine, CA 92697, USA |
2. | Department of Engineering Mathematics, University of Bristol, Bristol BS8 1UB, UK |
A class of two-fast, one-slow multiple timescale dynamical systems is considered that contains the system of ordinary differential equations obtained from seeking travelling-wave solutions to the FitzHugh-Nagumo equations in one space dimension. The question addressed is the mechanism by which a small-amplitude periodic orbit, created in a Hopf bifurcation, undergoes rapid amplitude growth in a small parameter interval, akin to a canard explosion. The presence of a saddle-focus structure around the slow manifold implies that a single periodic orbit undergoes a sequence of folds as the amplitude grows. An analysis is performed under some general hypotheses using a combination ideas from the theory of canard explosion and Shilnikov analysis. An asymptotic formula is obtained for the dependence of the parameter location of the folds on the singular parameter and parameters that control the saddle focus eigenvalues. The analysis is shown to agree with numerical results both for a synthetic normal-form example and the FitzHugh-Nagumo system.
References:
[1] |
E. Benoît, J.-L. Callot, F. Diener and M. Diener,
Chasse au canard, Collect. Math., 32 (1981), 37-119.
|
[2] |
P. Carter,
Spike-adding canard explosion in a class of square-wave bursters, J. Nonlinear Sci., 30 (2020), 2613-2669.
doi: 10.1007/s00332-020-09631-y. |
[3] |
P. Carter, J. D. M. Rademacher and B. Sandstede,
Pulse replication and accumulation of eigenvalues, SIAM J. Math. Anal., 53 (2021), 3520-3576.
doi: 10.1137/20M1340113. |
[4] |
P. Carter and B. Sandstede,
Fast pulses with oscillatory tails in the FitzHugh–Nagumo system, SIAM J. Math. Anal., 47 (2015), 3393-3441.
doi: 10.1137/140999177. |
[5] |
P. Carter and B. Sandstede,
Unpeeling a homoclinic banana in the FitzHugh–Nagumo system, SIAM J. Appl. Dyn. Syst., 17 (2018), 236-349.
doi: 10.1137/16M1080707. |
[6] |
P. Carter and A. Scheel,
Wave train selection by invasion fronts in the FitzHugh–Nagumo equation, Nonlinearity, 31 (2018), 5536-5572.
doi: 10.1088/1361-6544/aae1db. |
[7] |
A. R. Champneys,
Homoclinic orbits in the dynamics of articulated pipes conveying fluid, Nonlinearity, 4 (1991), 747-774.
doi: 10.1088/0951-7715/4/3/007. |
[8] |
A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. Sneyd,
When shil'nikov meets hopf in excitable systems, SIAM J. Appl. Dyn. Syst., 6 (2007), 663-693.
doi: 10.1137/070682654. |
[9] |
M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga and M. Wechselberger,
Mixed-mode oscillations with multiple time scales, SIAM Rev., 54 (2012), 211-288.
doi: 10.1137/100791233. |
[10] |
M. Desroches, T. J. Kaper and M. Krupa, Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster, Chaos: An Interdisciplinary Journal of Nonlinear Science, 23 (2013), 046106, 13 pp.
doi: 10.1063/1.4827026. |
[11] |
E. Doedel, B. Oldeman et al., Auto-07p: Continuation and bifurcation software for ordinary differential equations, 2020, Latest version at https://github.com/auto-07p. |
[12] |
F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), no. 577,100 pp.
doi: 10.1090/memo/0577. |
[13] |
C. Fall, E. Marland, J. Wagner and J. Tyson, Computational Cell Biology, Springer-Verlag, New York, 2002. |
[14] |
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. |
[15] |
P. Gaspard, R. Kapral and G. Nicolis,
Bifurcation phenomena near homoclinic systems; A two-parameter analysis, J. Stat. Phys., 35 (1984), 697-727.
doi: 10.1007/BF01010829. |
[16] |
P. Glendinning and C. Sparrow,
Local and global behaviour near homoclinic orbits, J. Stat. Phys., 35 (1984), 645-696.
doi: 10.1007/BF01010828. |
[17] |
J. Guckenheimer and C. Kuehn,
Homoclinic orbits of the FitzHugh–Nagumo equation: The singular-limit, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 851-872.
doi: 10.3934/dcdss.2009.2.851. |
[18] |
J. Guckenheimer and C. Kuehn,
Homoclinic orbits of the FitzHugh–Nagumo equation: Bifurcations in the full system, SIAM J. Appl. Dyn. Syst., 9 (2010), 138-153.
doi: 10.1137/090758404. |
[19] |
S. P. Hastings,
On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations, Quart. J. Math. Oxford Ser., 27 (1976), 123-134.
doi: 10.1093/qmath/27.1.123. |
[20] |
S. P. Hastings,
Single and multiple pulse waves for the FitzHugh–Nagumo equations, SIAM J. Appl. Math., 42 (1982), 247-260.
doi: 10.1137/0142018. |
[21] |
C. Jones, N. Kopell and R. Langer, Construction of the FitzHugh-Nagumo pulse using differential forms, in Patterns and Dynamics in Reactive Media, Springer, 1991,101–115.
doi: 10.1007/978-1-4612-3206-3_7. |
[22] |
T. Kaper and C. Jones, A primer on the exchange lemma for fast-slow systems, vol. 122 of The IMA Volumes in Mathematics and its Applications, 65–87, Springer, New York, 2001.
doi: 10.1007/978-1-4613-0117-2_3. |
[23] |
J. Keener and J. Sneyd, Mathematical Physiology, 2nd edition, Springer-Verlag, New York, 2009. |
[24] |
M. Krupa, B. Sandstede and P. Szmolyan,
Fast and slow waves in the FitzHugh–Nagumo equation, J. Differential Equations, 133 (1997), 49-97.
doi: 10.1006/jdeq.1996.3198. |
[25] |
M. Krupa and P. Szmolyan,
Extending geometric singular perturbation theory to nonhyperbolic points–-fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), 286-314.
doi: 10.1137/S0036141099360919. |
[26] |
M. Krupa and P. Szmolyan,
Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.
doi: 10.1006/jdeq.2000.3929. |
[27] |
C. Kuehn, Multiple Time Series Dynamical Systems, Springer-Verlag, Heidelberg, 2015, Applied Mathematical Sciences, vol. 191. |
[28] |
Y. Kuznetsov and A. Panfilov, Stochastic waves in the FitzHugh-Nagumo system, 1981, Research Computing Centre, USSR Academy of Sciences, Pushchino. In Russian. |
[29] |
D. Linaro, A. Champneys, M. Desroches and M. Storace,
Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh–Rose burster, SIAM J. Appl. Dyn. Syst., 11 (2012), 939-962.
doi: 10.1137/110848931. |
[30] |
J. Nagumo, S. 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. |
[31] |
H. Osinga and K. Tsaneva-Atanasova,
Dynamics of plateau bursting depending on the location of its equilibrium, Journal of Neuroendocrinology, 22 (2010), 1301-1314.
doi: 10.1111/j.1365-2826.2010.02083.x. |
[32] |
J. D. M. Rademacher, Homoclinic Bifurcation from Heteroclinic Cycles with Periodic Orbits and Tracefiring of Pulses, Ph.D. thesis, University of Minnesota, 2004, http://www.math.uni-bremen.de/~jdmr/pub/dissMay7Web.pdf. |
[33] |
L. Shilnikov,
On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type, Math. USSR Sb., 6 (1968), 427-438.
|
[34] |
L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Method of Qualitative Theory of Nonlinear Dynamics: Part II, World Scientific, Singapore, 2001.
doi: 10.1142/9789812798558_0001. |
[35] |
C. Soto-Trevino, Geometric Methods for Periodic Orbits in Singularly Perturbed Systems, Ph.D. Thesis, Boston University, 1998. |
[36] |
P. Szmolyan and M. Wechselberger,
Canards in $\mathbb{R}^3$, J. Differential Equations, 177 (2001), 419-453.
doi: 10.1006/jdeq.2001.4001. |
[37] |
D. Terman,
Chaotic spikes arising from a model of bursting in excitable membranes, SIAM J. Appl. Math., 51 (1991), 1418-1450.
doi: 10.1137/0151071. |
[38] |
M. Wechselberger,
Existence and bifurcation of canards in $\mathbb{R}^3$ in the case of a folded node, SIAM J. Appl. Dyn. Syst., 4 (2005), 101-139.
doi: 10.1137/030601995. |
show all references
References:
[1] |
E. Benoît, J.-L. Callot, F. Diener and M. Diener,
Chasse au canard, Collect. Math., 32 (1981), 37-119.
|
[2] |
P. Carter,
Spike-adding canard explosion in a class of square-wave bursters, J. Nonlinear Sci., 30 (2020), 2613-2669.
doi: 10.1007/s00332-020-09631-y. |
[3] |
P. Carter, J. D. M. Rademacher and B. Sandstede,
Pulse replication and accumulation of eigenvalues, SIAM J. Math. Anal., 53 (2021), 3520-3576.
doi: 10.1137/20M1340113. |
[4] |
P. Carter and B. Sandstede,
Fast pulses with oscillatory tails in the FitzHugh–Nagumo system, SIAM J. Math. Anal., 47 (2015), 3393-3441.
doi: 10.1137/140999177. |
[5] |
P. Carter and B. Sandstede,
Unpeeling a homoclinic banana in the FitzHugh–Nagumo system, SIAM J. Appl. Dyn. Syst., 17 (2018), 236-349.
doi: 10.1137/16M1080707. |
[6] |
P. Carter and A. Scheel,
Wave train selection by invasion fronts in the FitzHugh–Nagumo equation, Nonlinearity, 31 (2018), 5536-5572.
doi: 10.1088/1361-6544/aae1db. |
[7] |
A. R. Champneys,
Homoclinic orbits in the dynamics of articulated pipes conveying fluid, Nonlinearity, 4 (1991), 747-774.
doi: 10.1088/0951-7715/4/3/007. |
[8] |
A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. Sneyd,
When shil'nikov meets hopf in excitable systems, SIAM J. Appl. Dyn. Syst., 6 (2007), 663-693.
doi: 10.1137/070682654. |
[9] |
M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga and M. Wechselberger,
Mixed-mode oscillations with multiple time scales, SIAM Rev., 54 (2012), 211-288.
doi: 10.1137/100791233. |
[10] |
M. Desroches, T. J. Kaper and M. Krupa, Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster, Chaos: An Interdisciplinary Journal of Nonlinear Science, 23 (2013), 046106, 13 pp.
doi: 10.1063/1.4827026. |
[11] |
E. Doedel, B. Oldeman et al., Auto-07p: Continuation and bifurcation software for ordinary differential equations, 2020, Latest version at https://github.com/auto-07p. |
[12] |
F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), no. 577,100 pp.
doi: 10.1090/memo/0577. |
[13] |
C. Fall, E. Marland, J. Wagner and J. Tyson, Computational Cell Biology, Springer-Verlag, New York, 2002. |
[14] |
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. |
[15] |
P. Gaspard, R. Kapral and G. Nicolis,
Bifurcation phenomena near homoclinic systems; A two-parameter analysis, J. Stat. Phys., 35 (1984), 697-727.
doi: 10.1007/BF01010829. |
[16] |
P. Glendinning and C. Sparrow,
Local and global behaviour near homoclinic orbits, J. Stat. Phys., 35 (1984), 645-696.
doi: 10.1007/BF01010828. |
[17] |
J. Guckenheimer and C. Kuehn,
Homoclinic orbits of the FitzHugh–Nagumo equation: The singular-limit, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 851-872.
doi: 10.3934/dcdss.2009.2.851. |
[18] |
J. Guckenheimer and C. Kuehn,
Homoclinic orbits of the FitzHugh–Nagumo equation: Bifurcations in the full system, SIAM J. Appl. Dyn. Syst., 9 (2010), 138-153.
doi: 10.1137/090758404. |
[19] |
S. P. Hastings,
On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations, Quart. J. Math. Oxford Ser., 27 (1976), 123-134.
doi: 10.1093/qmath/27.1.123. |
[20] |
S. P. Hastings,
Single and multiple pulse waves for the FitzHugh–Nagumo equations, SIAM J. Appl. Math., 42 (1982), 247-260.
doi: 10.1137/0142018. |
[21] |
C. Jones, N. Kopell and R. Langer, Construction of the FitzHugh-Nagumo pulse using differential forms, in Patterns and Dynamics in Reactive Media, Springer, 1991,101–115.
doi: 10.1007/978-1-4612-3206-3_7. |
[22] |
T. Kaper and C. Jones, A primer on the exchange lemma for fast-slow systems, vol. 122 of The IMA Volumes in Mathematics and its Applications, 65–87, Springer, New York, 2001.
doi: 10.1007/978-1-4613-0117-2_3. |
[23] |
J. Keener and J. Sneyd, Mathematical Physiology, 2nd edition, Springer-Verlag, New York, 2009. |
[24] |
M. Krupa, B. Sandstede and P. Szmolyan,
Fast and slow waves in the FitzHugh–Nagumo equation, J. Differential Equations, 133 (1997), 49-97.
doi: 10.1006/jdeq.1996.3198. |
[25] |
M. Krupa and P. Szmolyan,
Extending geometric singular perturbation theory to nonhyperbolic points–-fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), 286-314.
doi: 10.1137/S0036141099360919. |
[26] |
M. Krupa and P. Szmolyan,
Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.
doi: 10.1006/jdeq.2000.3929. |
[27] |
C. Kuehn, Multiple Time Series Dynamical Systems, Springer-Verlag, Heidelberg, 2015, Applied Mathematical Sciences, vol. 191. |
[28] |
Y. Kuznetsov and A. Panfilov, Stochastic waves in the FitzHugh-Nagumo system, 1981, Research Computing Centre, USSR Academy of Sciences, Pushchino. In Russian. |
[29] |
D. Linaro, A. Champneys, M. Desroches and M. Storace,
Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh–Rose burster, SIAM J. Appl. Dyn. Syst., 11 (2012), 939-962.
doi: 10.1137/110848931. |
[30] |
J. Nagumo, S. 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. |
[31] |
H. Osinga and K. Tsaneva-Atanasova,
Dynamics of plateau bursting depending on the location of its equilibrium, Journal of Neuroendocrinology, 22 (2010), 1301-1314.
doi: 10.1111/j.1365-2826.2010.02083.x. |
[32] |
J. D. M. Rademacher, Homoclinic Bifurcation from Heteroclinic Cycles with Periodic Orbits and Tracefiring of Pulses, Ph.D. thesis, University of Minnesota, 2004, http://www.math.uni-bremen.de/~jdmr/pub/dissMay7Web.pdf. |
[33] |
L. Shilnikov,
On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type, Math. USSR Sb., 6 (1968), 427-438.
|
[34] |
L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Method of Qualitative Theory of Nonlinear Dynamics: Part II, World Scientific, Singapore, 2001.
doi: 10.1142/9789812798558_0001. |
[35] |
C. Soto-Trevino, Geometric Methods for Periodic Orbits in Singularly Perturbed Systems, Ph.D. Thesis, Boston University, 1998. |
[36] |
P. Szmolyan and M. Wechselberger,
Canards in $\mathbb{R}^3$, J. Differential Equations, 177 (2001), 419-453.
doi: 10.1006/jdeq.2001.4001. |
[37] |
D. Terman,
Chaotic spikes arising from a model of bursting in excitable membranes, SIAM J. Appl. Math., 51 (1991), 1418-1450.
doi: 10.1137/0151071. |
[38] |
M. Wechselberger,
Existence and bifurcation of canards in $\mathbb{R}^3$ in the case of a folded node, SIAM J. Appl. Dyn. Syst., 4 (2005), 101-139.
doi: 10.1137/030601995. |



















abs. error | |||||
2 | 0.007800 | 0.009120 | 0.297041 | -3.134495 | -3.312145 |
3 | 0.007810 | 0.008331 | 0.143856 | -2.766146 | -2.914168 |
4 | 0.007659 | 0.007844 | 0.088228 | -0.752243 | -2.656183 |
abs. error | |||||
2 | 0.007800 | 0.009120 | 0.297041 | -3.134495 | -3.312145 |
3 | 0.007810 | 0.008331 | 0.143856 | -2.766146 | -2.914168 |
4 | 0.007659 | 0.007844 | 0.088228 | -0.752243 | -2.656183 |
abs. error | |||||
2 | 0.010474 | 0.013585 | 0.229015 | -2.670591 | -3.312145 |
3 | 0.010851 | 0.012412 | 0.125764 | -2.500844 | -2.914168 |
4 | 0.010742 | 0.011690 | 0.081075 | -1.963004 | -2.656183 |
5 | 0.010570 | 0.011208 | 0.056862 | -2.607696 | -2.474712 |
6 | 0.010398 | 0.010861 | 0.042625 | 0.821494 | -2.338275 |
abs. error | |||||
2 | 0.010474 | 0.013585 | 0.229015 | -2.670591 | -3.312145 |
3 | 0.010851 | 0.012412 | 0.125764 | -2.500844 | -2.914168 |
4 | 0.010742 | 0.011690 | 0.081075 | -1.963004 | -2.656183 |
5 | 0.010570 | 0.011208 | 0.056862 | -2.607696 | -2.474712 |
6 | 0.010398 | 0.010861 | 0.042625 | 0.821494 | -2.338275 |
[1] |
John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851 |
[2] |
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 |
[3] |
John M. Hong, Cheng-Hsiung Hsu, Bo-Chih Huang, Tzi-Sheng Yang. Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1501-1526. doi: 10.3934/cpaa.2013.12.1501 |
[4] |
Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172 |
[5] |
Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150 |
[6] |
Ilona Gucwa, Peter Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 783-806. doi: 10.3934/dcdss.2009.2.783 |
[7] |
Hua Chen, Ling-Jun Wang. A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation. Kinetic and Related Models, 2012, 5 (2) : 261-281. doi: 10.3934/krm.2012.5.261 |
[8] |
John Boyd. Strongly nonlinear perturbation theory for solitary waves and bions. Evolution Equations and Control Theory, 2019, 8 (1) : 1-29. doi: 10.3934/eect.2019001 |
[9] |
Stefano Scrobogna. Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 5979-6034. doi: 10.3934/dcds.2017259 |
[10] |
Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305 |
[11] |
Kai Wang, Hongyong Zhao, Hao Wang. Geometric singular perturbation of a nonlocal partially degenerate model for Aedes aegypti. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022122 |
[12] |
Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106 |
[13] |
Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101 |
[14] |
Wenqiang Zhao. Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3453-3474. doi: 10.3934/dcdsb.2018251 |
[15] |
Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441 |
[16] |
Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639 |
[17] |
Carlos Arnoldo Morales. A note on periodic orbits for singular-hyperbolic flows. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 615-619. doi: 10.3934/dcds.2004.11.615 |
[18] |
Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807 |
[19] |
Jyoti Mishra. Analysis of the Fitzhugh Nagumo model with a new numerical scheme. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 781-795. doi: 10.3934/dcdss.2020044 |
[20] |
Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]