doi: 10.3934/dcdsb.2022063
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Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model

1. 

Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy

2. 

School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, UK

* Corresponding author: Maria Carmela Lombardo

Received  July 2021 Revised  January 2022 Early access March 2022

We investigate the formation of stationary patterns in the FitzHugh-Nagumo reaction-diffusion system with linear cross-diffusion terms. We focus our analysis on the effects of cross-diffusion on the Turing mechanism. Linear stability analysis indicates that positive values of the inhibitor cross-diffusion enlarge the region in the parameter space where a Turing instability is excited. A sufficiently large cross-diffusion coefficient of the inhibitor removes the requirement imposed by the classical Turing mechanism that the inhibitor must diffuse faster than the activator. In an extended region of the parameter space a new phenomenon occurs, namely the existence of a double bifurcation threshold of the inhibitor/activator diffusivity ratio for the onset of patterning instabilities: for large values of inhibitor/activator diffusivity ratio, classical Turing patterns emerge where the two species are in-phase, while, for small values of the diffusion ratio, the analysis predicts the formation of out-of-phase spatial structures (named cross-Turing patterns). In addition, for increasingly large values of the inhibitor cross-diffusion, the upper and lower bifurcation thresholds merge, so that the instability develops independently on the value of the diffusion ratio, whose magnitude selects Turing or cross-Turing patterns. Finally, the pattern selection problem is addressed through a weakly nonlinear analysis.

Citation: Gaetana Gambino, Valeria Giunta, Maria Carmela Lombardo, Gianfranco Rubino. Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022063
References:
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[2]

M. BachirG. Sonnino and M. Tlidi, Predicted formation of localized superlattices in spatially distributed reaction-diffusion solutions, Physical Review E, 86 (2012), 045103. 

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N. Boudiba and M. Pierre, Global existence for coupled reaction-diffusion systems, J. Math. Anal. Appl., 250 (2000), 1-12.  doi: 10.1006/jmaa.2000.6895.

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G. ConsoloC. Currò and G. Valenti, Pattern formation and modulation in a hyperbolic vegetation model for semiarid environments, Appl. Math. Model., 43 (2017), 372-392.  doi: 10.1016/j.apm.2016.11.031.

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G. ConsoloC. Currò and G. Valenti, Supercritical and subcritical Turing pattern formation in a hyperbolic vegetation model for flat arid environments, Physica D: Nonlinear Phenomena, 398 (2019), 141-163.  doi: 10.1016/j.physd.2019.03.006.

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G. DewelS. MétensM. HilaliP. Borckmans and C. B. Price, Resonant patterns through coupling with a zero mode, Phys. Rev. Lett., 74 (1995), 4647-4650.  doi: 10.1103/PhysRevLett.74.4647.

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E. DulosJ. BoissonadeJ. PerraudB. Rudovics and P. Kepper, Chemical morphogenesis: Turing patterns in an experimental chemical system, Acta Biotheoretica, 44 (1996), 249-261.  doi: 10.1007/BF00046531.

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V. Giunta and M.C. Lombardo and M. Sammartino, Pattern formation and transition to chaos in a chemotaxis model of acute inflammation, SIAM Journal on Applied Dynamical Systems, 20 (2021), 1844-1881. doi: 10.1137/20M1358104.

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R. Han and B. Dai, Spatiotemporal pattern formation and selection induced by nonlinear cross-diffusion in a toxic-phytoplankton-zooplankton model with Allee effect, Nonlinear Anal. Real World Appl., 45 (2019), 822-853.  doi: 10.1016/j.nonrwa.2018.05.018.

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D. KarigK. Michael MartiniT. LuN. DeLateurN. Goldenfeld and R. Weiss, Stochastic Turing patterns in a synthetic bacterial population, Proceedings of the National Academy of Sciences of the United States of America, 115 (2018), 6572-6577.  doi: 10.1073/pnas.1720770115.

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B. LiuR. Wu and L. Chen, Patterns induced by super cross-diffusion in a predator-prey system with Michaelis-Menten type harvesting, Math. Biosci., 298 (2018), 71-79.  doi: 10.1016/j.mbs.2018.02.002.

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A. MadzvamuseH. Ndakwo and R. Barreira, Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion, Discrete Contin. Dyn. Syst., 36 (2016), 2133-2170.  doi: 10.3934/dcds.2016.36.2133.

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show all references

References:
[1]

M. BachirS. MétensP. Borckmans and G. Dewel, Formation of rhombic and superlattice patterns in bistable systems, Europhysics Letters, 54 (2001), 612-618.  doi: 10.1209/epl/i2001-00336-3.

[2]

M. BachirG. Sonnino and M. Tlidi, Predicted formation of localized superlattices in spatially distributed reaction-diffusion solutions, Physical Review E, 86 (2012), 045103. 

[3]

N. Boudiba and M. Pierre, Global existence for coupled reaction-diffusion systems, J. Math. Anal. Appl., 250 (2000), 1-12.  doi: 10.1006/jmaa.2000.6895.

[4]

G. ConsoloC. Currò and G. Valenti, Pattern formation and modulation in a hyperbolic vegetation model for semiarid environments, Appl. Math. Model., 43 (2017), 372-392.  doi: 10.1016/j.apm.2016.11.031.

[5]

G. ConsoloC. Currò and G. Valenti, Supercritical and subcritical Turing pattern formation in a hyperbolic vegetation model for flat arid environments, Physica D: Nonlinear Phenomena, 398 (2019), 141-163.  doi: 10.1016/j.physd.2019.03.006.

[6]

A. De Wit, Spatial Patterns and Spatiotemporal Dynamics in Chemical Systems, John Wiley & Sons, Ltd, 2007. doi: 10.1002/9780470141687.ch5.

[7]

G. DewelM. BachirP. Borckmans and S. Métens, Superlattice structures and quasipatterns in bistable systems, Comptes Rendus de l'Academie de Sciences - Serie IIb: Mecanique, 329 (2001), 411-416. 

[8]

G. DewelS. MétensM. HilaliP. Borckmans and C. B. Price, Resonant patterns through coupling with a zero mode, Phys. Rev. Lett., 74 (1995), 4647-4650.  doi: 10.1103/PhysRevLett.74.4647.

[9]

X. DiegoL. MarconP. Müller and J. Sharpe, Key features of Turing systems are determined purely by network topology, Phys. Rev. X, 8 (2018), 021071.  doi: 10.1103/PhysRevX.8.021071.

[10]

E. DulosJ. BoissonadeJ. PerraudB. Rudovics and P. Kepper, Chemical morphogenesis: Turing patterns in an experimental chemical system, Acta Biotheoretica, 44 (1996), 249-261.  doi: 10.1007/BF00046531.

[11]

R. FitzHugh, Thresholds and plateaus in the Hodgkin-Huxley nerve equations, J. Gen. Physiol, 43 (1960), 867-896.  doi: 10.1085/jgp.43.5.867.

[12]

G. GambinoM. LombardoS. Lupo and M. Sammartino, Super-critical and sub-critical bifurcations in a reaction-diffusion Schnakenberg model with linear cross-diffusion, Ric. Mat., 65 (2016), 449-467.  doi: 10.1007/s11587-016-0267-y.

[13]

G. Gambino, M. Lombardo, S. Lupo and M. Sammartino, Turing–Hopf bifurcation in the Schnakenberg model with cross-diffusion, Submitted.

[14]

G. GambinoM. LombardoG. Rubino and M. Sammartino, Pattern selection in the 2D Fitzhugh-Nagumo model, Ric. Mat., 68 (2019), 535-549.  doi: 10.1007/s11587-018-0424-6.

[15]

G. GambinoM. Lombardo and M. Sammartino, Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross–diffusion, Math. Comput. Simulation, 82 (2012), 1112-1132.  doi: 10.1016/j.matcom.2011.11.004.

[16]

G. GambinoM. Lombardo and M. Sammartino, Pattern formation driven by cross-diffusion in a 2D domain, Nonlinear Anal. Real World Appl., 14 (2013), 1755-1779.  doi: 10.1016/j.nonrwa.2012.11.009.

[17]

G. GambinoM. Lombardo and M. Sammartino, Cross-diffusion-induced subharmonic spatial resonances in a predator-prey system, Phys. Rev. E, 97 (2018), 012220.  doi: 10.1103/PhysRevE.97.012220.

[18]

X. GaoL. DongH. WangH. ZhangY. LiuW. LiuW. Fan and Y. Pan, Three-dimensional patterns in dielectric barrier discharge with "h" shaped gas gap, Physics of Plasmas, 23 (2016), 083526.  doi: 10.1063/1.4960831.

[19]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.

[20]

V. Giunta and M.C. Lombardo and M. Sammartino, Pattern formation and transition to chaos in a chemotaxis model of acute inflammation, SIAM Journal on Applied Dynamical Systems, 20 (2021), 1844-1881. doi: 10.1137/20M1358104.

[21]

A. Hagberg and E. Meron, Pattern formation in non-gradient reaction-diffusion systems: The effects of front bifurcations, Nonlinearity, 7 (1994), 805-835.  doi: 10.1088/0951-7715/7/3/006.

[22]

R. Han and B. Dai, Spatiotemporal pattern formation and selection induced by nonlinear cross-diffusion in a toxic-phytoplankton-zooplankton model with Allee effect, Nonlinear Anal. Real World Appl., 45 (2019), 822-853.  doi: 10.1016/j.nonrwa.2018.05.018.

[23]

M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, Universitext, Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011. doi: 10.1007/978-0-85729-112-7.

[24]

B. Henry and S. Wearne, Existence of Turing instabilities in a two-species fractional reaction-diffusion system, SIAM J. Appl. Math., 62 (2001/02), 870-887.  doi: 10.1137/S0036139900375227.

[25]

J. HorváthI. Szalai and P. De Kepper, An experimental design method leading to chemical Turing patterns, Science, 324 (2009), 772-775. 

[26]

J. IrazoquiA. Gladfelter and D. Lew, Scaffold-mediated symmetry breaking by Cdc42p, Nature Cell Biology, 5 (2003), 1062-1070.  doi: 10.1038/ncb1068.

[27]

D. KarigK. Michael MartiniT. LuN. DeLateurN. Goldenfeld and R. Weiss, Stochastic Turing patterns in a synthetic bacterial population, Proceedings of the National Academy of Sciences of the United States of America, 115 (2018), 6572-6577.  doi: 10.1073/pnas.1720770115.

[28]

A. LandgeB. JordanX. Diego and P. Müller, Pattern formation mechanisms of self-organizing reaction-diffusion systems, Developmental Biology, 460 (2020), 2-11. 

[29]

B. LiuR. Wu and L. Chen, Patterns induced by super cross-diffusion in a predator-prey system with Michaelis-Menten type harvesting, Math. Biosci., 298 (2018), 71-79.  doi: 10.1016/j.mbs.2018.02.002.

[30]

M. LooseE. Fischer-FriedrichJ. RiesK. Kruse and P. Schwille, Spatial regulators for bacterial cell division self-organize into surface waves in vitro, Science, 320 (2008), 789-792.  doi: 10.1126/science.1154413.

[31]

A. Madzvamuse and R. Barreira, Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2775-2801.  doi: 10.3934/dcdsb.2018163.

[32]

A. MadzvamuseH. Ndakwo and R. Barreira, Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion, Discrete Contin. Dyn. Syst., 36 (2016), 2133-2170.  doi: 10.3934/dcds.2016.36.2133.

[33]

A. MadzvamuseH. Ndakwo and R. Barreira, Cross-diffusion-driven instability for reaction-diffusion systems: Analysis and simulations, J. Math. Biol., 70 (2015), 709-743.  doi: 10.1007/s00285-014-0779-6.

[34]

B. J. Matkowsky, Nonlinear dynamic stability: A formal theory, SIAM Journal on Applied Mathematics, 18 (1970), 872-883. 

[35]

G. Maugin, The Thermomechanics of Nonlinear Irreversible Behaviours, vol. 27, World Scientific Series in Nonlinear Science, Singapore, 1999.

[36]

A. MedvinskyS. PetrovskiiI. TikhonovaH. Malchow and B.-L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev., 44 (2002), 311-370.  doi: 10.1137/S0036144502404442.

[37]

H. Meinhardt, Turing's theory of morphogenesis of 1952 and the subsequent discovery of the crucial role of local self enhancement and long-range inhibition, Interface Focus, 2 (2012), 407-416.  doi: 10.1098/rsfs.2011.0097.

[38]

V. Mendez, W. Horsthemke, E. P. Zemskov and J. C. Vazquez, Segregation and pursuit waves in activator-inhibitor systems, Phys. Rev. E, 76 (2007), 046222, 6 pp. doi: 10.1103/PhysRevE.76.046222.

[39]

S. Métens, P. Borckmans and G. Dewel, Large amplitude patterns in bistable reaction-diffusion systems, In Instabilities and Nonequilibrium Structures VI. Nonlinear Phenomena and Complex Systems, (eds. E. Tirapegui, J. Martinez and R. Tiemann), Springer, Dordrecht, 5 (2000), 153–163. doi: 10.1007/978-94-011-4247-2_5.

[40]

S. MétensG. DewelP. Borckmans and R. Engelhardt, Pattern selection in bistable systems, Europhysics Letters, 37 (1997), 109-114. 

[41]

J. D. Murray, Mathematical Biology, 3$^{rd}$ edition, Springer, New York, 2002.

[42]

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.

[43]

A. Nepomnyashchy, Mathematical modelling of subdiffusion-reaction systems, Math. Model. Nat. Phenom., 11 (2016), 26-36.  doi: 10.1051/mmnp/201611102.

[44]

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Figure 1.  Nullclines of the local FHN system (1.1) in the monostable, excitable and bistable regime. (a) The monostable case, with $ \varepsilon = 1, a = -0.05 $, $ \beta = 0.1 $ and $ \gamma = 1.1 $. (b) The excitable case, with $ \varepsilon = 1, a = -0.3 $, $ \beta = 0.1 $ and $ \gamma = 1.02 $. (c) The bistable case, with $ \varepsilon = 1, a = 0 $, $ \beta = 0.5 $ and $ \gamma = 0.7 $. The labels $ m $ and $ M $ indicate the minimum and the maximum of the $ u $-nullcline (2.3). The matrices give the signs of the derivatives $ f_u, f_v, g_u, g_v $ evaluated at the equilibrium point
Figure 2.  Geometrical representation of the conditions for the diffusive instability. (a)-(b) For two different choices of $ d_v $, the dark gray regions in the $ (d_u, d) $-plane delimited by the two straight lines $ d = d_u\, d_v $ (dashed line) and $ d = \bar{d} $ (dotted line) correspond to the fulfillment of conditions (2.10)-(2.16a). The other parameters are chosen as $ \beta = 0.1, a = 0.0001, \gamma = 1.02 , \varepsilon = 2 $, so that $ E^* = (0.0051, 0.0051) $. (a) $ d_v = 0.1 $, which gives $ \delta_u^{(2)} = 2.2051 , \delta_u^{(1)} = 2.0045, I_c = 2.2030 $. (b) $ d_v = 1 $, which gives $ \delta_u^{(2)} = 4.0421 , \delta_u^{(1)} = 2.0201, I_c = 4.0382 $. (c)-(d) For two different values of $ d_v $, the gray shaded areas represent the diffusive instability regions in the $ (d_u, d) $-plane, corresponding to the fulfillment of both (2.10)-(2.16a) and (2.16b). The boundaries of the Turing region are $ d = d_c $, or $ P(d) = 0 $, (solid line) and $ d = d_ud_v $ (dashed line). (c) The parameters are chosen as in (a). (d) The parameters are chosen as in (b)
Figure 3.  The kinetics parameters are chosen as in Fig. 2(a). (a) The dark-gray and light-gray region indicate the presence of Turing and cross-Turing patterns, respectively, separated by the dash-dot curve expressed by (2.34). The dashed line marks the boundary above which $ d>d_ud_v $. (b) Turing pattern obtained by the numerical simulation of the system (1.1), with $ d_v = 0.1 $, $ d_u = 2.0194 $ and $ d = 0.32> d_c^{+} = 0.3191 $ (so that $ k_c^{+} = 0.75 $), corresponding to the point marked by an asterisk in (a). The profile of the activator (inhibitor) is represented by a solid (dotted) line. (c) Cross-Turing pattern obtained by the numerical simulation of the system (1.1) with $ d_v = 0.1 $, $ d_u = 2.0194 $ and $ d = 0.20423<d_c^{-} = 0.20432 $ (so that $ k_c^{-} = 2 $), corresponding to the point marked by a plus sign in (a). The profile of the activator $ u $ (inhibitor $ v $) is represented by a solid (dotted) line
Figure 4.  (a)-(b)-(c) Turing patterns observed along the upper branch of the bifurcation parabola. (d)-(e)-(f) Cross-Turing patterns observed along the lower branch of the bifurcation parabola. The parameters are chosen as in Fig. 2(a). The profile of the activator $ u $ (inhibitor $ v $) is represented by a solid (dotted) line. (a) $ d_u = 2.012 $ and $ d = 0.3398>d_c^+ = 0.3364 $, so that $ k_c^+ = 0.7 $. (b) $ d_u = 2.0198 $ and $ d = 0.3213>d_c^+ = 0.3181 $, so that $ k_c^+ = 0.7568 $. (c) $ d_u = 2.0385 $ and $ d = 0.2527>d_c^+ = 0.2553 $, so that $ k_c^+ = 0.9 $. (d) $ d_u = 2.0198 $ and $ d = 0.201689<d_c^- = 0.2017 $, so that $ k_c^- = 3 $. (e) $ d_u = 2.0194 $ and $ d = 0.2040<d_c^- = 0.2044 $, so that $ k_c^- = 1.9881 $. (f) $ d_u = 2.0385 $ and $ d = 0.2319<d_c^- = 0.2324 $, so that $ k_c^- = 1.07 $
Figure 5.  (a) Bifurcation diagram of the system (1.1) in the $ (d_u, d) $-plane as given by the WNL theory. The black, gray and dark gray regions correspond to cross-Turing subcritical bifurcation, cross-Turing supercritical bifurcation and Turing supercritical bifurcation, respectively. The dashed line marks the boundary above which $ d>d_ud_v $. The dash-dot curve, expressed by (2.34), that separates Turing from cross-Turing patterns is also reported. The parameters are chosen as in Fig. 2. (b) Turing instability for $ d\ge d_c^+ $: graph of the coefficient $ L $ of the amplitude equation (3.7) versus $ d_u $ for different choices of $ d_v $. The kinetic parameter are chosen as in (a). (c) Cross-Turing instability for $ d\le d_c^- $: graph of the coefficient $ L $ of the amplitude equation (3.7) versus $ d_u $ for different choices of $ d_v $. The kinetic parameter are chosen as in (a)
Figure 6.  Cross-Turing stationary pattern supported by the system (1.1) at a regular bifurcation. The parameters are chosen as $ \beta = 0.1 $, $ \gamma = 1.02 $, $ \varepsilon = 2 $, $ a = 0.0001 $, so that $ E^* = (0.0051, 0.0051) $ and $ d_u = 2.035 $, $ d_v = 0.1 $ and $ d = 0.2176<d_c^-\approx 0.2198 $. In the square domain $ [0, {4\sqrt{2}\pi}/{k_c}]\times[0, {4\sqrt{2}\pi}/{k_c}] $ the Turing bifurcation is regular: the critical wavenumber $ k_c\approx 1.2356 $ corresponds to the unique couple of modes $ (m, n) = (4, 4) $ satisfying the condition (3.11). (a)-(b) Initial condition assigned as a small random perturbation of the homogeneous equilibrium. (b)-(c) Numerical solution of the system (1.1) computed via spectral methods. (d)-(e) Spectrum of the solution
Figure 7.  Cross-Turing stationary subcritical pattern supported by the system (1.1) at a regular bifurcation. The parameters of the reaction term are chosen as in Fig. 6 and $ d_v = 0.1 $, $ d_u = 2.0151 $, $ d = 0.2024< d_c^-\approx0.2026 $. In the square domain $ [0, \pi]\times[0, \sqrt{2}\pi] $ the most unstable mode $ k_c\approx 2.4495 $ corresponds to the unique couple of integers $ (m, n) = (2, 2) $ satisfying the condition (3.11). The simulation reveals that several modes, other than the critical one, are excited. (a)-(c) Numerical solution of the system (1.1) computed using spectral methods and assigning as initial condition a small random perturbation of the homogeneous equilibrium. (b)-(d) Spectrum of the solution
Figure 8.  Cross-Turing stationary mixed-mode pattern supported by the system (1.1) when the monostable equilibrium loses stability via a degerate bifurcation and non-resonance conditions (3.13) hold. The parameters of the reaction term are chosen as in Fig. 6 and $ d_u = 2.035 $ and $ d_v = 0.1 $, $ d = 0.219<d_c^-\approx 0.2198 $. In the rectangular domain $ \left[0, {8\pi}/{k_c}\right]\times\left[0, {8\pi}/{k_c}\right] $ the most unstable mode $ k_c\approx 1.2356 $ corresponds to the two couples of integers $ (m_1, n_1) = (0, 8) $ and $ (m_2, n_2) = (8, 0) $ satisfying the condition (3.11). (a)-(c) Numerical solution of the system (1.1) computed using spectral methods and assigning as initial condition a small random perturbation of the homogeneous equilibrium. (b)-(d) Spectrum of the solution
Figure 9.  Resonant degenerate cross-Turing bifurcation: bifurcation diagrams of the amplitude system (3.18). The parameters of the reaction term in (1.1) are chosen as in Fig. 6 and $ d_u = 2.035 $ and $ d_v = 0.1 $, $ d_c^-\approx 0.2198 $. A solid red (dashed black) line represents stable (unstable) branches of stationary equlibria. The branches labeled by $ H^\pm_s $ represent stable hexagons of the FHN system (1.1) that bifurcate subcritically from the homogeneous equilibrium at the point $ B_p $. A pair of rolls of (1.1) bifurcates at $ B_p $ in a saddle-node bifurcation so leading, on the left of $ B_p $, to coexistence of stable rolls and hexagons. (a) Bifurcation diagram of $ A_1 $. (b) Bifurcation diagram of $ A_2 $
Figure 10.  Resonant degenerate cross-Turing bifurcation: phase space diagrams of the amplitude system (3.18). (a) The parameters are chosen as in Fig. 9 with $ d = d_c^- +0.0002 = 0.22 $. The coordinates of equilibrium points are $ H^\pm_s = (\pm 0.0801882, -0.061473) $ (stable), $ H^\pm_u = (\pm 0.00185716, -0.00094344) $ (unstable). (b) The parameters are chosen as in Fig. 9 with $ d = d_c^- -0.0008 = 0.219 $. The coordinates of equilibrium points are $ R^+_s = (0, 0.02246) $ (stable), $ R^-_u = (0, -0.02246) $ (unstable), $ H^\pm_s = (\pm 0.088722, -0.06956) $ (stable), $ H^\pm_u = (\pm 0.003852, 0.008295) $ (unstable)
Figure 11.  Resonant degenerate cross-Turing bifurcation: stationary hexagonal pattern supported by the system (1.1) when resonance conditions (3.17) hold. The parameters are chosen as in Fig. 9 with $ d = d_c^- -0.0008 = 0.219 $. In the rectangular domain $ [0, {8\pi}/{k_c}]\times[0, 8\sqrt{3}\pi/{k_c}] $ the most unstable mode $ k_c\approx 0.8625 $ corresponds to the two couples of integers $ (m_1, n_1) = (4, 12) $ and $ (m_2, n_2) = (8, 0) $ satisfying the condition (3.11). (a)-(c) Numerical solution of the system (1.1) computed using spectral methods. The initial condition is chosen of the form (3.19), where $ (A_{1\infty}, A_{2\infty}) $ is in the basin of attraction of the stable equilibrium point $ H^+_s $. (b)-(d) Spectrum of the solution
Figure 12.  Resonant degenerate cross-Turing bifurcation: stationary roll pattern supported by the system (1.1) when resonance conditions (3.17) hold. The parameters and the domain are chosen as in Fig. 11. The initial condition is chosen of the form (3.19), where $ (A_{1\infty}, A_{2\infty}) $ is in the basin of attraction of the stable equilibrium point $ R^+_s = (0, 0.02246) $ and $ (m, n) = (8, 0) $. (a)-(c) Numerical solution of the system (1.1) computed via spectral methods. (b)-(d) Spectrum of the solution
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