×

Stability analysis of singular patterns in the 1D Gray-Scott model: a matched asymptotics approach. (English) Zbl 0943.34039

The authors develop a stability analysis in order to facilitate the understanding of the recently discovered phenomena of self replicating pulses. They analyze the stability of singular homoclinic stationary solutions and spatially periodic stationary ones in the 1D Gray-Scott model.
The paper is organized as follows. The introduction is followed by Section 2, Single pulse and spatially periodic stationary patterns , divided into two subsections: Single pulse homoclinic solutions and multiple pulse, spatially periodic stationary states. The paper continues with Section 3, The main stability results and with Section 4, Reduction of the singularly perturbed eigenvalue problem to a nonlocal eigenvalue problem, divided into The fast system , The slow system and Determination of \(c\) subsections. The 5th Section, The nonlocal eigenvalue problem is made of Explicit eigenvalue formulae and Hopf bifurcations and is followed by Section 6, Disappearance of one pulse homoclinic stationary states, divided into Asymptotically small \(d\) \((\beta<1)\), Asymptotically large \(d\) \((\beta>1)\) and \(d=O\) (1) but small \((\beta\sim 1)\). The study contains Numerical simulations , formed by The homoclinic one pulse pattern \((m = 1)\) and Spatially periodic \(N\) pulse patterns \((m > 1)\) and followed by conclusions in Section 8.

MSC:

34D20 Stability of solutions to ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alexander, J.; Gardner, R.A.; Jones, C.K.R.T., A topological invariant arising in the stability of travelling waves, J. reine angew. math., 410, 167-212, (1990) · Zbl 0705.35070
[2] Alhumaizi, K.; Aris, R., Surveying a dynamical system: A study of the gray-Scott reaction in a two-phase reactor, () · Zbl 0854.92024
[3] Blom, J.G.; Zegeling, P.A., Algorthm 731: A moving-grid interface for systems of one-dimensional time-dependent partial differential equations, ACM trans. math. software, 20, 194-214, (1994) · Zbl 0889.65099
[4] A. Doelman, W. Eckhaus, T.J. Kaper (1998), Submitted.
[5] A. Doelman, R.A. Gardner, T.J. Kaper, A stability index analysis of 1-D singular patterns in the Gray-Scott model, submitted. · Zbl 0994.35059
[6] Doelman, A.; Kaper, T.J.; Zegeling, P., Pattern formation in the one-dimensional gray-Scott model, Nonlinearity, 10, 523-563, (1997) · Zbl 0905.35044
[7] Eckhaus, W., Studies in nonlinear stability theory, (1965), Springer New York · Zbl 0125.33101
[8] Eckhaus, W., On modulation equations of the Ginzburg-Landau type, (), 83-98
[9] Fenichel, N., Persistence and smoothness of invariant manifolds for flows, Ind. univ. math. J., 21, 193-226, (1971) · Zbl 0246.58015
[10] Fenichel, N., Geometrical singular perturbation theory for ordinary differential equations, J. diff. eqs., 31, 53-98, (1979) · Zbl 0476.34034
[11] Gardner, R.A., On the structure of the spectra of periodic travelling waves, J. math. pure appl., 72, 415-439, (1993) · Zbl 0831.35077
[12] Gardner, R.A., Spectral analysis of long wavelength periodic waves and applications, J. reine angew. math., 49, 149-181, (1997) · Zbl 0883.35055
[13] Gardner, R.A.; Jones, C.K.R.T., Stability of the travelling wave solutions of diffusive predator-prey systems, Trans. AMS, 327, 465-524, (1991) · Zbl 0755.35056
[14] Gardner, R.A.; Zumbrun, K., The gap lemma and a geometric condition for the instability of viscous shock profiles, Comm. pure appl. math., 51, 797-855, (1998)
[15] Gray, P.; Scott, S.K., Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability, Chem. eng. sci., 38, 29-43, (1983)
[16] Gray, P.; Scott, S.K., 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)
[17] Gray, P.; Scott, S.K., Sustained oscillations and other exotic patterns of behavior in isothermal reactions, J. phys. chem., 89, 22-32, (1985)
[18] Hale, J.K.; Peletier, L.A.; Troy, W.C., Exact homoclinic and heteroclinic solutions of the gray-Scott model for autocatalysis, () · Zbl 0965.34037
[19] Henry, D., Geometric theory of semilinear parabolic equations, () · Zbl 0456.35001
[20] Jones, C.K.R.T., Stability of the travelling wave solution of the Fitzhugh-Nagumo system, Trans. AMS, 286, 431-469, (1984) · Zbl 0567.35044
[21] Jones, C.K.R.T., Geometric singular perturbation theory, () · Zbl 0779.35040
[22] Kerner, B.S.; Osipov, V.V., Autosolitons: A new approach to problems of self-organization and turbulence, (1994), Kluwer Dordrecht
[23] Lee, K.J.; Swinney, H.L., Lamellar structures and self-replicating spots in a reaction-diffusion system, Phys. rev. E, 51, 1899-1915, (1995)
[24] Lin, K.-J.; McCormick, W.D.; Pearson, J.E.; Swinney, H.L., Experimental observation of self-replicating spots in a reaction-diffusion system, Nature, 369, 6477, 215-218, (1994)
[25] MATHEMATICA^{{\scr}}, (1996), Wolfram Research, Inc, version 3.0
[26] Morse, P.M.; Feshbach, H., Methods of theoretical physics, (1953), McGraw-Hill New York · Zbl 0051.40603
[27] Nishiura, Y.; Fujii, H., Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. math. anal., 18, 1726-1770, (1987) · Zbl 0638.35010
[28] Nishiura, Y.; Ueyama, D., A skeleton structure for self-replication dynamics, (1997), preprint
[29] Osipov, V.V.; Severtsev, A.V., Theory of self-replication and granulation of spike autosolitons, Pla, 222, 400-404, (1996) · Zbl 1037.35501
[30] Pearson, J.E., Complex patterns in a simple system, Science, 261, 189-192, (1993)
[31] Petrov, V.; Scott, S.K.; Showalter, K., Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system, (), 631-642 · Zbl 0867.35047
[32] Reynolds, W.N.; Pearson, J.E.; Ponce-Dawson, S., Dynamics of self-replicating patterns in reaction diffusion systems, Phys. rev. lett., 72, 2797-2800, (1994)
[33] Reynolds, W.N.; Ponce-Dawson, S.; Pearson, J.E., Self-replicating spots in reaction-diffusion systems, Phys. rev. E, 56, 185-198, (1997)
[34] Zegeling, P.A.; Verwer, J.G.; Eijkeren, J.C.H v., Application of a moving-grid method to a class of 1D brine transport problems in porous media, Int. J. numer. methods fluids, 15, 175-191, (1992) · Zbl 0781.76063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.