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.


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
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