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Complex Attractors and Patterns in Reaction–Diffusion Systems

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Abstract

We consider semiflows generated by initial boundary value problems for reaction–diffusion systems. In these systems, reaction terms satisfy general conditions, which admit a transparent chemical interpretation. It is shown that the semiflows generated by these initial boundary value problems exhibit a complicated large time behavior. Any structurally stable finite dimensional dynamics (up to an orbital topological equivalence) can be realized by these semiflows by a choice of appropriate external sources and diffusion coefficients (nonlinear terms are fixed). Results can be applied to the morphogenesis and pattern formation problems.

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References

  1. Arnol’d, V.I.: Geometric Methods in Theory of Ordinary Differential Equations, 2nd edn. Springer, New York (1988)

    Book  MATH  Google Scholar 

  2. Carr, J., Pego, R.: Invariant manifolds for metastable patterns in \(u_t = \epsilon ^2 u_{xx} - f(u)\). Proc. R. Soc. Edimburgh 116A, 133–160 (1990)

    Article  MATH  Google Scholar 

  3. Carvalho, A.N., Langa, Jose A., Robinson, James C.: Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems. Springer, New York (2013)

    Book  MATH  Google Scholar 

  4. Constantin, P., Foias, C., Nicolaenko, B., Temam, R.: Integrable Manifolds and Inertial Manifolds for Dissipative Differential Equations. Springer, New York (1989)

    Book  MATH  Google Scholar 

  5. Dancer, E.N., Poláčik, P.: Realization of Vector Fields and Dynamics of Spatially Homogeneous Parabolic Equation, vol. 668. Memoirs of American Mathematical Society, Providence (1999)

    MATH  Google Scholar 

  6. Fiedler, B., Mallet-Paret, J.: The Poincare–Bendixson theorem for scalar reaction diffusion equations. Arch. Ration. Mech. Anal. 107, 325–345 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Guckenheimer, J., Holmes, P.: Nonlinear Osscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1981)

    Google Scholar 

  8. Hale, J.K.: Asymptotic Behavior of Dissipative Systems. American Mathematical Society, Providence (1988)

    MATH  Google Scholar 

  9. Henry, D.: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)

    Book  Google Scholar 

  10. Hirsch, M.W.: Stability and convergence in strongly monotone dynamical systems. J. Reine. Angew. Math. 383, 1–58 (1988)

    MathSciNet  MATH  Google Scholar 

  11. Hirsch, M.W., Smith, H.: Monotone dynamical systems. In: Barbu, V., Lefter, C., Bartsch, T., Szulkin, A., Crjá, O., Vrabie, I.I., Hirsch, M.W., Smith, H., López-Gómez, J., Ntouyas, S.K. (eds.) Handbook of Differential Equations: Ordinary Differential Equations, pp. 239–357. Elsevier B. V, Amsterdam (2005)

    Google Scholar 

  12. Il’yashenko, YuS: Weakly contracting systems and attractors of Galerkin approximation for Navier–Stokes equation on two-dimensional torus. Usp. Mech. 1, 31–63 (1982)

    MathSciNet  Google Scholar 

  13. Katok, A.B., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and Its Applications, vol. 54. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  14. Ladygenskaya, O.A.: Finding minimal global attractors for Navier–Stokes equations and other partial differential equations. Usp. Mat. Nauk 42, 25–60 (1987)

    MathSciNet  Google Scholar 

  15. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci 20, 130–141 (1963)

    Article  Google Scholar 

  16. Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhauser, Basel (1995)

    Book  MATH  Google Scholar 

  17. Matano, H.: Convergence of solutions of one-dimensional semilinear parabolic equations. J. Math. Kyoto Univ. 18, 221–227 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  18. Poláčik, P. (2002) Parabolic equations: Asymptotic behaviour and Dynamics on Invariant Manifolds, Ch.16, pp. 835-883, in: Handbook of dynamical systems, Vol 2., Edited by B. Fiedler

  19. Poláčik, P.: Realization of any finite jet in a scalar semilinear parabolic equation on the ball in \(R^2\). Annali Scuola Norm Pisa 17, 83–102 (1991)

    MathSciNet  MATH  Google Scholar 

  20. Poláčik, P.: Complicated dynamics in scalar semilinear parabolic equations in higher space dimensions. J. Differ. Equ. 89, 244–271 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Poláčik, P.: High dimensional \(\omega \)-limit sets and chaos in scalar parabolic equations. J. Differ. Equ. 119, 24–53 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Prigogine, I., Nicolis, G.: Self-Organization in Non-equilibrium Systems. Wiley, New York (1977)

    MATH  Google Scholar 

  23. Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics and Chaos. CRC Press, Taylor & Francis (1999)

    MATH  Google Scholar 

  24. Ruelle, D.: Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press, Boston (1989)

    MATH  Google Scholar 

  25. Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys 20, 167–192 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rybakowski, K.P.: Realization of arbitrary vector fields on center manifolds of parabolic Dirichlet BVP’s. J. Differ. Equ. 114, 199–221 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Simon, L.: Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. Math. 118, 525–571 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  28. Smoller, J.: Shock Waves and Reaction–Diffusion Systems. Springer, New York (1983)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  30. Tucker, W.: A Rigorous ODE Solver and Smale’s 14th Problem. Found. Comp. Math. 2, 53–117. http://www.math.uu.se/~warwick/main/rodes.html (2002)

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

    Article  MathSciNet  Google Scholar 

  32. Vakulenko, S.A.: Reaction–diffusion systems with prescribed large time behaviour. Annales de L’Institut H Poincarè Physique Théorique 66, 373–410 (1997)

    MathSciNet  MATH  Google Scholar 

  33. Vakulenko, S.A.: Dissipative systems generating any structurally stable chaos. Adv. Differ. Equ. 5(7–9), 1139–1178 (2000)

    MathSciNet  MATH  Google Scholar 

  34. Vakulenko, S., Radulescu, O., Reinitz, J.: Size regulation in the segmentation of Drosophila: interacting interfaces between localized domains of gene expression ensure robust spatial patterning. Phys. Rev. Lett. 103, 168102–168106 (2009)

    Article  Google Scholar 

  35. Vakulenko, S., Grigoriev, D., Weber, A.: Reduction methods and chaos for quadratic systems of differential equations. Stud. Appl. Math. 135, 225–247 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Vanderbauwhede, A., Ioss, G.: Center manifold theory in infinite dimensions. In Dynamics Reported: Expositions in Dynamical systems, Springer, Berlin, pp. 125–163, (1992)

  37. Wiggins, S.: Normally Hyperbolic Invariant Manifolds in Dynamical Systems. Spinger, New York (1994)

    Book  MATH  Google Scholar 

  38. Wolpert, L.: Positional information and pattern formation in development. Dev. Genet. 15, 485–490 (1994)

    Article  Google Scholar 

  39. Yagi, A.: Abstract Parabolic Evolution Equations and Their Applications. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  40. Zelenyak, T.I.: Stabilization of solution of boundary nonlinear problems for a second order parabolic equations with one space variable. Differ. Equ. 4, 17–22 (1968)

    MATH  Google Scholar 

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Acknowledgments

The author is grateful to Referee for useful remarks which helped to improve the paper.

The author was financially supported by Government of Russian Federation, Grant 074-U01, also supported in part by Grant RO1 OD010936 (formerly RR07801) from the US NIH and by Grant 16-01-00648 of Russian Fund of Basic Research.

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Authors and Affiliations

  1. Institute for Mechanical Engineering Problems, Bolshoy pr. V. O.61, Saint Petersburg, Russia

    Sergey Vakulenko

  2. ITMO University, Saint Petersburg, Russia

    Sergey Vakulenko

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  1. Sergey Vakulenko
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Correspondence to Sergey Vakulenko.

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Vakulenko, S. Complex Attractors and Patterns in Reaction–Diffusion Systems. J Dyn Diff Equat 30, 175–207 (2018). https://doi.org/10.1007/s10884-016-9552-4

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  • DOI: https://doi.org/10.1007/s10884-016-9552-4

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