Top

Published in:

2019 | OriginalPaper | Chapter

“Strong” Turing-Hopf Instability for Reaction-Diffusion Systems

Authors : Giani Egaña Fernández, J Sarría González, Mariano Rodríguez Ricard

Published in: Analysis and Partial Differential Equations: Perspectives from Developing Countries

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

Turing-Hopf instabilities for reaction-diffusion systems provide spatially inhomogeneous time-periodic patterns of chemical concentrations. In this presentation, it is shown the parameter space in which the reaction-diffusion system modelling glycolysis and the Lengyel-Epstein model could show twinkling patterns. To do so, we follow the Ricard-Mischler procedure in Ricard and Mischler (J Nonlinear Sci 19(5):467–496, 2009, [18]), i.e., considering this phenomenom as a consequence of the instability generated by diffusion on the limit cycle which appears due to a Hopf bifurcation about the spatially homogeneous steady state.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Convergence of Fourier-Walsh Double Series in Weighted

next chapter Correspondence Between Multiscale Frame Shrinkage and High-Order Nonlinear Diffusion

Baurmann, M., Gross, T., Feudel, U.: Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations. J. Theor. Biol. 245(2), 220–229 (2007)MathSciNetCrossRef

Bogoliubov, N.N., Mitropolski, Y.A.: Asymptotic Methods in the Theory of Nonlinear Oscillations. Gordon and Breach Science Publishers, New York (1961)

Díaz Rodrígues, L.A., Mistro, D.C., Petrovskii, S.: Pattern formation, long-term transients, and the Turing-Hopf Bifurcation in a space- and time-discrete predator-prey system. Bull. Math. Biol. (2010). https://doi.org/10.1007/s11538-010-9593-5MathSciNetCrossRef

Egaña Fernández, G., Rodríguez Ricard, M.: Emergence and collapse of limit cycles in the glycolysis model. Revista Investigación Operacional 39(1), 23–32 (2018)MathSciNet

Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, New York (1981)CrossRef

Just, W., Bose, M., Bose, S., Engel, H., Schöll, E.: Spatiotemporal dynamics near a supercritical Turing-Hopf bifurcation in a two-dimensional reaction-diffusion system. Phys. Rev. E. 64(026219), 1–12 (2001). https://doi.org/10.1103/PhysRevE.64.026219CrossRef

Kuznetsov, YuA: Elements of Applied Bifurcation Theory. Applied Mathematical Sciences, vol. 112, 3rd edn. Springer Science+Business Media, New York (2004)CrossRef

Lacitignola, D., Bozzini, B., Sgura,I.: Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay. Eur. J. Appl. Math. (2014). https://doi.org/10.1017/S0956792514000370MathSciNetCrossRef

Leiva, H.: Stability of a periodic solution for a system of parabolic equations. Appl. Anal. 60, 277–300 (1996)MathSciNetCrossRef

10.

Lengyel, I., Epstein, I.R.: Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model. Sciences 251, 650–652 (1991)CrossRef

11.

Lengyel, I., Epstein, I.R.: A chemical approach to designing Turing patterns in reaction-diffusion system. Proc. Natl. Acad. Sci. USA 89, 3977–3979 (1992)CrossRef

12.

Liu, P., Shi, J., Wang, Y., Feng, X.: Bifurcation analysis of reaction-diffusion Schnakenberg model. J. Math. Chem. 51, 2001–2019 (2013). https://doi.org/10.1007/s10910-013-0196-xMathSciNetCrossRef

13.

Marsden, J.E., McCracken, M.: The Hopf Bifurcation and its Applications. Springer, New York (1976)CrossRef

14.

Meixner, M., De Wit, A., Bose, S., Schöll, E.: Generic spatiotemporal dynamics near codimension-two Turing-Hopf bifurcations. Phys. Rev. E 55(6), 6690–6697 (1997)MathSciNetCrossRef

15.

Murray, J.D.: Mathematical Biology I: An Introduction. Interdisciplinary Applied Mathematics, vol. 17, 3rd edn. Springer, New York (2001)

16.

Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications. Interdisciplinary Applied Mathematics, vol. 18, 3rd edn. Springer, New York (2003)MATH

17.

Ricard, M.R.: On degenerate planar Hopf bifurcations. J. Phys. A. Math. Theor. 44, 065202 (2011)MathSciNetCrossRef

18.

Ricard, M.R., Mischler, S.: Turing instabilities at Hopf bifurcation. J. Nonlinear Sci. 19(5), 467–496 (2009). https://doi.org/10.1007/s00332-009-9041-6MathSciNetCrossRef

19.

Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976)MATH

20.

Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems. Applied Mathematical Sciences, vol. 59, 2nd edn. Springer, New York (2007)MATH

21.

Sarría-González, J., Ricard, M.R.: Twinkling patterns for the Lengyel-Epstein reaction-diffusion model (2018) (in preparation)

22.

Settani, G., Sgura, I.: Devising efficient numerical methods for oscillating patterns in reaction-diffusion systems. J. Comput. Appl. Math. (2015). https://doi.org/10.1016/j.cam.2015.04.044MathSciNetCrossRef

23.

Sgura, I., Bozzini, B., Lacitignola, D.: Numerical approximation of Turing patterns in electrodeposition by ADI methods. J. Comput. Appl. Math 236, 4132–4147 (2012)MathSciNetCrossRef

24.

Sgura, I., Bozzini, B., Lacitignola, D.: Numerical approximation of oscillating Turing patterns in a reaction-diffusion model for electrochemical material growth. In: AIP Conference Proceedings, vol. 1493, pp. 896–903. Melville, New York (2012). https://doi.org/10.1063/1.4765594

25.

Strier, D.E., Ponce, D.S.: Turing patterns inside cells. PLoS ONE 2, (10), e1053 (2007). https://doi.org/10.1371/journal.pone.0001053CrossRef

26.

Turing, A.M.: The chemical basis for morphogenesis. Philos. Trans. R. Soc. Lond. B 237, 37–72 (1952)

27.

Wang, L., Zhao, H.: Hopf bifurcation and Turing instability of 2-D Lengyel-Epstein system with reaction-diffusion terms. Appl. Math. Comput. 21, 9229–9244 (2013)MathSciNetMATH

28.

Wu, X.P., Eshete, M.: Bautin bifurcation for the Lengyel-Epstein system. J. Math. Chem. 52, 2570–2580 (2014). https://doi.org/10.1007/s10910-014-0401-6MathSciNetCrossRefMATH

29.

Yang, L., Berenstein, I., Epstein, I.R.: Segmented Waves from a Spatiotemporal Transverse Wave Instability. Phys. Rev. Lett. 95, 3, 038303 (2005). https://doi.org/10.1103/PhysRevLett.95.038303

30.

Yi, F., Wei, J., Shi, J.: Diffusion-driven instability and bifurcation in the Lengyel-Epstein system. Nonlinear Anal. Real World Appl. 9, 1038–1051 (2008)MathSciNetCrossRef

31.

Zhou, J., Shi, J.: Pattern formation in a general glycolysis reaction-diffusion system. IMA J. Appl. Math. 13, 1–36 (2015). https://doi.org/10.1093/imamat/hxv013MathSciNetCrossRefMATH

Title: “Strong” Turing-Hopf Instability for Reaction-Diffusion Systems
Authors: Giani Egaña Fernández
J Sarría González
Mariano Rodríguez Ricard
Publisher: Springer International Publishing
Book: Analysis and Partial Differential Equations: Perspectives from Developing Countries
Print ISBN: 978-3-030-05656-8

Electronic ISBN: 978-3-030-05657-5

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-3-030-05657-5_9

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner