Localized patterns in a three-component Fitzhugh-Nagumo model revisited via an action functional. (English) Zbl 1402.35026
The authors introduce a variational approach to revisit the study of stationary pulse solutions of a singularly perturbed FitzHugh-Nagumo system. They find some action function whose (minimizers) critical points are (stable) stationary solutions. This allows them to formally recover earlier results from the literature while avoiding most of the technical machinery (Evans function and nonlocal eigenvalue problem approaches), and to formulate new conjectures on the stability of various types of pulses (asymmetric or with several peaks).
Reviewer: Thomas Giletti (Vandœuvre-lès-Nancy)
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35B35 | Stability in context of PDEs |
| 35K57 | Reaction-diffusion equations |
| 49J40 | Variational inequalities |
| 35J20 | Variational methods for second-order elliptic equations |
References:
| [1] | Bode, M; Liehr, AW; Schenk, CP; Purwins, HG, Interaction of dissipative solitons: particle-like behaviour of localized structures in a three-component reaction-diffusion system, Physica D, 161, 45-66, (2002) · Zbl 0985.35096 · doi:10.1016/S0167-2789(01)00360-8 |
| [2] | Chen, CN; Choi, YS, Standing pulse solutions to Fitzhugh-Nagumo equations, Arch. Ration. Mech. Anal., 206, 741-777, (2012) · Zbl 1264.35119 · doi:10.1007/s00205-012-0542-3 |
| [3] | Chen, CN; Choi, YS, Traveling pulse solutions to Fitzhugh-Nagumo equations, Calc. Var. Partial Difer. Equ., 54, 1-45, (2015) · Zbl 1328.34037 · doi:10.1007/s00526-014-0776-z |
| [4] | Chen, CN; Hu, X, Maslov index for homoclinic orbits of Hamiltonian systems, Ann. I. H. Poincare-An., 24, 589-603, (2007) · Zbl 1202.37081 · doi:10.1016/j.anihpc.2006.06.002 |
| [5] | Chen, CN; Hu, X, Stability analysis for standing pulse solutions to Fitzhugh-Nagumo equations, Calc. Var. Partial Differ. Equ., 49, 827-845, (2014) · Zbl 1315.35030 · doi:10.1007/s00526-013-0601-0 |
| [6] | Chen, CN; Tanaka, K, A variational approach for standing waves of Fitzhugh-Nagumo type systems, J. Differ. Equ., 257, 109-144, (2014) · Zbl 1292.35117 · doi:10.1016/j.jde.2014.03.013 |
| [7] | Chen, CN; Ei, SI; Lin, YP; Kung, SY, Standing waves joining with Turing patterns in Fitzhugh-Nagumo type systems, Commun. Partial Differ. Equ., 36, 998-1015, (2011) · Zbl 1233.35114 · doi:10.1080/03605302.2010.509769 |
| [8] | Chen, CN; Kung, SY; Morita, Y, Planar standing wavefronts in the Fitzhugh-Nagumo equations, SIAM J. Math. Anal., 46, 657-690, (2014) · Zbl 1301.34060 · doi:10.1137/130907793 |
| [9] | Chirilus-Bruckner, M; Doelman, A; Heijster, P; Rademacher, JDM, Butterfly catastrophe for fronts in a three-component reaction-diffusion system, J. Nonlinear Sci., 25, 87-129, (2015) · Zbl 1325.35088 · doi:10.1007/s00332-014-9222-9 |
| [10] | Corwin, L., Szczarba, R.H.: Multivariable Calculus. Monographs and Textbooks in Pure and Applied Mathematics, vol. 64. Marcel Dekker, New York (1982) · Zbl 0494.26003 |
| [11] | Rijk, B; Doelman, A; Rademacher, JDM, Spectra and stability of spatially periodic pulse patterns: Evans function factorization via Riccati transformation, SIAM J. Math. Anal., 48, 61-121, (2016) · Zbl 1336.35042 · doi:10.1137/15M1007264 |
| [12] | Doedel, EJ; Krauskopf, B (ed.); Osinga, HM (ed.); Galn-Vioque, J (ed.), Lecture notes on numerical analysis of nonlinear equations, 1-49, (2007), Dordrecht · Zbl 1130.65119 |
| [13] | Doelman, A; Gardner, RA; Kaper, TJ, Stability analysis of singular patterns in the 1D gray-Scott model: a matched asymptotics approach, Physica D, 122, 1-36, (1998) · Zbl 0943.34039 · doi:10.1016/S0167-2789(98)00180-8 |
| [14] | Doelman, A; Gardner, RA; Kaper, TJ, Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J., 50, 443-507, (2001) · Zbl 0994.35058 · doi:10.1512/iumj.2001.50.1873 |
| [15] | Doelman, A., Gardner, R.A., Kaper, T.J.: A stability index analysis of 1-D patterns of the Gray-Scott model. Mem. Am. Math. Soc. 155, xii+64 (2002) · Zbl 0994.35059 |
| [16] | Doelman, A; Heijster, P; Kaper, T, Pulse dynamics in a three-component system: existence analysis, J. Dyn. Differ. Equ., 21, 73-115, (2009) · Zbl 1173.35068 · doi:10.1007/s10884-008-9125-2 |
| [17] | Doelman, A; Heijster, P; Xie, F, A geometric approach to stationary defect solutions in one space dimension, SIAM J. Appl. Dyn. Syst., 15, 655-712, (2016) · Zbl 1343.34041 · doi:10.1137/15M1026742 |
| [18] | Fenichel, N, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31, 53-98, (1979) · Zbl 0476.34034 · doi:10.1016/0022-0396(79)90152-9 |
| [19] | Gurevich, SV; Amiranashvili, S; Purwins, HG, Breathing dissipative solitons in three-component reaction-diffusion system, Phys. Rev. E, 74, 066201, (2006) · doi:10.1103/PhysRevE.74.066201 |
| [20] | Jones, CKRT; Arnold, L (ed.), Geometric singular perturbation theory, No. 1609, 44-118, (1995), Berlin · Zbl 0840.58040 · doi:10.1007/BFb0095239 |
| [21] | Kaper, TJ; OMalley, RE (ed.); Cronin, J (ed.), An introduction to geometric methods and dynamical systems theory for singular perturbation problems, No. 56, 85-131, (1999), Providence, RI · doi:10.1090/psapm/056/1718893 |
| [22] | Kapitula, T., Promislow, K.: Spectral and Dynamical Stability of Nonlinear Waves. Springer, New York (2013) · Zbl 1297.37001 · doi:10.1007/978-1-4614-6995-7 |
| [23] | Keller, HB, Numerical solution of bifurcation and nonlinear eigenvalue problems, Appl. Bifurcation Theory, 1, 359-384, (1977) · Zbl 0581.65043 |
| [24] | Nishiura, Y.: Coexistence Of Infinitely Many Stable Solutions to Reaction Diffusion Systems in the Singular Limit. Dynamics Reported: Exposition in Dynamical Systems New Series, vol. 3, pp. 25-103. Springer, New York (1994) · Zbl 0806.35079 |
| [25] | Nishiura, Y; Fujii, H, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18, 1726-1770, (1987) · Zbl 0638.35010 · doi:10.1137/0518124 |
| [26] | Nishiura, Y; Ikeda, H; Suzuki, H, Stability of traveling waves and a relation between the Evans function and the SLEP equation, J. Reine Angew. Math., 475, 1-37, (1996) · Zbl 0847.35014 |
| [27] | Nishiura, Y; Teramoto, T; Ueda, KI, Dynamic transitions through scattors in dissipative systems, Chaos, 13, 962-972, (2003) · doi:10.1063/1.1592131 |
| [28] | Nishiura, Y; Teramoto, T; Yuan, X; Ueda, KI, Dynamics of traveling pulses in heterogeneous media, Chaos, 17, 037104, (2007) · Zbl 1163.37356 · doi:10.1063/1.2778553 |
| [29] | Or-Guil, M; Bode, M; Schenk, CP; Purwins, HG, Spot bifurcations in three-component reaction-diffusion systems: the onset of propagation, Phys. Rev. E, 57, 6432-6437, (1998) · doi:10.1103/PhysRevE.57.6432 |
| [30] | Promislow, K, A renormalization method for modulational stability of quasi-steady patterns in dispersive systems, SIAM J. Math. Anal., 33, 1455-1482, (2002) · Zbl 1019.34057 · doi:10.1137/S0036141000377547 |
| [31] | Purwins, HG; Stollenwerk, L, Synergetic aspects of gas-discharge: lateral patterns in dc systems with a high ohmic barrier, Plasma Phys. Controlled Fusion, 56, 123001, (2014) · doi:10.1088/0741-3335/56/12/123001 |
| [32] | Rademacher, JDM, First and second order semi-strong interaction in reaction-diffusion systems, SIAM J. Appl. Dyn. Syst., 12, 175-203, (2013) · Zbl 1282.35029 · doi:10.1137/110850165 |
| [33] | Reinecke, C; Sweers, G, A positive solution on \(\mathbb{R}^n\) to a equations of Fitzhugh-Nagumo type, J. Differ. Equ., 153, 292-312, (1999) · Zbl 0929.35042 · doi:10.1006/jdeq.1998.3560 |
| [34] | Robinson, C, Sustained resonance for a nonlinear system with slowly-varying coefficients, SIAM J. Math. Anal., 14, 847-860, (1983) · Zbl 0523.34035 · doi:10.1137/0514066 |
| [35] | Schenk, CP; Or-Guil, M; Bode, M; Purwins, HG, Interacting pulses in three-component reaction-diffusion systems on two-dimensional domains, Phys. Rev. Lett., 78, 3781-3784, (1997) · doi:10.1103/PhysRevLett.78.3781 |
| [36] | Seydel, R.: Practical Bifurcation and Stability Analysis, vol. 5. Springer, New York (2009) · Zbl 1195.34004 |
| [37] | Ploeg, H; Doelman, A, Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations, Indiana Univ. Math. J., 54, 1219-1301, (2005) · Zbl 1083.35011 · doi:10.1512/iumj.2005.54.2792 |
| [38] | Heijster, P; Doelman, A; Kaper, T, Pulse dynamics in a three-component system: stability and bifurcations, Physica D, 237, 3335-3368, (2008) · Zbl 1153.37437 · doi:10.1016/j.physd.2008.07.014 |
| [39] | Heijster, P; Doelman, A; Kaper, TJ; Promislow, K, Front interactions in a three-component system, SIAM J. Appl. Dyn. Syst., 9, 292-332, (2010) · Zbl 1197.35047 · doi:10.1137/080744785 |
| [40] | Heijster, P; Doelman, A; Kaper, T; Nishiura, Y; Ueda, KI, Pinned fronts in heterogeneous media of jump type, Nonlinearity, 24, 127-157, (2011) · Zbl 1208.35010 · doi:10.1088/0951-7715/24/1/007 |
| [41] | Heijster, P; Sandstede, B, Planar radial spots in a three-component Fitzhugh-Nagumo system, J. Nonlinear Sci., 21, 705-745, (2011) · Zbl 1233.35012 · doi:10.1007/s00332-011-9098-x |
| [42] | Heijster, P; Sandstede, B, Coexistence of stable spots and fronts in a three-component Fitzhugh-Nagumo system, RIMS Kokyuroku Bessatsu, B31, 135-155, (2012) · Zbl 1258.35123 |
| [43] | Heijster, P; Sandstede, B, Bifurcations to travelling planar spots in a three-component Fitzhugh-Nagumo system, Physica D, 275, 19-34, (2014) · Zbl 1341.37029 · doi:10.1016/j.physd.2014.02.001 |
| [44] | Vanag, VK; Epstein, IR, Localized patterns in reaction-diffusion systems, Chaos, 17, 037110, (2007) · Zbl 1163.37381 · doi:10.1063/1.2752494 |
| [45] | Yadome, M; Nishiura, Y; Teramoto, T, Robust pulse generators in an excitable medium with jump-type heterogeneity, SIAM J. Appl. Dyn. Syst., 13, 1168-1201, (2014) · Zbl 1331.35040 · doi:10.1137/13091261X |
| [46] | Yuan, X; Teramoto, T; Nishiura, Y, Heterogeneity-induced defect bifurcation and pulse dynamics for a three-component reaction-diffusion system, Phys. Rev. E, 75, 036220, (2007) · doi:10.1103/PhysRevE.75.036220 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
