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  • 1
    Digitale Medien
    Digitale Medien
    A Direct Approach to Homoclinic Orbits in the Fast Dynamics of Singularly Perturbed Systems (1999)
    Springer
    Journal of dynamics and differential equations 11 (1999), S. 671-709 
    Details
    ISSN: 1572-9222
    Schlagwort(e): Homoclinic orbits ; singular perturbations ; bifurcation ; FitzHugh–Nagumo system
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract Homoclinic orbits in the fast dynamics of singular perturbation problems are usually analyzed by a combination of Fenichel's invariant manifold theory with general transversality arguments (see Ref. 29 and the Exchange Lemma in Ref. 16). In this paper an alternative direct approach is developed which uses a two-time scaling and a contraction argument in exponentially weighted spaces. Homoclinic orbits with one last transition are treated and it is shown how ε-expansions can be extracted rigorously from this approach. The result is applied to a singularity perturbed Bogdanov point in the FitzHugh–Nagumo system.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1022663512855
    Bibliothek Standort Signatur Band/Heft/Jahr Verfügbarkeit
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  • 2
    Digitale Medien
    Digitale Medien
    Quasi-steady-state approximation for chemical reaction networks (1998)
    Springer
    Journal of mathematical biology 36 (1998), S. 593-609 
    Details
    ISSN: 1432-1416
    Schlagwort(e): Key words: Quasi-steady-state ; Singular perturbation ; Dictyostelium ; Maltose transport
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Biologie , Mathematik
    Notizen: Abstract.  The parameter embedding leading to the quasi-steady-state approximation of Heinrich [9] is investigated within the theory of invariant manifolds of Fenichel [4] in order to clarify the essential assumptions needed for this reduction to a low dimensional system. In particular, the concept of pool-variables can be avoided in this generalized approach. Moreover, the dominating influence of the slow subnetwork over the complementary fast subnetwork is interpreted geometrically and in chemical terms and this can be seen as an “enslaving” of the fast subsystem by the slow subsystem. Finally, the results are applied to a system of slime mould communication [6, 7, 13] and to a maltose transport system [2, 3].
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s002850050116
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