\(2^n\)-splitting or edge-splitting? A manner of splitting in dissipative systems. (English) Zbl 0983.35061

The splitting of pulses in reaction-diffusion systems, especially for the Gray-Scott model, is investigated. In experiments and numerical calculations it has been observed that self-replicating patterns are produced by splitting of the pulses at the boundary, leading to the creation of 2 new pulses in every step for the one-dimensional case. In this paper the transient dynamics is investigated by reducing the system to a local invariant manifold close to the bifurcation point and investigating the resulting ODE. Starting from a single pulse it is shown that in this model only the pulses at the boundary split.


35K57 Reaction-diffusion equations
35B32 Bifurcations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B42 Inertial manifolds


HomCont; AUTO
Full Text: DOI


[1] P.De Kepper, J.J. Perraud, B. Rudovics and E. Dulos, Experimental study of stationary Turing patterns and their interaction with traveling waves in a chemical system. Int. J. Bifurcation and Chaos,4, No. 5 (1994), 1215–1231. · Zbl 0877.92032
[2] E.J. Doedel, A.R. Champneys, T.F. Fairgrieve, Y.A. Kuznetsov, B. Sandstede and X. Wang, AUTO97: Continuation and bifurcation software for ordinary differential equations (with HomCont). ftp://ftp.cs.concordia.ca/pub/doedel/auto, 1997.
[3] A. Doelman, T.J. Kaper and P.A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model. Nonlinearity,10 (1997), 523–563. · Zbl 0905.35044
[4] A. Doelman, R.A. Gardner and T.J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotic approach. Physica D,122 (1998), 1–36. · Zbl 0943.34039
[5] A. Doelman, W. Eckhaus and T.J. Kaper, Slowly-modulated two pulse solutions in the Gray-Scott model I, II. SIAM J. Appl. Math, (in press). · Zbl 0979.35074
[6] S. Ei, The motion of weakly interacting pulses in reaction diffusion systems. To appear in J. D. D. E. · Zbl 1007.35039
[7] S. Ei, Y. Nishiura and B. Sandstede, Pulse interaction approach to self-replicating dynamics in reaction diffusion systems. Preprint, 2000.
[8] H. Fujii, M. Mimura and Y. Nishiura, A picture of the global bifurcation diagram in ecological interacting and diffusing systems. Physica,5D (1982), 1–42.
[9] P. Gray and S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the systemA + 2B 3B,B . Chem. Eng. Sci.,39 (1984), 1087–1097.
[10] D. Henry, Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics840, Springer-Verlag, 1981. · Zbl 0456.35001
[11] Y. Kuznetsov, Elements of Applied Bifurcation Theory. Springer-Verlag, 1995. · Zbl 0829.58029
[12] K.J. Lee, W.D. McCormick, J.E. Pearson and H.L. Swinney, Experimental observation of self-replicating spots in a reaction-diffusion system. Nature,369 (1994), 215–218.
[13] K.J. Lee and H.L. Swinney, Lamellar structures and self-replicating spots in a reactiondiffusion system. Phys. Rev. E,51 (1995), 1899–1915.
[14] W. Mazin, K.E. Rasmussen, E. Mosekilde, P. Borckmans and G. Dewel, Pattern formation in the bistable Gray-Scott model. Mathematics and Computers in Simulation,40 (1996), 371–396. · Zbl 05475328
[15] C.B. Muratov and V.V. Osipov, Spike autosolitons in the Gray-Scott model. Preprint. · Zbl 0986.34023
[16] Y. Nishiura and D. Ueyama, A skeleton structure of self-replicating dynamics. Physica D,130 (1999), 73–104. · Zbl 0936.35090
[17] Y. Nishiura and D. Ueyama, Spatio-temporal chaos for the Gray-Scott model. Physica D (in press). · Zbl 0981.35022
[18] E. Ott, Chaos in Dynamical Systems. Cambridge Univ. Press, 1993. · Zbl 0792.58014
[19] J.E. Pearson, Complex patterns in a simple system. Science,216 (1993), 189–192.
[20] V. Petrov, S.K. Scott and K. Showalter, Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system. Phil. Trans. Roy. Soc. Lond. A,347 (1994), 631–642. · Zbl 0867.35047
[21] K.E. Rasmussen, W. Mazin, E. Mosekilde, G. Dewel and P. Borckmans, Wave-splitting in the bistable Gray-Scott model. Int. J. Bifurcation and Chaos,6, No. 6 (1996), 1077–1092. · Zbl 0881.92038
[22] W.N. Reynolds, J.E. Pearson and S. Ponce-Dawson, Dynamics of self-replicating patterns in reaction diffusion systems. Phys. Rev. Lett.,72, No. 17 (1994), 1120–1123.
[23] W.N. Reynolds, S. Ponce-Dawson and J.E. Pearson, Self-replicating spots in reactiondiffusion systems. Phys. Rev. E,56, No. 1 (1997), 185–198.
[24] D. Ueyama, Dynamics of self-replicating dynamics in the one-dimensional Gray-Scott model. PhD thesis. · Zbl 0987.34031
[25] J. Wei, On two dimensional Gray-Scott model: existence of single pulse solutions and their stability. Preprint, 1999.
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