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Diffusion chaos in the reaction–diffusion boundary problem in the dumbbell domain

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Abstract

We consider a boundary problem of the reaction–diffusion type in the domain that consists of two rectangular areas connected by a bridge. The bridge width is a bifurcation parameter of the problem and is changed in such way that the measure of the domain is preserved. The conditions on chaotic oscillations emergence have been studied and the dependence of invariant characteristics of the attractor on the bridge width has been constructed. The diffusion parameter has been chosen such that, in the case of widest possible bridge (corresponding to a rectangular domain), the spatially homogeneous cycle of the problem is orbitally asymptotically stable. By decreasing the bridge width, the homogeneous cycle loses stability; then, a spatially inhomogeneous chaotic attractor emerges. For the obtained attractor, we have calculated the Lyapunov exponents and Lyapunov dimension and observed that the dimension grows as the parameter decreases, but is bounded. We have shown that the dimension growth is connected with the growing complexity of the distribution of stable solutions with respect to the space variable.

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References

  1. Turing, A.M., The chemical basis of morphogenesis, Philos. Trans. R. Soc., B, 1952, vol. 237, no. 641, pp. 37–72.

    Article  MathSciNet  Google Scholar 

  2. Nicolis, G. and Prigogine, I., Self-Organization in Non-Equilibrium Systems, Wiley, 1977.

    MATH  Google Scholar 

  3. Kuramoto, Y., Diffusion-induced chaos in reaction systems, Prog. Theor. Phys. Suppl., 1978, no. 64, pp. 346–367. doi 10.1143/PTPS.64.346

    Article  Google Scholar 

  4. Arrieta, J.M., Neumann eigenvalue problems on exterior perturbations of the domain, J. Differ. Equations, 1995, vol. 118, no. 1, pp. 54–103.

    Article  MathSciNet  MATH  Google Scholar 

  5. Arrieta, J.M., Rates of eigenvalues on a dumbbell domain. Simple eigenvalue case, Trans. Am. Math. Soc., 1995, vol. 347, no. 9, pp. 3503–3531.

    Article  MathSciNet  MATH  Google Scholar 

  6. Arrieta, J.M., Carvalho, A.N., and Lozada-Cruz, G., Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differ. Equations, 2006, vol. 231, no. 2, pp. 551–597.

    Article  MathSciNet  MATH  Google Scholar 

  7. Arrieta, J.M., Carvalho, A.N., and Lozada-Cruz, G., Dynamics in dumbbell domains II.The limiting problem, J. Differ. Equations, 2009, vol. 247, no. 1, pp. 174–202.

    Article  MathSciNet  MATH  Google Scholar 

  8. Arrieta, J.M., Carvalho, A.N., and Lozada-Cruz, G., Dynamics in dumbbell domains III. Continuity of attractors, J. Differ. Equations, 2009, vol. 247, no. 1, pp. 225–259.

    Article  MathSciNet  MATH  Google Scholar 

  9. Vasil’eva, A.B., Kashchenko, S.A., Kolesov, Yu.S., and Rozov, N.Kh., Bifurcation of self-oscillations of nonlinear parabolic equations with small diffusion, Mat. Sb., 1987, vol. 58, no. 2, pp. 491–503.

    Article  MATH  Google Scholar 

  10. Dormand, J.R. and Prince, P.J., A family of embedded Runge–Kutta formulae, J. Comp. Appl. Math., 1980, vol. 6, pp. 19–26.

    Article  MathSciNet  MATH  Google Scholar 

  11. Oseledec, V.I., A multiplicative ergodic theorem. Characteristic Lyapunov exponents of dynamical systems, Tr. Mosk. Mat. O-va, 1968, vol. 19, pp. 179–210.

    MathSciNet  Google Scholar 

  12. Benettin, G., Galgani, L., and Strelcyn, J.M., Kolmogorov entropy and numerical experiments, Phys. Rev., 1976, vol. A14, pp. 2338–2345.

    Article  Google Scholar 

  13. Wolf, A., Swift, J.B., Swinney, H.L., and Vastano, J.A., Determining Lyapunov exponents from a time series, Phys. D, 1985, vol. D16, pp. 285–317.

    Article  MathSciNet  MATH  Google Scholar 

  14. Glyzin, D.S., Glyzin, S.D., Kolesov, A.Yu., and Rozov, N.Kh., The dynamic renormalization method for finding the maximum Lyapunov exponent of a chaotic attractor, Differ. Equations, 2005, vol. 41, no. 2, pp 268–273.

    Article  MathSciNet  MATH  Google Scholar 

  15. Frederickson, P., Kaplan, J., and Yorke, J., The Lyapunov dimension of strange attractors, J. Differ. Equaions, 1983, vol. 49, no. 2, pp. 185–207.

    Article  MATH  Google Scholar 

  16. Glyzin, S.D., Difference approximations of reaction–diffusion equation on a segment, Model. Anal. Inf. Syst., 2009, vol. 16, no. 3, pp. 96–116.

    Google Scholar 

  17. Glyzin, S.D., Kolesov, A.Yu., and Rozov, N.Kh., Finite-dimensional models of diffusion chaos, Comput. Math. Math. Phys., 2010, vol. 50, no. 5, pp. 816–830.

    Article  MathSciNet  MATH  Google Scholar 

  18. Glyzin, S.D., Dimensional characteristics of diffusion chaos, Model. Anal. Inf. Syst., 2013, vol. 20, no. 1, pp. 30–51.

    Google Scholar 

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Authors and Affiliations

  1. Scientific Center in Chernogolovka Russian Academy of Sciences, Chernogolovka, Moscow region, 142432, Russia

    S. D. Glyzin

  2. Demidov Yaroslavl State University, Yaroslavl, 150000, Russia

    S. D. Glyzin & P. L. Shokin

Authors
  1. S. D. Glyzin
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  2. P. L. Shokin
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Correspondence to S. D. Glyzin.

Additional information

Original Russian Text © S.D. Glyzin, P.L. Shokin, 2013, published in Modelirovanie i Analiz Informatsionnykh Sistem, 2013, Vol. 20, No. 3, pp. 43–57.

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Glyzin, S.D., Shokin, P.L. Diffusion chaos in the reaction–diffusion boundary problem in the dumbbell domain. Aut. Control Comp. Sci. 50, 625–635 (2016). https://doi.org/10.3103/S0146411616070075

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  • DOI: https://doi.org/10.3103/S0146411616070075

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