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doi: 10.3934/dcdss.2021119
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The amplitude system for a Simultaneous short-wave Turing and long-wave Hopf instability

1. 

IADM, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

2. 

Department of Mathematics, Brunel University London, Uxbridge UB8 3PH, United Kingdom

*Corresponding author: Guido Schneider

Received  March 2021 Revised  July 2021 Early access November 2021

We consider reaction-diffusion systems for which the trivial solution simultaneously becomes unstable via a short-wave Turing and a long-wave Hopf instability. The Brusseletor, Gierer-Meinhardt system and Schnakenberg model are prototype biological pattern forming systems which show this kind of behavior for certain parameter regimes. In this paper we prove the validity of the amplitude system associated to this kind of instability. Our analytical approach is based on the use of mode filters and normal form transformations. The amplitude system allows us an efficient numerical simulation of the original multiple scaling problems close to the instability.

Citation: Guido Schneider, Matthias Winter. The amplitude system for a Simultaneous short-wave Turing and long-wave Hopf instability. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021119
References:
[1]

D. ArmbrusterG. Dangelmayr and W. Güttinger, Imperfection sensitivity of interacting Hopf and steady-state bifurcations and their classification, Physica D, 16 (1985), 99-123.  doi: 10.1016/0167-2789(85)90087-9.  Google Scholar

[2]

S. BaumstarkG. SchneiderK. Schratz and D. Zimmermann, Effective slow dynamics models for a class of dispersive systems, J. Dynam. Differential Equations, 32 (2020), 1867-1899.  doi: 10.1007/s10884-019-09791-w.  Google Scholar

[3]

P. Collet and J.-P. Eckmann, The time dependent amplitude equation for the Swift-Hohenberg problem, Comm. Math. Phys., 132 (1990), 139-153.  doi: 10.1007/BF02278004.  Google Scholar

[4]

A. De WitD. LimaG. Dewel and P. Borckmans, Spatiotemporal dynamics near a codimension-two point, Phys. Rev. E, 54 (1996), 261-271.  doi: 10.1103/PhysRevE.54.261.  Google Scholar

[5]

W. Eckhaus, The Ginzburg-Landau manifold is an attractor, J. Nonlinear Sci., 3 (1993), 329-348.  doi: 10.1007/BF02429869.  Google Scholar

[6]

E. Faou and K. Schratz, Asymptotic preserving schemes for the Klein-Gordon equation in the non-relativistic limit regime, Numer. Math., 126 (2014), 441-469.  doi: 10.1007/s00211-013-0567-z.  Google Scholar

[7]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.   Google Scholar

[8]

T. Haas and G. Schneider, Failure of the $n$-wave interaction approximation without imposing periodic boundary conditions, ZAMM Z. Angew. Math. Mech., 100 (2020), 16pp. doi: 10.1002/zamm.201900230.  Google Scholar

[9]

H. Haken and H. Olbrich, Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis, J. Math. Biol., 6 (1978), 317-331.  doi: 10.1007/BF02462997.  Google Scholar

[10]

W. Just, M. Bose, S. Bose, H. Engel and E. Schöll, Spatiotemporal dynamics near a supercritical turing-hopf bifurcation in a two-dimensional reaction-diffusion system, Phys. Rev. E, 64 (2001). doi: 10.1103/PhysRevE.64.026219.  Google Scholar

[11]

Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer-Verlag New York, NY, 1984. doi: 10.1007/978-3-642-69689-3.  Google Scholar

[12]

W. F. Langford and G. Iooss, Interactions of Hopf and pitchfork bifurcations, In Bifurcation Problems and Their Numerical Solution (Proc. Workshop, Univ. Dortmund, Dortmund, 1980), Internat. Ser. Numer. Math., Birkhäuser, Basel-Boston, Mass., 54 (1980), 103–134.  Google Scholar

[13]

M. MeixnerA. De WitS. Bose and E. Schöll, Generic spatiotemporal dynamics near codimension-two Turing-Hopf bifurcations, Phys. Rev. E, 55 (1997), 6690-6697.  doi: 10.1103/PhysRevE.55.6690.  Google Scholar

[14]

A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains – existence and comparison, Nonlinearity, 8 (1995), 743-768.  doi: 10.1088/0951-7715/8/5/006.  Google Scholar

[15]

F. Paquin-LefebvreW. Nagata and M. J. Ward, Pattern formation and oscillatory dynamics in a two-dimensional coupled bulk-surface reaction-diffusion system, SIAM J. Appl. Math., 18 (2019), 1334-1390.  doi: 10.1137/18M1213737.  Google Scholar

[16]

J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol., 81 (1979), 389–400, URLhttps://www.sciencedirect.com/science/article/pii/0022519379900420. doi: 10.1016/0022-5193(79)90042-0.  Google Scholar

[17]

G. Schneider, A new estimate for the Ginzburg-Landau approximation on the real axis, J. Nonlinear Sci., 4 (1994), 23-34.  doi: 10.1007/BF02430625.  Google Scholar

[18]

G. Schneider, Error estimates for the Ginzburg-Landau approximation, Z. Angew. Math. Phys., 45 (1994), 433-457.  doi: 10.1007/BF00945930.  Google Scholar

[19]

G. Schneider, Validity and limitation of the Newell-Whitehead equation, Math. Nachr., 176 (1995), 249-263.  doi: 10.1002/mana.19951760118.  Google Scholar

[20]

G. Schneider, Hopf bifurcation in spatially extended reaction-diffusion systems, J. Nonlinear Sci., 8 (1998), 17-41.  doi: 10.1007/s003329900042.  Google Scholar

[21]

G. SchneiderD. A. Sunny and D. Zimmermann, The NLS approximation makes wrong predictions for the water wave problem in case of small surface tension and spatially periodic boundary conditions, J. Dynam. Differential Equations, 27 (2015), 1077-1099.  doi: 10.1007/s10884-014-9350-9.  Google Scholar

[22]

G. Schneider and H. Uecker, Nonlinear PDEs. A Dynamical Systems Approach., Graduate Studies in Mathematics, vol. 182, American Mathematical Society, Providence, RI, 2017. doi: 10.1090/gsm/182.  Google Scholar

[23]

A. van Harten, On the validity of the Ginzburg-Landau equation, J. Nonlinear Sci., 1 (1991), 397-422.  doi: 10.1007/BF02429847.  Google Scholar

show all references

References:
[1]

D. ArmbrusterG. Dangelmayr and W. Güttinger, Imperfection sensitivity of interacting Hopf and steady-state bifurcations and their classification, Physica D, 16 (1985), 99-123.  doi: 10.1016/0167-2789(85)90087-9.  Google Scholar

[2]

S. BaumstarkG. SchneiderK. Schratz and D. Zimmermann, Effective slow dynamics models for a class of dispersive systems, J. Dynam. Differential Equations, 32 (2020), 1867-1899.  doi: 10.1007/s10884-019-09791-w.  Google Scholar

[3]

P. Collet and J.-P. Eckmann, The time dependent amplitude equation for the Swift-Hohenberg problem, Comm. Math. Phys., 132 (1990), 139-153.  doi: 10.1007/BF02278004.  Google Scholar

[4]

A. De WitD. LimaG. Dewel and P. Borckmans, Spatiotemporal dynamics near a codimension-two point, Phys. Rev. E, 54 (1996), 261-271.  doi: 10.1103/PhysRevE.54.261.  Google Scholar

[5]

W. Eckhaus, The Ginzburg-Landau manifold is an attractor, J. Nonlinear Sci., 3 (1993), 329-348.  doi: 10.1007/BF02429869.  Google Scholar

[6]

E. Faou and K. Schratz, Asymptotic preserving schemes for the Klein-Gordon equation in the non-relativistic limit regime, Numer. Math., 126 (2014), 441-469.  doi: 10.1007/s00211-013-0567-z.  Google Scholar

[7]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.   Google Scholar

[8]

T. Haas and G. Schneider, Failure of the $n$-wave interaction approximation without imposing periodic boundary conditions, ZAMM Z. Angew. Math. Mech., 100 (2020), 16pp. doi: 10.1002/zamm.201900230.  Google Scholar

[9]

H. Haken and H. Olbrich, Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis, J. Math. Biol., 6 (1978), 317-331.  doi: 10.1007/BF02462997.  Google Scholar

[10]

W. Just, M. Bose, S. Bose, H. Engel and E. Schöll, Spatiotemporal dynamics near a supercritical turing-hopf bifurcation in a two-dimensional reaction-diffusion system, Phys. Rev. E, 64 (2001). doi: 10.1103/PhysRevE.64.026219.  Google Scholar

[11]

Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer-Verlag New York, NY, 1984. doi: 10.1007/978-3-642-69689-3.  Google Scholar

[12]

W. F. Langford and G. Iooss, Interactions of Hopf and pitchfork bifurcations, In Bifurcation Problems and Their Numerical Solution (Proc. Workshop, Univ. Dortmund, Dortmund, 1980), Internat. Ser. Numer. Math., Birkhäuser, Basel-Boston, Mass., 54 (1980), 103–134.  Google Scholar

[13]

M. MeixnerA. De WitS. Bose and E. Schöll, Generic spatiotemporal dynamics near codimension-two Turing-Hopf bifurcations, Phys. Rev. E, 55 (1997), 6690-6697.  doi: 10.1103/PhysRevE.55.6690.  Google Scholar

[14]

A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains – existence and comparison, Nonlinearity, 8 (1995), 743-768.  doi: 10.1088/0951-7715/8/5/006.  Google Scholar

[15]

F. Paquin-LefebvreW. Nagata and M. J. Ward, Pattern formation and oscillatory dynamics in a two-dimensional coupled bulk-surface reaction-diffusion system, SIAM J. Appl. Math., 18 (2019), 1334-1390.  doi: 10.1137/18M1213737.  Google Scholar

[16]

J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol., 81 (1979), 389–400, URLhttps://www.sciencedirect.com/science/article/pii/0022519379900420. doi: 10.1016/0022-5193(79)90042-0.  Google Scholar

[17]

G. Schneider, A new estimate for the Ginzburg-Landau approximation on the real axis, J. Nonlinear Sci., 4 (1994), 23-34.  doi: 10.1007/BF02430625.  Google Scholar

[18]

G. Schneider, Error estimates for the Ginzburg-Landau approximation, Z. Angew. Math. Phys., 45 (1994), 433-457.  doi: 10.1007/BF00945930.  Google Scholar

[19]

G. Schneider, Validity and limitation of the Newell-Whitehead equation, Math. Nachr., 176 (1995), 249-263.  doi: 10.1002/mana.19951760118.  Google Scholar

[20]

G. Schneider, Hopf bifurcation in spatially extended reaction-diffusion systems, J. Nonlinear Sci., 8 (1998), 17-41.  doi: 10.1007/s003329900042.  Google Scholar

[21]

G. SchneiderD. A. Sunny and D. Zimmermann, The NLS approximation makes wrong predictions for the water wave problem in case of small surface tension and spatially periodic boundary conditions, J. Dynam. Differential Equations, 27 (2015), 1077-1099.  doi: 10.1007/s10884-014-9350-9.  Google Scholar

[22]

G. Schneider and H. Uecker, Nonlinear PDEs. A Dynamical Systems Approach., Graduate Studies in Mathematics, vol. 182, American Mathematical Society, Providence, RI, 2017. doi: 10.1090/gsm/182.  Google Scholar

[23]

A. van Harten, On the validity of the Ginzburg-Landau equation, J. Nonlinear Sci., 1 (1991), 397-422.  doi: 10.1007/BF02429847.  Google Scholar

Figure 1.  The spectrum of the linearization around the trivial solution becoming simultaneously unstable via a short-wave Turing instability (thick lines) and a long-wave Hopf instability.
Figure 1.  The spectrum of the linearization around the trivial solution becoming simultaneously unstable via a short-wave Turing instability (red) and a long-wave Hopf instability (blue).
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