Coupling-induced Turing instability and $ O(3) $-equivariant mode interaction in a bulk–surface system with fractional surface diffusion

Duc Hau Nguyen; Duc Hau Nguyen

doi:10.3934/math.20261130

AIMS Mathematics

2026, Volume 11, Issue 9: 28383-28408. doi: 10.3934/math.20261130

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Research article

Coupling-induced Turing instability and $ O(3) $-equivariant mode interaction in a bulk–surface system with fractional surface diffusion

Duc Hau Nguyen ^,

Thuyloi University, Hanoi, Vietnam

Received: 02 August 2026 Revised: 25 August 2026 Accepted: 27 August 2026 Published: 04 September 2026
MSC : 35B32, 35K57, 35R11, 37G40, 65M60

We study a reaction–diffusion system in a three-dimensional ball and on its spherical boundary, coupled by species-dependent Robin exchange and fractional membrane transport. Both isolated compartments are linearly stable. A normalized spectral fractional Laplace–Beltrami operator fixes the damping of a reference spherical harmonic degree, allowing the fractional order to change spectral slope without merely rescaling diffusivity. Modewise boundary-response analysis reveals a coupling-induced instability band and a switch of the preferred degree from $ \ell = 7 $ to $ \ell = 6 $. At the codimension-two crossing, we derive the full cubic $ O(3) $-equivariant normal form on the 28-dimensional center space $ V_6\oplus V_7 $, containing three independent quadratic and twenty independent cubic Gaunt channels. Independent spectral checks and exact-sphere simulations verify the reduction and show reproducible degree-six selection approaching saturation together with angularly resolved finite-time degree-seven selection at $ t = 400 $ from random multimode data. At $ t = 800 $, degree seven remains dominant, but its modal energy fraction still changes with cutoff refinement, so the calculations do not identify a stationary degree-seven state. Coefficient convergence, equivariance residuals, trajectory comparisons, perturbation tests, and mesh/cutoff studies are used to separate robust conclusions from numerical and geometric limitations. The results connect compartment exchange, anomalous surface transport, symmetry-constrained mode interaction, and finite-amplitude pattern selection.
- bulk–surface system,
- Turing instability,
- fractional diffusion,
- $ O(3) $ symmetry,
- mode interaction,
- spherical harmonics
Citation: Duc Hau Nguyen. Coupling-induced Turing instability and $ O(3) $-equivariant mode interaction in a bulk–surface system with fractional surface diffusion[J]. AIMS Mathematics, 2026, 11(9): 28383-28408. doi: 10.3934/math.20261130

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Abstract

We study a reaction–diffusion system in a three-dimensional ball and on its spherical boundary, coupled by species-dependent Robin exchange and fractional membrane transport. Both isolated compartments are linearly stable. A normalized spectral fractional Laplace–Beltrami operator fixes the damping of a reference spherical harmonic degree, allowing the fractional order to change spectral slope without merely rescaling diffusivity. Modewise boundary-response analysis reveals a coupling-induced instability band and a switch of the preferred degree from $ \ell = 7 $ to $ \ell = 6 $. At the codimension-two crossing, we derive the full cubic $ O(3) $-equivariant normal form on the 28-dimensional center space $ V_6\oplus V_7 $, containing three independent quadratic and twenty independent cubic Gaunt channels. Independent spectral checks and exact-sphere simulations verify the reduction and show reproducible degree-six selection approaching saturation together with angularly resolved finite-time degree-seven selection at $ t = 400 $ from random multimode data. At $ t = 800 $, degree seven remains dominant, but its modal energy fraction still changes with cutoff refinement, so the calculations do not identify a stationary degree-seven state. Coefficient convergence, equivariance residuals, trajectory comparisons, perturbation tests, and mesh/cutoff studies are used to separate robust conclusions from numerical and geometric limitations. The results connect compartment exchange, anomalous surface transport, symmetry-constrained mode interaction, and finite-amplitude pattern selection.

References

[1]	A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. B, 237 (1952), 37–72. http://doi.org/10.1098/rstb.1952.0012 doi: 10.1098/rstb.1952.0012
[2]	J. D. Murray, Mathematical biology Ⅱ: spatial models and biomedical Applications, 3 Eds., New York: Springer, 2003. http://doi.org/10.1007/b98869
[3]	M. A. J. Chaplain, M. Ganesh, I. G. Graham, Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth, J. Math. Biol., 42 (2001), 387–423. http://doi.org/10.1007/s002850000067 doi: 10.1007/s002850000067
[4]	T. K. Callahan, Turing patterns with $O(3)$ symmetry, Physica D, 188 (2004), 65–91. http://doi.org/10.1016/S0167-2789(03)00286-0 doi: 10.1016/S0167-2789(03)00286-0
[5]	J. G. Borgqvist, P. Gerlee, C. Lundholm, Turing pattern formation on the sphere is robust to the removal of a hole, J. Math. Biol., 88 (2024), 23. http://doi.org/10.1007/s00285-023-02034-z doi: 10.1007/s00285-023-02034-z
[6]	P. Chossat, R. Lauterbach, I. Melbourne, Steady-state bifurcation with $O(3)$-symmetry, Arch. Rational Mech. Anal., 113 (1991), 313–376. http://doi.org/10.1007/BF00374697 doi: 10.1007/BF00374697
[7]	G. Iooss, M. Rossi, Hopf bifurcation in the presence of spherical symmetry: analytical results, SIAM J. Math. Anal., 20 (1989), 511–532. http://doi.org/10.1137/0520036 doi: 10.1137/0520036
[8]	F. Antoneli, A. P. S. Dias, P. C. Matthews, Invariants, equivariants and characters in symmetric bifurcation theory, P. Roy. Soc. Edinb. A, 138 (2008), 477–512. http://doi.org/10.1017/S0308210506001119 doi: 10.1017/S0308210506001119
[9]	C. Chen, H. B. Wang, W. H. Jiang, Pattern formation from equivariant Turing bifurcations in reaction–diffusion systems on a square domain, Z. Angew. Math. Phys., 76 (2025), 212. http://doi.org/10.1007/s00033-025-02593-9 doi: 10.1007/s00033-025-02593-9
[10]	A. Madzvamuse, A. H. W. Chung, C. Venkataraman, Stability analysis and simulations of coupled bulk–surface reaction–diffusion systems, Proc. R. Soc. A, 471 (2015), 20140546. http://doi.org/10.1098/rspa.2014.0546 doi: 10.1098/rspa.2014.0546
[11]	A. Rätz, Turing-type instabilities in bulk–surface reaction–diffusion systems, J. Comput. Appl. Math., 289 (2015), 142–152. http://doi.org/10.1016/j.cam.2015.02.050 doi: 10.1016/j.cam.2015.02.050
[12]	F. Paquin-Lefebvre, W. Nagata, M. J. Ward, Pattern formation and oscillatory dynamics in a two-dimensional coupled bulk–surface reaction–diffusion system, SIAM J. Appl. Dyn. Syst., 18 (2019), 1334–1390. http://doi.org/10.1137/18M1213737 doi: 10.1137/18M1213737
[13]	E. Villar-Sepúlveda, A. R. Champneys, D. Cusseddu, A. Madzvamuse, Pattern formation of bulk–surface reaction–diffusion systems in a ball, SIAM J. Appl. Math., 86 (2026), 21–51. http://doi.org/10.1137/24M1671037 doi: 10.1137/24M1671037
[14]	M. Frittelli, I. Sgura, B. Bozzini, Turing patterns in a 3D morpho-chemical bulk–surface reaction–diffusion system for battery modeling, Math. Eng., 6 (2024), 363–393. http://doi.org/10.3934/mine.2024015 doi: 10.3934/mine.2024015
[15]	M. Frittelli, A. Madzvamuse, I. Sgura, VEMcomp: A virtual elements MATLAB package for bulk–surface PDEs in 2D and 3D, Numer. Algor., 99 (2025), 1393–1428. http://doi.org/10.1007/s11075-024-01919-4 doi: 10.1007/s11075-024-01919-4
[16]	Z. C. Tang, Z. J. Fu, M. Chen, L. Ling, A novel localized least-squares collocation method for coupled bulk–surface problems, Appl. Math. Comput., 492 (2025), 129250. http://doi.org/10.1016/j.amc.2024.129250 doi: 10.1016/j.amc.2024.129250
[17]	J. Morgan, B. Q. Tang, Global well-posedness for volume–surface reaction–diffusion systems, Commun. Contemp. Math., 25 (2023), 2250002. http://doi.org/10.1142/S021919972250002X doi: 10.1142/S021919972250002X
[18]	D. Krapf, Mechanisms underlying anomalous diffusion in the plasma membrane, Curr. Top. Membr., 75 (2015), 167–207. http://doi.org/10.1016/bs.ctm.2015.03.002 doi: 10.1016/bs.ctm.2015.03.002
[19]	B. I. Henry, S. L. Wearne, Existence of Turing instabilities in a two-species fractional reaction-diffusion system, SIAM J. Appl. Math., 62 (2002), 870–887. http://doi.org/10.1137/S0036139900375227 doi: 10.1137/S0036139900375227
[20]	Y. Nec, A. A. Nepomnyashchy, Turing instability of anomalous reaction–anomalous diffusion systems, Eur. J. Appl. Math., 19 (2008), 329–349. http://doi.org/10.1017/S0956792508007389 doi: 10.1017/S0956792508007389
[21]	A. Bonito, W. Y. Lei, Approximation of the spectral fractional powers of the Laplace–Beltrami operator, Numer. Math. Theory Me., 15 (2022), 1193–1218. http://doi.org/10.4208/nmtma.OA-2022-0005s doi: 10.4208/nmtma.OA-2022-0005s
[22]	J. Chen, Y. Ye, Y. Zhao, Fractional diffusion induced pattern formation in an epidemic model with nonlinear incidence rate, Nonlinear Dyn., 113 (2025), 31815–31840. http://doi.org/10.1007/s11071-025-11675-2 doi: 10.1007/s11071-025-11675-2
[23]	D. Armbruster, J. Guckenheimer, P. Holmes, Kuramoto–Sivashinsky dynamics on the center–unstable manifold, SIAM J. Appl. Math., 49 (1989), 676–691. http://doi.org/10.1137/0149039 doi: 10.1137/0149039
[24]	G. Dziuk, C. M. Elliott, Finite element methods for surface PDEs, Acta Numer., 22 (2013), 289–396. http://doi.org/10.1017/S0962492913000056 doi: 10.1017/S0962492913000056

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