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18
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85033042454
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note
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Rather than constrain the class of equations at the outset, we can always modify the equations by a judicious, if somewhat ad hoc, application of Heaviside functions, as was done in ref 2. A more elegant general solution is sought here.
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19
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85033059962
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note
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max may be infinite in the case of distributed delays.
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25
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85033036497
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Ph.D. Thesis, University of Toronto
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Roussel, M. R. Ph.D. Thesis, University of Toronto, 1994.
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Roussel, M.R.1
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31
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85033043123
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note
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A very simple strategy was adopted for numerical integration: The integrator was a straightforward constant-step-size implicit trapezoidal code. Values obtained by integration were stored in a circular buffer. The routine which returns the function to be integrated extracts the history-dependent , terms from the buffer by linear interpolation. To verify the plausibility of the trajectories obtained, the results of the integration were compared to analytic results whenever possible.
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33
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85033072412
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note
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These are functions with a random component rather than truly random functions. These functions were constructed in the following manner: Since a DDE integrator uses stored points and interpolation to compute the history-dependent terms, it is necessary only to store a set of points to initialize the integrator. Random initial values for the concentrations are chosen in some positive interval. These values are then incremented by random amounts uniformly selected in a small interval at each mesh point. Negative values, when they arise, are replaced by small positive values in order to maintain the nonnegativity constraint imposed on these models. Finally, the stored values are scaled to satisfy any additional normalization conditions.
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34
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0003983878
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Springer-Verlag: New York
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This procedure is similar to numerical methods for constructing the slow manifold of enzymic mechanisms; see Roussel, M. R. M.Sc. Thesis, University of Toronto, 1990. It is also closely related to seminumerical methods for constructing integral manifolds of PDE systems. The boundary of the manifold is constructed (analogous to finding an initial function which generates the slow manifold), and the manifold is then found by integration from the boundary; see Constantin, P.; Foias, C.; Nicolaenko, B.; Temam, R. Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations; Springer-Verlag: New York, 1989.
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Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations
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Constantin, P.1
Foias, C.2
Nicolaenko, B.3
Temam, R.4
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35
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0000766018
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Wischert, W.; Wunderlin, A.; Pelster, A.; Olivier, M.; Groslambert, J. Phys. Rev. E 1994, 49, 203-219.
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Wischert, W.1
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Olivier, M.4
Groslambert, J.5
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39
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85033043836
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note
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This scaling was chosen to make the steady-state values of the concentration variables of unit magnitude.
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40
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85033069798
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note
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Because of the extreme stiffness of the Oregonator and delayed Oregonator equations, a slightly different numerical integration strategy had to be adopted than for the other models studied here. A third-order Gear method was used with a conservative step-size adjustment policy: The step size was decreased whenever either the solution threatened to become negative or the functional iteration routine at the core of the integrator failed to converge reasonably quickly. After a suitably large number of steps, the step size was increased again. Otherwise, the method was exactly as described in note 31 above.
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44
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0004104171
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Wiley: New York
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The universal acceptance of this idea is relatively modern. In the early days of enzymology, it was believed by many that enzymes catalyzed only exothermic reactions. For a discussion, see Fruton, J. S. Molecules and Life; Wiley: New York, 1972; p 83.
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(1972)
Molecules and Life
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Fruton, J.S.1
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