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1
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0003562886
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See, e.g, (Dover, New York, 1962); W. Kaplan, Ordinary Differential Equations (Addison‐Wesley, London, 1958); V. I. Arnold, Ordinary Differential Equations (MIT, Cambridge).
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(1978)
Introduction to Nonlinear Differential and Integral Equations
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Davis, H.T.1
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3
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0039235350
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This is an early application of time‐dependent perturbation theory to steady‐state chemical kinetics.
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(1957)
J. Chem. Phys.
, vol.26
, pp. 271
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Hirschfelder, J.O.1
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7
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0003791868
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See, e.g, edited by K. Kustin (Academic, New York, 1969), Vol.; C. F. Bernasconi, Relaxation Kinetics (Academic, New York, 1976); K. Hiromi, Kinetics of Fast Enzyme Reactions: Theory and Practice (Wiley, New York), and references therein.
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(1979)
Methods in Enzymology
, vol.16
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8
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85034899079
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See, e.g, (McGraw‐Hill, New York, 1960); K. J. Laidler, Chemical Kinetics (Harper and Row, New York, 1987). For enzyme kinetics see, e.g, S. Ainsworth, Steady‐State Enzyme Kinetics (University Park, London, 1977); J. Tze‐Fei Wong, Kinetics of Enzyme Mechanisms (Academic, New York, 1975); I. H. Segel, Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady‐State Enzyme Systems (Wiley, New York, 1975); K. J. Laidler and P. S. Bunting, The Chemical Kinetics of Enzyme Action (Clarendon, Oxford).
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(1973)
The Foundations of Chemical Kinetics
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Benson, S.W.1
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15
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0003459267
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General discussions of the application of perturbation theory to dynamical problems are given in, e.g, Appl. Math. Sci. Ser. 8 (Springer, New York, 1972); A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York).
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(1981)
Perturbation Methods in Non‐linear Systems
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Giacaglia, G.E.O.1
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19
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0014115961
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The uniqueness of this stable equilibrium for closed systems has been proved by,;
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(1967)
J. Theor. Biol.
, vol.16
, pp. 212
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Shear, D.1
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21
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0004168084
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For discussion of bifurcations and manifold theory see, e.g, Appl. Math. Sci. 35 (Springer, New York, 1981); and J. E. Marsden and M. McCracken, The Hopf Bifurcation and its ApplicationsAppl. Math. Sci. 19 (Springer, New York, 1976); D. R. J. Chillingworth, Differential Topology with a View to Applications (Pitman, San Francisco).
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(1976)
Applications of Center Manifold Theory
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Carr, J.1
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24
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84950889747
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V. Henri, Lois Générales del’ Action des Diastases (Hermann, Paris).
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(1903)
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31
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85034890113
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practice the flow contraction takes place in the physical significant part of phase space, where all the concentrations are positive in the complete system of ODEs. Because of the reduction using constants of the motion, this region may be restricted to be some smaller positive region of the reduced phase space used here. Also, for the simple flows described here, there is a repeller at infinity and an attractor at the origin for [Formula Omitted] and one branch of M “attracts” the flow and another branch in the unphysical part of Γ “repels” the flow. For [Formula Omitted] these branches of M become branches of E: The stable branch is a true one‐dimensional attractor and the unstable branch is a true one‐dimensional repeller.
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32
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85034907708
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Regarding M as the “attracting” slow manifold of a flow, we note that for plane autonomous systems this flow is into the region, containing Mbetween [Formula Omitted] and [Formula Omitted] Since [Formula Omitted] and [Formula Omitted] coalesce at infinity for scheme (M), M is “pinched” between them, and its asymptotic solution, obtained from Eq. (3.3), approaches [Formula Omitted] very rapidly at large 8 as discussed in Ref. 9.
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33
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84950919201
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The uniqueness of M as a solution of the functional equation (3.3) is ensured by the contraction mapping theorem, since (locally), for sufficiently small [Formula Omitted] and low frequency (small λ) Fourier components in the starting function, the iterative procedure is a contraction. This small‐λ condition implies smoothness of the starting function. See Appendix A. For a discussion of contraction mappings see, e.g, P. Roman, Some Modern Mathematics for Physicists and Other OutsidersVol. I (Pergamon, New York, 1975), especially pp. -; P. Henrici, Elements of Numerical Analysis (Wiley, New York, 1964); G. F. Simmons, Introduction to Topology and Modern Analysis (McGraw‐Hill, London).
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(1963)
, pp. 280-289
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34
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85034902478
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Note that [Formula Omitted] numerically, though we will continue to use the superscript (*) to indicate fixed points of the functional equation algebra, and the subscript M to indicate the corresponding geometrical manifold in [Formula Omitted]
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35
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85034899816
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This is the King and Altman solution. See Ref. 7, and the discussion in Ref. 9.
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37
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85034899692
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principle Aitken’s extrapolation procedure does not strictly require the iteration process to be a contraction (see Ref. 24), because it annihilates a growing eigenvalues. However, we have never encountered this case.
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38
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0003758310
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See e.g, (McGraw‐Hill, New York, 1968), p.; A. Messiah, Quantum Mechanics (North‐Holland, Amsterdam), Vol. I, p. 183.
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(1970)
Quantum Mechanics
, pp. 56
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Schiff, L.I.1
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