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Global Attractor of a Coupled Two-Cell Brusselator Model

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Infinite Dimensional Dynamical Systems

Part of the book series: Fields Institute Communications ((FIC,volume 64))

Abstract

In this work the existence of a global attractor for the solution semiflow of the coupled two-cell Brusselator model equations is proved. A grouping estimation method and a new decomposition approach are introduced to deal with the challenges in proving the absorbing property and the asymptotic compactness of this type of four-variable reaction-diffusion systems with cubic autocatalytic nonlinearity and with linear coupling. It is also proved that the Hausdorff dimension and the fractal dimension of the global attractor are finite.

Mathematics Subject Classification (2010): Primary 37L30, 35B40, 35K55, 35Q80; Secondary 80A32, 92B05

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References

  1. M. Ashkenazi, H.G. Othmer, Spatial patterns in coupled biochemical oscillators. J. Math. Biology 5, 305–350 (1978)

    MathSciNet  MATH  Google Scholar 

  2. J.F.G. Auchmuty, G. Nicolis, Bifurcation analysis of nonlinear reaction-diffusion equations I: Evolution equations and the steady state solutions. Bull. Math. Biol. 37, 323–365 (1975)

    MathSciNet  MATH  Google Scholar 

  3. A.Yu. Berezin, A. Gainoval, Yu.G. Matushkin, V.A. Likhoshval, S.I. Fadeev, Numerical study of mathematical models described dynamics of gene nets functioning: software package STEP, BGRS 2000, 243–245

    Google Scholar 

  4. K.J. Brown, F.A. Davidson, Global bifurcation in the Brusselator system. Nonlinear Anal. 24, 1713–1725 (1995)

    Article  MathSciNet  Google Scholar 

  5. X. Chen, Y. Qi, Sharp estimates on minimum traveling wave speed of reaction-diffusion systems modeling autocatalysis. SIAM J. Math. Anal. 39, 437–448 (2007)

    Article  MathSciNet  Google Scholar 

  6. V.V. Chepyzhov, M.I. Vishik, Attractors for Equations of Mathematical Physics, vol. 49 (AMS Colloquium Publications, Providence, 2002)

    MATH  Google Scholar 

  7. G. Dangelmayr, Degenerate bifurcation near a double eigenvalue in the Brusselator. J. Austral. Math. Soc. Ser. B 28, 486–535 (1987)

    Article  MathSciNet  Google Scholar 

  8. A. Doelman, T.J. Kaper, P.A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model. Nonlinearity 10, 523–563 (1997)

    Article  MathSciNet  Google Scholar 

  9. I.R. Epstein, Complex dynamical behavior in simple chemical systems. J. Phys. Chemistry 88, 187–198 (1984)

    Article  Google Scholar 

  10. T. Erneux, E. Reiss, Brusselator isolas. SIAM J. Appl. Math. 43, 1240–1246 (1983)

    Article  Google Scholar 

  11. Z. Fu, X. Xu, H. Wang, Q. Ouyang, Stochastic simulation of Turing patterns. Chin. Phys. Lett. 25, 1220–1223 (2008)

    Article  Google Scholar 

  12. P. Gormley, K. Li, G.W. Irwin, Modeling molecular interaction pathways using a two-stage identification algorithm. Syst. Synthetic Biol. 1, 145–160 (2007)

    Article  Google Scholar 

  13. P. Gray, S.K. Scott, Autocatalytic reactions in the isothermal continuous stirred tank reactor: isolas and other forms of multistability. Chem. Eng. Sci. 38, 29–43 (1983)

    Article  Google Scholar 

  14. P. Gray, S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A + 2B → 3B, B → C. Chem. Eng. Sci. 39, 1087–1097 (1984)

    Article  Google Scholar 

  15. J.K. Hale, Asymptotic Behavior of Dissipative Systems. (Amer. Math. Soc., Providence, 1988)

    Google Scholar 

  16. S. Ishihara, K. Kanedo, Turing pattern with proportion preservation. J. Theor. Biol. 238, 683–693 (2006)

    Article  MathSciNet  Google Scholar 

  17. I. Karafyllis, P.D. Christofides, P. Daoutidis, Dynamical analysis of a reaction-diffusion system with Brusselator kinetics under feedback control. Proc. Amer. Control Conference, Albuquerque, NM, June 1997, pp. 2213–2217

    Google Scholar 

  18. M. Kawato, R. Suzuki, Two coupled neural oscillators as a model of the circadian pacemaker. J. Theor. Biol. 86, 547–575 (1980)

    Article  MathSciNet  Google Scholar 

  19. H. Kitano, Systems biology: a brief overview. Science 295, 1662–1664 (2002)

    Google Scholar 

  20. A. Klic̆, Period doubling bifurcations in a two-box model of the Brusselator. Aplikace Matematiky 28, 335–343 (1983)

    Google Scholar 

  21. T. Kolokolnikov, T. Erneux, J. Wei, Mesa-type patterns in one-dimensional Brusselator and their stability. Physica D 214(1), 63–77 (2006)

    Article  MathSciNet  Google Scholar 

  22. K.J. Lee, W.D. McCormick, Q. Ouyang, H. Swinney, Pattern formation by interacting chemical fronts. Science 261, 192–194 (1993)

    Article  Google Scholar 

  23. J.E. Pearson, Complex patterns in a simple system. Science 261, 189–192 (1993)

    Article  Google Scholar 

  24. R. Peng, M. Wang, Pattern formation in the Brusselator system. J. Math. Anal. Appl. 309, 151–166 (2005)

    Article  MathSciNet  Google Scholar 

  25. I. Prigogine, R. Lefever, Symmetry-breaking instabilities in dissipative systems. J. Chem. Phys. 48, 1665–1700 (1968)

    Article  Google Scholar 

  26. B. Peña, C. Pérez-García, Stability of Turing patterns in the Brusselator model. Phys. Review E 64(5), (2001)

    Google Scholar 

  27. Y. Qi, The development of traveling waves in cubic auto-catalysis with different rates of diffusion. Physica D 226, 129–135 (2007)

    Article  MathSciNet  Google Scholar 

  28. T. Rauber, G. Runger, Aspects of a distributed solution of the Brusselator equation. Proc. of the First Aizu International Symposium on Parallel Algorithms and Architecture Syntheses, 114–120, 1995

    Google Scholar 

  29. W. Reynolds, J.E. Pearson, S. Ponce-Dawson, Dynamics of self-replicating patterns in reaction-diffusion systems. Phys. Rev. E 56, 185–198 (1997)

    Article  MathSciNet  Google Scholar 

  30. J. Schnackenberg, Simple chemical reaction systems with limit cycle behavior. J. Theor. Biol. 81, 389–400 (1979)

    Article  Google Scholar 

  31. I. Schreiber, M. Marek, Strange attractors in coupled reaction-diffusion cells. Physica D 5, 258–272 (1982)

    Article  MathSciNet  Google Scholar 

  32. S.K. Scott, K. Showalter, Simple and complex reaction-diffusion fronts, in Chemical Waves and Patterns, ed. by R. Kapral, K. Showalter (Kluwer Acad. Publ., Dordrecht, 1995), pp. 485–516

    Chapter  Google Scholar 

  33. E.E. Sel’kov, Self-oscillations in glycolysis: a simple kinetic model. Euro. J. Biochem. 4, 79–86 (1968)

    Google Scholar 

  34. G.R. Sell, Y. You, Dynamics of Evolutionary Equations (Springer, New York, 2002)

    Book  Google Scholar 

  35. S.H. Strogatz, Nonlinear Dynamics and Chaos (Westview Press, 1994)

    Google Scholar 

  36. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics (Springer, New York, 1988)

    Book  Google Scholar 

  37. J.J. Tyson, K. Chen, B. Novak, Network dynamics and cell physiology. Nature Rev. Mol. Cell Biol. 2, 908–916 (2001)

    Article  Google Scholar 

  38. J. Wei, M. Winter, Asymmetric spotty patterns for the Gray-Scott model in ℜ2. Stud. Appl. math. 110, 63–102 (2003)

    Article  MathSciNet  Google Scholar 

  39. A. De Wit, D. Lima, G. Dewel, P. Borckmans, Spatiotemporal dynamics near a codimension-two point. Phys. Review E 54(1), (1996)

    Google Scholar 

  40. L. Yang, A.M. Zhabotinsky, I.R. Epstein, Stable squares and other oscillatory Turing patterns in a reaction-diffusion model. Phys. Rev. Lett. 92, 198303:1–4 (2004)

    Google Scholar 

  41. Y. You, Global dynamics of the Brusselator equations. Dynamics of PDE 4, 167–196 (2007)

    MathSciNet  MATH  Google Scholar 

  42. Y. You, Global attractor of the Gray-Scott equations. Comm. Pure Appl. Anal. 7, 947–970 (2008)

    Article  MathSciNet  Google Scholar 

  43. Y. You, Asymptotical dynamics of Selkov equations. Discrete Continuous Dyn. Syst. Ser. S 2, 193–219 (2009)

    Article  MathSciNet  Google Scholar 

  44. Y. You, Asymptotical dynamics of the modified Schnackenberg equations. Discrete and Continuous Dynamical Systems–Supplement, 857–868 (2009)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of South Florida, Tampa, FL, 33620, USA

    Yuncheng You

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  1. Yuncheng You
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Correspondence to Yuncheng You .

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Editors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, USA

    John Mallet-Paret

  2. Department of Mathematics and Statistics, York University, Toronto, ON, Canada

    Jianhong Wu  & Huaiping Zhu  & 

  3. Georgia Institute of Technology, School of Mathematics, Atlanta, GA, USA

    Yingfei Yi

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You, Y. (2013). Global Attractor of a Coupled Two-Cell Brusselator Model. In: Mallet-Paret, J., Wu, J., Yi, Y., Zhu, H. (eds) Infinite Dimensional Dynamical Systems. Fields Institute Communications, vol 64. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4523-4_13

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