# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022085
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## Rotating spirals in oscillatory media with nonlocal interactions and their normal form

 Department of Mathematics, University of Houston, 3551 Cullen Blvd., Houston, TX 77204-3008, USA

* Corresponding author: Gabriela Jaramillo

Received  March 2021 Revised  January 2022 Early access April 2022

Fund Project: This work is supported by NSF grant DMS-1911742

Biological and physical systems that can be classified as oscillatory media give rise to interesting phenomena like target patterns and spiral waves. The existence of these structures has been proven in the case of systems with local diffusive interactions. In this paper the more general case of oscillatory media with nonlocal coupling is considered. We model these systems using evolution equations where the nonlocal interactions are expressed via a diffusive convolution kernel, and prove the existence of rotating wave solutions for these systems. Since the nonlocal nature of the equations precludes the use of standard techniques from spatial dynamics, the method we use relies instead on a combination of a multiple-scales analysis and a construction similar to Lyapunov-Schmidt. This approach then allows us to derive a normal form, or reduced equation, that captures the leading order behavior of these solutions.

Citation: Gabriela Jaramillo. Rotating spirals in oscillatory media with nonlocal interactions and their normal form. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022085
##### References:
 [1] R. A. Adams and J. J. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. [2] F. Alliot and C. Amrouche, Weak solutions for the exterior Stokes problem in weighted {S}obolev spaces, Math. Methods Appl. Sci., 23 (2000), 575–600, https://onlinelibrary.wiley.com/doi/abs/10.1002/. doi: 10.1002/(SICI)1099-1476(200004)23:6<575::AID-MMA128>3.0.CO;2-4. [3] F. Andreu-Vaillo, J. J. Toledo-Melero, J. M. Mazon and J. D. Rossi, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010. doi: 10.1090/surv/165. [4] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Fifth edition, Harcourt/Academic Press, Burlington, MA, 2001. [5] C. Bachmair and E. Schöll, Nonlocal control of pulse propagation in excitable media, Eur. Phys. J. B, 87 (2014), Art. 276, 10 pp. doi: 10.1140/epjb/e2014-50339-2. [6] P. W. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428–440, http://www.sciencedirect.com/science/article/pii/S0022247X06009863. doi: 10.1016/j.jmaa.2006.09.007. [7] P. C. Bressloff, Spatiotemporal dynamics of continuum neural fields, J. Phys. A, 45 (2011), 033001,109 pp. doi: 10.1088/1751-8113/45/3/033001. [8] J. Christoph and M. Eiswirth, Theory of electrochemical pattern formation, Chaos: An Interdisciplinary Journal of Nonlinear Science, 12 (2002), 215-230.  doi: 10.1063/1.1449956. [9] P. Colet, M. A. Matías, L. Gelens and D. Gomila, Formation of localized structures in bistable systems through nonlocal spatial coupling. I. General framework, Phys. Rev. E, 89 (2014), 012914.  doi: 10.1103/PhysRevE.89.012914. [10] S. Coombes, Waves, bumps, and patterns in neural field theories, Biol. Cybernet., 93 (2005), 91-108.  doi: 10.1007/s00422-005-0574-y. [11] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727–755, https://hal.inrae.fr/hal-02659746. doi: 10.1017/S0308210504000721. [12] J. M. Davidenko, A. V. Pertsov, R. Salomonsz, W. Baxter and J. Jalife, Stationary and drifting spiral waves of excitation in isolated cardiac muscle, Nature, 355 (1992), 349-351.  doi: 10.1038/355349a0. [13] G. Faye and A. Scheel, Center manifolds without a phase space, Trans. Amer. Math. Soc., 370 (2018), 5843-5885.  doi: 10.1090/tran/7190. [14] V. García-Morales and K. Krischer, Nonlocal complex Ginzburg–Landau equation for electrochemical systems, Phys. Rev. Lett., 100 (2008), 054101.  doi: 10.1103/PhysRevLett.100.054101. [15] L. Gelens, M. A. Matías, D. Gomila, T. Dorissen and P. Colet, Formation of localized structures in bistable systems through nonlocal spatial coupling. II. The nonlocal Ginzburg–Landau equation, Phys. Rev. E, 89 (2014), 012915.  doi: 10.1103/PhysRevE.89.012915. [16] V. Girault and A. Sequeira, A well–posed problem for the exterior Stokes equations in two and three dimensions, Arch. Rational Mech. Anal., 114 (1991), 313-333.  doi: 10.1007/BF00376137. [17] J. M. Greenberg, Spiral waves for $\lambda$- $\omega$ systems, SIAM J. Appl. Math., 39 (1980), 301–309, http://www.jstor.org/stable/2101052. doi: 10.1137/0139026. [18] J. M. Greenberg, Spiral waves for $\lambda$- $\omega$ systems. II., Adv. in Appl. Math., 2 (1981), 450-454.  doi: 10.1016/0196-8858(81)90044-0. [19] P. S. Hagan, Spiral waves in reaction-diffusion equations, SIAM J. Appl. Math., 42 (1982), 762-786.  doi: 10.1137/0142054. [20] X. Huang, W. C. Troy, Q. Yang, H. Ma, C. R. Laing, S. J. Schiff and J.-Y. Wu, Spiral waves in disinhibited mammalian neocortex, Journal of Neuroscience, 24 (2004), 9897–9902, https://www.jneurosci.org/content/24/44/9897. doi: 10.1523/JNEUROSCI.2705-04.2004. [21] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1. [22] G. Jaramillo, Inhomogeneities in 3 dimensional oscillatory media, Netw. Heterog. Media, 10 (2015), 387-399.  doi: 10.3934/nhm.2015.10.387. [23] G. Jaramillo and A. Scheel, Deformation of striped patterns by inhomogeneities, Math. Methods Appl. Sci., 38 (2015), 51-65.  doi: 10.1002/mma.3049. [24] G. Jaramillo and A. Scheel, Pacemakers in large arrays of oscillators with nonlocal coupling, J. Differential Equations, 260 (2016), 2060–2090, https://www.sciencedirect.com/science/article/pii/S0022039615005288. doi: 10.1016/j.jde.2015.09.054. [25] G. Jaramillo, A. Scheel and Q. Wu, The effect of impurities on striped phases, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 131-168.  doi: 10.1017/S0308210518000197. [26] G. Jaramillo and S. C. Venkataramani, Target patterns in a 2D array of oscillators with nonlocal coupling, Nonlinearity, 31 (2018), 4162-4201.  doi: 10.1088/1361-6544/aac9a6. [27] J. P. Keener, The dynamics of three-dimensional scroll waves in excitable media, Physica D: Nonlinear Phenomena, 31 (1988), 269-276.  doi: 10.1016/0167-2789(88)90080-2. [28] J. P. Keener and J. J. Tyson, The dynamics of scroll waves in excitable media, SIAM Review, 34 (1992), 1-39.  doi: 10.1137/1034001. [29] P. Kirrmann, G. Schneider and A. Mielke, The validity of modulation equations for extended systems with cubic nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 85-91.  doi: 10.1017/S0308210500020989. [30] A. N. Kolmogorov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. Moskow, Ser. Internat., Sec. A, 1 (1937), 1-25. [31] V. A. Kondratév, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskovskogo Matematicheskogo Obshchestva, 16 (1967), 209-292. [32] N. Kopell and L. N. Howard, Target pattern and spiral solutions to reaction-diffusion equations with more than one space dimension, Adv. in Appl. Math., 2 (1981), 417-449.  doi: 10.1016/0196-8858(81)90043-9. [33] C. Kuehn and S. Throm, Validity of amplitude equations for nonlocal nonlinearities, J. Math. Phys., 59 (2018), 071510, 17 pp. doi: 10.1063/1.4993112. [34] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer Series in Synergetics, 19. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3. [35] R. C. McOwen, The behavior of the Laplacian on weighted Sobolev spaces, Comm. Pure Appl. Math., 32 (1979), 783-795.  doi: 10.1002/cpa.3160320604. [36] S. Nettesheim, A. von Oertzen, H. H. Rotermund and G. Ertl, Reaction–diffusion patterns in the catalytic CO-oxidation on Pt(110): Front propagation and spiral waves, J. Chem. Phys., 98 (1993), 9977-9985.  doi: 10.1063/1.464323. [37] E. M. Nicola, M. Bär and H. Engel, Wave instability induced by nonlocal spatial coupling in a model of the light-sensitive Belousov–Zhabotinsky reaction, Phys. Rev. E, 73 (2006), 066225.  doi: 10.1103/PhysRevE.73.066225. [38] E. M. Nicola, M. Or-Guil, W. Wolf and M. Bär, Drifting pattern domains in a reaction-diffusion system with nonlocal coupling, Phys. Rev. E, 65 (2002), 055101.  doi: 10.1103/PhysRevE.65.055101. [39] F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl and M. A. McClain, NIST Digital Library of Mathematical Functions, Release 1.0.28 of 2020-09-15, http://dlmf.nist.gov/. [40] A. M. Pertsov, J. M. Davidenko, R. Salomonsz, W. T. Baxter and J. Jalife, Spiral waves of excitation underlie reentrant activity in isolated cardiac muscle, Circulation Research, 72 (1993), 631-650.  doi: 10.1161/01.RES.72.3.631. [41] D. J. Pinto and G. B. Ermentrout, Spatially structured activity in synaptically coupled neuronal networks: I. Traveling fronts and pulses, SIAM J. Appl. Math., 62 (2001), 206-225.  doi: 10.1137/S0036139900346453. [42] F. Plenge, H. Varela and K. Krischer, Asymmetric target patterns in one-dimensional oscillatory media with genuine nonlocal coupling, Phys. Rev. Lett., 94 (2005), 198301.  doi: 10.1103/PhysRevLett.94.198301. [43] M. Reed and B. Simon, II: Fourier Analysis, Self-Adjointness, vol. 2, Elsevier, 1975. [44] A. J. Roberts, Macroscale, slowly varying, models emerge from the microscale dynamics, IMA J. Appl. Math., 80 (2015), 1492-1518.  doi: 10.1093/imamat/hxv004. [45] A. Scheel, Bifurcation to spiral waves in reaction-diffusion systems, SIAM J. Math. Anal., 29 (1998), 1399-1418.  doi: 10.1137/S0036141097318948. [46] G. Schneider, Validity and limitation of the Newell–Whitehead equation, Math. Nachr., 176 (1995), 249-263.  doi: 10.1002/mana.19951760118. [47] G. Schneider, The validity of generalized Ginzburg–Landau equations, Math. Methods Appl. Sci., 19 (1996), 717-736.  doi: 10.1002/(SICI)1099-1476(199606)19:9<717::AID-MMA792>3.0.CO;2-Z. [48] G. Schneider, Justification of modulation equations for hyperbolic systems via normal forms, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 69-82.  doi: 10.1007/s000300050034. [49] M. Sheintuch and O. Nekhamkina, Reaction-diffusion patterns on a disk or a square in a model with long-range interaction, J. Chem. Phys., 107 (1997), 8165-8174.  doi: 10.1063/1.3427649. [50] S.-i. Shima and Y. Kuramoto, Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators, Phys. Rev. E, 69 (2004), 036213.  doi: 10.1103/PhysRevE.69.036213. [51] F. Siegert and C. J. Weijer, Spiral and concentric waves organize multicellular dictyostelium mounds, Current Biology, 5 (1995), 937-943.  doi: 10.1016/S0960-9822(95)00184-9. [52] J. Siebert, S. Alonso, M. Bär and E. Schöll, Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection, Phys. Rev. E, 89 (2014), 052909.  doi: 10.1103/PhysRevE.89.052909. [53] M. Specovius-Neugebauer and W. Wendland, Exterior Stokes problems and decay at infinity, Math. Methods Appl. Sci., 8 (1986), 351-367.  doi: 10.1002/mma.1670080124. [54] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Vol. 32, Princeton Mathematical Series. [55] D. Tanaka and Y. Kuramoto, Complex Ginzburg–Landau equation with nonlocal coupling, Phys. Rev. E, 68 (2003), 026219.  doi: 10.1103/PhysRevE.68.026219. [56] A. van Harten, On the validity of the Ginzburg–Landau equation, J. Nonlinear Sci., 1 (1991), 397-422.  doi: 10.1007/BF02429847.

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##### References:
 [1] R. A. Adams and J. J. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. [2] F. Alliot and C. Amrouche, Weak solutions for the exterior Stokes problem in weighted {S}obolev spaces, Math. Methods Appl. Sci., 23 (2000), 575–600, https://onlinelibrary.wiley.com/doi/abs/10.1002/. doi: 10.1002/(SICI)1099-1476(200004)23:6<575::AID-MMA128>3.0.CO;2-4. [3] F. Andreu-Vaillo, J. J. Toledo-Melero, J. M. Mazon and J. D. Rossi, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010. doi: 10.1090/surv/165. [4] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Fifth edition, Harcourt/Academic Press, Burlington, MA, 2001. [5] C. Bachmair and E. Schöll, Nonlocal control of pulse propagation in excitable media, Eur. Phys. J. B, 87 (2014), Art. 276, 10 pp. doi: 10.1140/epjb/e2014-50339-2. [6] P. W. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428–440, http://www.sciencedirect.com/science/article/pii/S0022247X06009863. doi: 10.1016/j.jmaa.2006.09.007. [7] P. C. Bressloff, Spatiotemporal dynamics of continuum neural fields, J. Phys. A, 45 (2011), 033001,109 pp. doi: 10.1088/1751-8113/45/3/033001. [8] J. Christoph and M. Eiswirth, Theory of electrochemical pattern formation, Chaos: An Interdisciplinary Journal of Nonlinear Science, 12 (2002), 215-230.  doi: 10.1063/1.1449956. [9] P. Colet, M. A. Matías, L. Gelens and D. Gomila, Formation of localized structures in bistable systems through nonlocal spatial coupling. I. General framework, Phys. Rev. E, 89 (2014), 012914.  doi: 10.1103/PhysRevE.89.012914. [10] S. Coombes, Waves, bumps, and patterns in neural field theories, Biol. Cybernet., 93 (2005), 91-108.  doi: 10.1007/s00422-005-0574-y. [11] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727–755, https://hal.inrae.fr/hal-02659746. doi: 10.1017/S0308210504000721. [12] J. M. Davidenko, A. V. Pertsov, R. Salomonsz, W. Baxter and J. Jalife, Stationary and drifting spiral waves of excitation in isolated cardiac muscle, Nature, 355 (1992), 349-351.  doi: 10.1038/355349a0. [13] G. Faye and A. Scheel, Center manifolds without a phase space, Trans. Amer. Math. Soc., 370 (2018), 5843-5885.  doi: 10.1090/tran/7190. [14] V. García-Morales and K. Krischer, Nonlocal complex Ginzburg–Landau equation for electrochemical systems, Phys. Rev. Lett., 100 (2008), 054101.  doi: 10.1103/PhysRevLett.100.054101. [15] L. Gelens, M. A. Matías, D. Gomila, T. Dorissen and P. Colet, Formation of localized structures in bistable systems through nonlocal spatial coupling. II. The nonlocal Ginzburg–Landau equation, Phys. Rev. E, 89 (2014), 012915.  doi: 10.1103/PhysRevE.89.012915. [16] V. Girault and A. Sequeira, A well–posed problem for the exterior Stokes equations in two and three dimensions, Arch. Rational Mech. Anal., 114 (1991), 313-333.  doi: 10.1007/BF00376137. [17] J. M. Greenberg, Spiral waves for $\lambda$- $\omega$ systems, SIAM J. Appl. Math., 39 (1980), 301–309, http://www.jstor.org/stable/2101052. doi: 10.1137/0139026. [18] J. M. Greenberg, Spiral waves for $\lambda$- $\omega$ systems. II., Adv. in Appl. Math., 2 (1981), 450-454.  doi: 10.1016/0196-8858(81)90044-0. [19] P. S. Hagan, Spiral waves in reaction-diffusion equations, SIAM J. Appl. Math., 42 (1982), 762-786.  doi: 10.1137/0142054. [20] X. Huang, W. C. Troy, Q. Yang, H. Ma, C. R. Laing, S. J. Schiff and J.-Y. Wu, Spiral waves in disinhibited mammalian neocortex, Journal of Neuroscience, 24 (2004), 9897–9902, https://www.jneurosci.org/content/24/44/9897. doi: 10.1523/JNEUROSCI.2705-04.2004. [21] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1. [22] G. Jaramillo, Inhomogeneities in 3 dimensional oscillatory media, Netw. Heterog. Media, 10 (2015), 387-399.  doi: 10.3934/nhm.2015.10.387. [23] G. Jaramillo and A. Scheel, Deformation of striped patterns by inhomogeneities, Math. Methods Appl. Sci., 38 (2015), 51-65.  doi: 10.1002/mma.3049. [24] G. Jaramillo and A. Scheel, Pacemakers in large arrays of oscillators with nonlocal coupling, J. Differential Equations, 260 (2016), 2060–2090, https://www.sciencedirect.com/science/article/pii/S0022039615005288. doi: 10.1016/j.jde.2015.09.054. [25] G. Jaramillo, A. Scheel and Q. Wu, The effect of impurities on striped phases, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 131-168.  doi: 10.1017/S0308210518000197. [26] G. Jaramillo and S. C. Venkataramani, Target patterns in a 2D array of oscillators with nonlocal coupling, Nonlinearity, 31 (2018), 4162-4201.  doi: 10.1088/1361-6544/aac9a6. [27] J. P. Keener, The dynamics of three-dimensional scroll waves in excitable media, Physica D: Nonlinear Phenomena, 31 (1988), 269-276.  doi: 10.1016/0167-2789(88)90080-2. [28] J. P. Keener and J. J. Tyson, The dynamics of scroll waves in excitable media, SIAM Review, 34 (1992), 1-39.  doi: 10.1137/1034001. [29] P. Kirrmann, G. Schneider and A. Mielke, The validity of modulation equations for extended systems with cubic nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 85-91.  doi: 10.1017/S0308210500020989. [30] A. N. Kolmogorov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. Moskow, Ser. Internat., Sec. A, 1 (1937), 1-25. [31] V. A. Kondratév, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskovskogo Matematicheskogo Obshchestva, 16 (1967), 209-292. [32] N. Kopell and L. N. Howard, Target pattern and spiral solutions to reaction-diffusion equations with more than one space dimension, Adv. in Appl. Math., 2 (1981), 417-449.  doi: 10.1016/0196-8858(81)90043-9. [33] C. Kuehn and S. Throm, Validity of amplitude equations for nonlocal nonlinearities, J. Math. Phys., 59 (2018), 071510, 17 pp. doi: 10.1063/1.4993112. [34] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer Series in Synergetics, 19. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3. [35] R. C. McOwen, The behavior of the Laplacian on weighted Sobolev spaces, Comm. Pure Appl. Math., 32 (1979), 783-795.  doi: 10.1002/cpa.3160320604. [36] S. Nettesheim, A. von Oertzen, H. H. Rotermund and G. Ertl, Reaction–diffusion patterns in the catalytic CO-oxidation on Pt(110): Front propagation and spiral waves, J. Chem. Phys., 98 (1993), 9977-9985.  doi: 10.1063/1.464323. [37] E. M. Nicola, M. Bär and H. Engel, Wave instability induced by nonlocal spatial coupling in a model of the light-sensitive Belousov–Zhabotinsky reaction, Phys. Rev. E, 73 (2006), 066225.  doi: 10.1103/PhysRevE.73.066225. [38] E. M. Nicola, M. Or-Guil, W. Wolf and M. Bär, Drifting pattern domains in a reaction-diffusion system with nonlocal coupling, Phys. Rev. E, 65 (2002), 055101.  doi: 10.1103/PhysRevE.65.055101. [39] F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl and M. A. McClain, NIST Digital Library of Mathematical Functions, Release 1.0.28 of 2020-09-15, http://dlmf.nist.gov/. [40] A. M. Pertsov, J. M. Davidenko, R. Salomonsz, W. T. Baxter and J. Jalife, Spiral waves of excitation underlie reentrant activity in isolated cardiac muscle, Circulation Research, 72 (1993), 631-650.  doi: 10.1161/01.RES.72.3.631. [41] D. J. Pinto and G. B. Ermentrout, Spatially structured activity in synaptically coupled neuronal networks: I. Traveling fronts and pulses, SIAM J. Appl. Math., 62 (2001), 206-225.  doi: 10.1137/S0036139900346453. [42] F. Plenge, H. Varela and K. Krischer, Asymmetric target patterns in one-dimensional oscillatory media with genuine nonlocal coupling, Phys. Rev. Lett., 94 (2005), 198301.  doi: 10.1103/PhysRevLett.94.198301. [43] M. Reed and B. Simon, II: Fourier Analysis, Self-Adjointness, vol. 2, Elsevier, 1975. [44] A. J. Roberts, Macroscale, slowly varying, models emerge from the microscale dynamics, IMA J. Appl. Math., 80 (2015), 1492-1518.  doi: 10.1093/imamat/hxv004. [45] A. Scheel, Bifurcation to spiral waves in reaction-diffusion systems, SIAM J. Math. Anal., 29 (1998), 1399-1418.  doi: 10.1137/S0036141097318948. [46] G. Schneider, Validity and limitation of the Newell–Whitehead equation, Math. Nachr., 176 (1995), 249-263.  doi: 10.1002/mana.19951760118. [47] G. Schneider, The validity of generalized Ginzburg–Landau equations, Math. Methods Appl. Sci., 19 (1996), 717-736.  doi: 10.1002/(SICI)1099-1476(199606)19:9<717::AID-MMA792>3.0.CO;2-Z. [48] G. Schneider, Justification of modulation equations for hyperbolic systems via normal forms, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 69-82.  doi: 10.1007/s000300050034. [49] M. Sheintuch and O. Nekhamkina, Reaction-diffusion patterns on a disk or a square in a model with long-range interaction, J. Chem. Phys., 107 (1997), 8165-8174.  doi: 10.1063/1.3427649. [50] S.-i. Shima and Y. Kuramoto, Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators, Phys. Rev. E, 69 (2004), 036213.  doi: 10.1103/PhysRevE.69.036213. [51] F. Siegert and C. J. Weijer, Spiral and concentric waves organize multicellular dictyostelium mounds, Current Biology, 5 (1995), 937-943.  doi: 10.1016/S0960-9822(95)00184-9. [52] J. Siebert, S. Alonso, M. Bär and E. Schöll, Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection, Phys. Rev. E, 89 (2014), 052909.  doi: 10.1103/PhysRevE.89.052909. [53] M. Specovius-Neugebauer and W. Wendland, Exterior Stokes problems and decay at infinity, Math. Methods Appl. Sci., 8 (1986), 351-367.  doi: 10.1002/mma.1670080124. [54] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Vol. 32, Princeton Mathematical Series. [55] D. Tanaka and Y. Kuramoto, Complex Ginzburg–Landau equation with nonlocal coupling, Phys. Rev. E, 68 (2003), 026219.  doi: 10.1103/PhysRevE.68.026219. [56] A. van Harten, On the validity of the Ginzburg–Landau equation, J. Nonlinear Sci., 1 (1991), 397-422.  doi: 10.1007/BF02429847.
Spirals obtained by numerical simulations of a FitzHugh-Nagumo system (see Section 6) with $\delta = 0.2, \beta = 1, \tau = 0.1$, and $K \ast u = \frac{\sigma}{D} [ - u + K_0(|x|/ \sqrt{D}) \ast u]$. Here $K_0$ is the Modified Bessel function of the second kind. In figures a) and b) the choice of $\sigma = 5$ and $D = 0.5$, results in a spiral pattern. In contrast, in figures c) and d) the choice of $\sigma = 5$ and $D = 1$ results in a spiral chimera pattern (incoherent core). The figures on the right zoom in into the core of the spirals appearing on the left
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