Slow manifold for a bimolecular association mechanism

Journal of Chemical Physics

Volumn 120, Issue 7, 2004, Pages 3075-3085

  Fraser, Simon J ^a  

^a UNIVERSITY OF TORONTO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; ASYMPTOTIC STABILITY; CHEMICAL RELAXATION; COMPUTER SIMULATION; CONVERGENCE OF NUMERICAL METHODS; FUNCTIONS; LINEAR EQUATIONS; MATHEMATICAL MODELS; ORDINARY DIFFERENTIAL EQUATIONS;

STEADY-STATE APPROXIMATION (SSA); TRAJECTORY DIFFERENTIAL EQUATION;

MOLECULAR PHYSICS;

BIOPOLYMER;

ALGORITHM; ARTICLE; BINDING SITE; CHEMICAL MODEL; CHEMISTRY; COMPUTER SIMULATION; KINETICS;

ALGORITHMS; BINDING SITES; BIOPOLYMERS; COMPUTER SIMULATION; KINETICS; MODELS, CHEMICAL;

EID: 1642331556     PISSN: 00219606     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1634555     Document Type: Article

References (80)

1. A. S. Tomlin, M. J. Pilling, T. Turányi, J. H. Merkin, and J. Brindley, Combust. Flame 91, 107 (1992); G. Li, A. S. Tomlin, and H. Rabitz, J. Chem. Phys. 99, 3562 (1993).
2. A. S. Tomlin, M. J. Pilling, T. Turányi, J. H. Merkin, and J. Brindley, Combust. Flame 91, 107 (1992); G. Li, A. S. Tomlin, and H. Rabitz, J. Chem. Phys. 99, 3562 (1993).
3. For chemical applications see J. C. Giddings and H. K. Shin, Trans. Faraday Soc. 51, 468 (1961); H. K. Shin and J. C. Giddings, J. Phys. Chem. 65, 1164 (1961); J. R. Bowen, A. Acrivos, and A. K. Oppenheim, Chem. Eng. Sci. 18, 177 (1963); F. G. Heineken, H. M. Tsuchiya, and R. Aris, Math. Biosci. 1, 95 (1967); S. I. Rubinow and J. L. Lebowitz, J. Am. Chem. Soc. 92, 3888 (1970). For mathematical background see: A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974), Chaps. 9 and 10; J. D. Murray, Lectures on Nonlinear-Differential-Equation Models in Biology (Clarendon, Oxford, 1977), Chap. 1; J. Kervorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Appl. Math. Ser. (Springer, New York, 1996), Vol. 114; R. E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Ser.; Appl. Math. Sci. (Springer, New York, 1991), Vol. 89.
4. For chemical applications see J. C. Giddings and H. K. Shin, Trans. Faraday Soc. 51, 468 (1961); H. K. Shin and J. C. Giddings, J. Phys. Chem. 65, 1164 (1961); J. R. Bowen, A. Acrivos, and A. K. Oppenheim, Chem. Eng. Sci. 18, 177 (1963); F. G. Heineken, H. M. Tsuchiya, and R. Aris, Math. Biosci. 1, 95 (1967); S. I. Rubinow and J. L. Lebowitz, J. Am. Chem. Soc. 92, 3888 (1970). For mathematical background see: A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974), Chaps. 9 and 10; J. D. Murray, Lectures on Nonlinear-Differential-Equation Models in Biology (Clarendon, Oxford, 1977), Chap. 1; J. Kervorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Appl. Math. Ser. (Springer, New York, 1996), Vol. 114; R. E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Ser.; Appl. Math. Sci. (Springer, New York, 1991), Vol. 89.
5. For chemical applications see J. C. Giddings and H. K. Shin, Trans. Faraday Soc. 51, 468 (1961); H. K. Shin and J. C. Giddings, J. Phys. Chem. 65, 1164 (1961); J. R. Bowen, A. Acrivos, and A. K. Oppenheim, Chem. Eng. Sci. 18, 177 (1963); F. G. Heineken, H. M. Tsuchiya, and R. Aris, Math. Biosci. 1, 95 (1967); S. I. Rubinow and J. L. Lebowitz, J. Am. Chem. Soc. 92, 3888 (1970). For mathematical background see: A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974), Chaps. 9 and 10; J. D. Murray, Lectures on Nonlinear-Differential-Equation Models in Biology (Clarendon, Oxford, 1977), Chap. 1; J. Kervorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Appl. Math. Ser. (Springer, New York, 1996), Vol. 114; R. E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Ser.; Appl. Math. Sci. (Springer, New York, 1991), Vol. 89.
6. For chemical applications see J. C. Giddings and H. K. Shin, Trans. Faraday Soc. 51, 468 (1961); H. K. Shin and J. C. Giddings, J. Phys. Chem. 65, 1164 (1961); J. R. Bowen, A. Acrivos, and A. K. Oppenheim, Chem. Eng. Sci. 18, 177 (1963); F. G. Heineken, H. M. Tsuchiya, and R. Aris, Math. Biosci. 1, 95 (1967); S. I. Rubinow and J. L. Lebowitz, J. Am. Chem. Soc. 92, 3888 (1970). For mathematical background see: A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974), Chaps. 9 and 10; J. D. Murray, Lectures on Nonlinear-Differential-Equation Models in Biology (Clarendon, Oxford, 1977), Chap. 1; J. Kervorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Appl. Math. Ser. (Springer, New York, 1996), Vol. 114; R. E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Ser.; Appl. Math. Sci. (Springer, New York, 1991), Vol. 89.
7. For chemical applications see J. C. Giddings and H. K. Shin, Trans. Faraday Soc. 51, 468 (1961); H. K. Shin and J. C. Giddings, J. Phys. Chem. 65, 1164 (1961); J. R. Bowen, A. Acrivos, and A. K. Oppenheim, Chem. Eng. Sci. 18, 177 (1963); F. G. Heineken, H. M. Tsuchiya, and R. Aris, Math. Biosci. 1, 95 (1967); S. I. Rubinow and J. L. Lebowitz, J. Am. Chem. Soc. 92, 3888 (1970). For mathematical background see: A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974), Chaps. 9 and 10; J. D. Murray, Lectures on Nonlinear-Differential-Equation Models in Biology (Clarendon, Oxford, 1977), Chap. 1; J. Kervorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Appl. Math. Ser. (Springer, New York, 1996), Vol. 114; R. E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Ser.; Appl. Math. Sci. (Springer, New York, 1991), Vol. 89.
8. For chemical applications see J. C. Giddings and H. K. Shin, Trans. Faraday Soc. 51, 468 (1961); H. K. Shin and J. C. Giddings, J. Phys. Chem. 65, 1164 (1961); J. R. Bowen, A. Acrivos, and A. K. Oppenheim, Chem. Eng. Sci. 18, 177 (1963); F. G. Heineken, H. M. Tsuchiya, and R. Aris, Math. Biosci. 1, 95 (1967); S. I. Rubinow and J. L. Lebowitz, J. Am. Chem. Soc. 92, 3888 (1970). For mathematical background see: A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974), Chaps. 9 and 10; J. D. Murray, Lectures on Nonlinear-Differential-Equation Models in Biology (Clarendon, Oxford, 1977), Chap. 1; J. Kervorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Appl. Math. Ser. (Springer, New York, 1996), Vol. 114; R. E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Ser.; Appl. Math. Sci. (Springer, New York, 1991), Vol. 89.
9. For chemical applications see J. C. Giddings and H. K. Shin, Trans. Faraday Soc. 51, 468 (1961); H. K. Shin and J. C. Giddings, J. Phys. Chem. 65, 1164 (1961); J. R. Bowen, A. Acrivos, and A. K. Oppenheim, Chem. Eng. Sci. 18, 177 (1963); F. G. Heineken, H. M. Tsuchiya, and R. Aris, Math. Biosci. 1, 95 (1967); S. I. Rubinow and J. L. Lebowitz, J. Am. Chem. Soc. 92, 3888 (1970). For mathematical background see: A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974), Chaps. 9 and 10; J. D. Murray, Lectures on Nonlinear-Differential-Equation Models in Biology (Clarendon, Oxford, 1977), Chap. 1; J. Kervorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Appl. Math. Ser. (Springer, New York, 1996), Vol. 114; R. E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Ser.; Appl. Math. Sci. (Springer, New York, 1991), Vol. 89.
10. For chemical applications see J. C. Giddings and H. K. Shin, Trans. Faraday Soc. 51, 468 (1961); H. K. Shin and J. C. Giddings, J. Phys. Chem. 65, 1164 (1961); J. R. Bowen, A. Acrivos, and A. K. Oppenheim, Chem. Eng. Sci. 18, 177 (1963); F. G. Heineken, H. M. Tsuchiya, and R. Aris, Math. Biosci. 1, 95 (1967); S. I. Rubinow and J. L. Lebowitz, J. Am. Chem. Soc. 92, 3888 (1970). For mathematical background see: A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974), Chaps. 9 and 10; J. D. Murray, Lectures on Nonlinear-Differential-Equation Models in Biology (Clarendon, Oxford, 1977), Chap. 1; J. Kervorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Appl. Math. Ser. (Springer, New York, 1996), Vol. 114; R. E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Ser.; Appl. Math. Sci. (Springer, New York, 1991), Vol. 89.
11. For chemical applications see J. C. Giddings and H. K. Shin, Trans. Faraday Soc. 51, 468 (1961); H. K. Shin and J. C. Giddings, J. Phys. Chem. 65, 1164 (1961); J. R. Bowen, A. Acrivos, and A. K. Oppenheim, Chem. Eng. Sci. 18, 177 (1963); F. G. Heineken, H. M. Tsuchiya, and R. Aris, Math. Biosci. 1, 95 (1967); S. I. Rubinow and J. L. Lebowitz, J. Am. Chem. Soc. 92, 3888 (1970). For mathematical background see: A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974), Chaps. 9 and 10; J. D. Murray, Lectures on Nonlinear-Differential-Equation Models in Biology (Clarendon, Oxford, 1977), Chap. 1; J. Kervorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Appl. Math. Ser. (Springer, New York, 1996), Vol. 114; R. E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Ser.; Appl. Math. Sci. (Springer, New York, 1991), Vol. 89.
12. For chemical applications see J. C. Giddings and H. K. Shin, Trans. Faraday Soc. 51, 468 (1961); H. K. Shin and J. C. Giddings, J. Phys. Chem. 65, 1164 (1961); J. R. Bowen, A. Acrivos, and A. K. Oppenheim, Chem. Eng. Sci. 18, 177 (1963); F. G. Heineken, H. M. Tsuchiya, and R. Aris, Math. Biosci. 1, 95 (1967); S. I. Rubinow and J. L. Lebowitz, J. Am. Chem. Soc. 92, 3888 (1970). For mathematical background see: A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974), Chaps. 9 and 10; J. D. Murray, Lectures on Nonlinear-Differential-Equation Models in Biology (Clarendon, Oxford, 1977), Chap. 1; J. Kervorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Appl. Math. Ser. (Springer, New York, 1996), Vol. 114; R. E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Ser.; Appl. Math. Sci. (Springer, New York, 1991), Vol. 89.
13. J. Carr, Applications of Center Manifold Theory, Appl. Math. Sci. (Springer, New York, 1981), Vol, 35. In Chap. 1 of Carr's book there is a discussion of an indirect way of obtaining the equation of motion on the slow manifold by augmenting the system of ODEs to make parameters formal variables. The system can then be separated into a finite eigenvalue set and a zero eigenvalue set corresponding to the slow (center) manifold.
14. J. E. Marsden and M. McCracken, The Hopf Bifurcation and its Applications, Appl. Math. Sciences (Springer, New York, 1976), Vol. 19.
15. A. N. Gorban and I. V. Karlin, Chem. Eng. Sci. 58, 4751 (2003).
16. N. Chafee, J. Diff. Eqns. 4, 661 (1968); J. Hale, Ordinary Differential Equations (Wiley, New York, 1969), especially Chap. 7; C, Foias, G. R. Sell, and R. Temam, J. Diff. Eqns. 73, 309 (1988); M. S. Jolly, ibid. 78, 220 (1989); A. N. Yannacopoulios, A. S. Tomlin, J. Brindley, J. H. Merkin, and M. J. Pilling, Physica D 83, 421 (1995).
17. N. Chafee, J. Diff. Eqns. 4, 661 (1968); J. Hale, Ordinary Differential Equations (Wiley, New York, 1969), especially Chap. 7; C, Foias, G. R. Sell, and R. Temam, J. Diff. Eqns. 73, 309 (1988); M. S. Jolly, ibid. 78, 220 (1989); A. N. Yannacopoulios, A. S. Tomlin, J. Brindley, J. H. Merkin, and M. J. Pilling, Physica D 83, 421 (1995).
18. N. Chafee, J. Diff. Eqns. 4, 661 (1968); J. Hale, Ordinary Differential Equations (Wiley, New York, 1969), especially Chap. 7; C, Foias, G. R. Sell, and R. Temam, J. Diff. Eqns. 73, 309 (1988); M. S. Jolly, ibid. 78, 220 (1989); A. N. Yannacopoulios, A. S. Tomlin, J. Brindley, J. H. Merkin, and M. J. Pilling, Physica D 83, 421 (1995).
19. N. Chafee, J. Diff. Eqns. 4, 661 (1968); J. Hale, Ordinary Differential Equations (Wiley, New York, 1969), especially Chap. 7; C, Foias, G. R. Sell, and R. Temam, J. Diff. Eqns. 73, 309 (1988); M. S. Jolly, ibid. 78, 220 (1989); A. N. Yannacopoulios, A. S. Tomlin, J. Brindley, J. H. Merkin, and M. J. Pilling, Physica D 83, 421 (1995).
20. N. Chafee, J. Diff. Eqns. 4, 661 (1968); J. Hale, Ordinary Differential Equations (Wiley, New York, 1969), especially Chap. 7; C, Foias, G. R. Sell, and R. Temam, J. Diff. Eqns. 73, 309 (1988); M. S. Jolly, ibid. 78, 220 (1989); A. N. Yannacopoulios, A. S. Tomlin, J. Brindley, J. H. Merkin, and M. J. Pilling, Physica D 83, 421 (1995).
21. U. Maas and S. B. Pope, Combust. Flame 88, 239 (1992).
22. S. J. Fraser, J. Chem. Phys. 88, 4732 (1988).
23. See, e.g., T. L. Saaty, Modern Nonlinear Equations (Dover, New York, 1981), for a general discussion of the convergence of iterative procedures.
24. A. C. Aitken, Proc. R. Soc. Edinburgh 46, 289 (1925); J. F. Steffensen, Skandinavisk Aktuarietidskrift 16, 64 (1933). The Aitken-Steffensen accelerated convergence method can also prevent divergence because it removes local exponential growth by annihilating the corresponding unstable eigenvector from the local linearization of the function space.
25. A. C. Aitken, Proc. R. Soc. Edinburgh 46, 289 (1925); J. F. Steffensen, Skandinavisk Aktuarietidskrift 16, 64 (1933). The Aitken-Steffensen accelerated convergence method can also prevent divergence because it removes local exponential growth by annihilating the corresponding unstable eigenvector from the local linearization of the function space.
26. M. R. Roussel and S. J. Fraser, J. Chem. Phys. 93, 1072 (1990). The accelerated convergence methods in Ref. 10 are used here.
27. n]/[1 +S(x)], related to the stabilized point mappings discussed in R. Thomas, J. Richelle, and R. d'Ari, Bull. Cl. Sci., Acad. R. Belg. 73, 62 (1987). The original and stabilized iteration schemes have the same fixed point but the stabilized scheme can converge to this fixed point even when the original scheme diverges. The extension of point mapping schemes to functional mapping schemes is discussed in G. D. Birkhoff and O. D. Kellogg, Trans. Am. Math. Soc. 23, 96 (1922). The functional equation method for finding the slow manifold, first used in Ref. 8, contains the unbounded differential operator on the rhs and this can lead to iterative instability.
28. n]/[1 +S(x)], related to the stabilized point mappings discussed in R. Thomas, J. Richelle, and R. d'Ari, Bull. Cl. Sci., Acad. R. Belg. 73, 62 (1987). The original and stabilized iteration schemes have the same fixed point but the stabilized scheme can converge to this fixed point even when the original scheme diverges. The extension of point mapping schemes to functional mapping schemes is discussed in G. D. Birkhoff and O. D. Kellogg, Trans. Am. Math. Soc. 23, 96 (1922). The functional equation method for finding the slow manifold, first used in Ref. 8, contains the unbounded differential operator on the rhs and this can lead to iterative instability.
29. See M. R. Roussel, J. Math. Chem. 21, 385 (1997) for an example of stabilization, by the method in Ref. 12, of the iterative scheme for an unstable case of mechanism (M). The method for choosing S(x) in Ref. 12 is discussed in the current reference.
30. M. R. Roussel and S. J. Fraser, J. Chem. Phys. 94, 7106 (1991).
31. M. J. Davis and R. T. Skodje, J. Chem. Phys. 111, 859 (1999).
32. H. G. Kaper and T. J. Kaper, Physica D 165, 66 (2002).
33. R. de la Llave (private communication) has found a model ODE that generates "exponentially" divergent series. The author suggested to R. de la Llave that the chemical example of bimolecular association mechanism (B) was likely to produce divergent series because of its separating nullclines.
34. See, e.g., K. J. Laidler, Chemical Kinetics, 3rd ed. (Harper and Row, New York, 1987), Chap. 6, for a discussion of many reactions in solution conforming to this simple bimolecular association model.
35. Since the last step in all the mechanisms discussed is irreversible, P denotes one or more products.
36. See, e.g., J. Hine, Physical Organic Chemistry (McGraw-Hill, New York, 1959), especially Chap. 17 on nucleophilic aromatic substitution; for specific cases see: J. F. Bunnett and R. E. Zahler, Chem. Rev. 49, 273 (1951); C. W. L. Sevan, J. Chem. Soc. 1951, 2340; G. S. Hammond and L. R. Parks, J. Am. Chem. Soc. 77, 340 (1955).
37. See, e.g., J. Hine, Physical Organic Chemistry (McGraw-Hill, New York, 1959), especially Chap. 17 on nucleophilic aromatic substitution; for specific cases see: J. F. Bunnett and R. E. Zahler, Chem. Rev. 49, 273 (1951); C. W. L. Sevan, J. Chem. Soc. 1951, 2340; G. S. Hammond and L. R. Parks, J. Am. Chem. Soc. 77, 340 (1955).
38. See, e.g., J. Hine, Physical Organic Chemistry (McGraw-Hill, New York, 1959), especially Chap. 17 on nucleophilic aromatic substitution; for specific cases see: J. F. Bunnett and R. E. Zahler, Chem. Rev. 49, 273 (1951); C. W. L. Sevan, J. Chem. Soc. 1951, 2340; G. S. Hammond and L. R. Parks, J. Am. Chem. Soc. 77, 340 (1955).
39. See, e.g., J. Hine, Physical Organic Chemistry (McGraw-Hill, New York, 1959), especially Chap. 17 on nucleophilic aromatic substitution; for specific cases see: J. F. Bunnett and R. E. Zahler, Chem. Rev. 49, 273 (1951); C. W. L. Sevan, J. Chem. Soc. 1951, 2340; G. S. Hammond and L. R. Parks, J. Am. Chem. Soc. 77, 340 (1955).
40. V. Henri, C. R. Acad. Sci. (Paris) 135, 916 (1902); Lois Générales de l'Action des Diastases (Hermann, Paris, 1903); L. Michaelis and M. L. Menten, Biochem. Z. 49, 333 (1913).
41. V. Henri, C. R. Acad. Sci. (Paris) 135, 916 (1902); Lois Générales de l'Action des Diastases (Hermann, Paris, 1903); L. Michaelis and M. L. Menten, Biochem. Z. 49, 333 (1913).
42. V. Henri, C. R. Acad. Sci. (Paris) 135, 916 (1902); Lois Générales de l'Action des Diastases (Hermann, Paris, 1903); L. Michaelis and M. L. Menten, Biochem. Z. 49, 333 (1913).
43. F. A. Lindemann, Trans. Faraday Soc. 17, 598 (1922).
44. J. A. Christiansen, Ph.D. thesis, University of Copenhagen, 1921.
45. See, e.g., Ref. 18, especially Sec. 5.3 for a discussion of unimolecular reactions.
46. S. J. Fraser, J. Chem. Phys. 109, 411 (1998). Formal two-parameter series methods are discussed here.
47. Some of the slow-manifold equations can even be reduced to a single parameter equation.
48. The trajectory equations and functional equations can all be written in the same generic form, so that only the detailed functional differences determine their mathematical behavior.
49. In general, the term "equilibrium" is used to mean a point at which the velocity field of a system of ODEs vanishes; there is only one (stable) equilibrium in the positive quadrant of the phase plane.
50. For a discussion of contraction mappings see e.g., Ref. 9; P. Roman, Some Modern Mathematics for Physicists and Other Outsiders, (Pergamon, New York, 1975), Vol. I, pp. 280-289; P. Henrici, Elements of Numerical Analysis (Wiley, New York, 1964); G. F. Simmons, Introduction to Topology and Modern Analysis (McGraw-Hill, London, 1963). Contraction mapping in relation to the functional equations used here has been previously discussed in Ref. 8.
51. For a discussion of contraction mappings see e.g., Ref. 9; P. Roman, Some Modern Mathematics for Physicists and Other Outsiders, (Pergamon, New York, 1975), Vol. I, pp. 280-289; P. Henrici, Elements of Numerical Analysis (Wiley, New York, 1964); G. F. Simmons, Introduction to Topology and Modern Analysis (McGraw-Hill, London, 1963). Contraction mapping in relation to the functional equations used here has been previously discussed in Ref. 8.
52. For a discussion of contraction mappings see e.g., Ref. 9; P. Roman, Some Modern Mathematics for Physicists and Other Outsiders, (Pergamon, New York, 1975), Vol. I, pp. 280-289; P. Henrici, Elements of Numerical Analysis (Wiley, New York, 1964); G. F. Simmons, Introduction to Topology and Modern Analysis (McGraw-Hill, London, 1963). Contraction mapping in relation to the functional equations used here has been previously discussed in Ref. 8.
53. What is true of the planar flows studied here is also true of the parent four-dimensional phase flows since the eliminated dynamical variables can be recovered from the two linear constants of the motion.
54. Iteration of the functional equation can occasionally be unstable and result in local buckling of the slow manifold; such unstable iteration schemes can be stabilized using the schemes in Refs. 10 and 12.
55. See, e.g., A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956); J. D. Murray, Asymptotic Analysis, Ser.: Appl. Math. Sci. (Springer, New York, 1984), Vol. 48.
56. See, e.g., A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956); J. D. Murray, Asymptotic Analysis, Ser.: Appl. Math. Sci. (Springer, New York, 1984), Vol. 48.
57. A. H. Nguyen and S. J. Fraser, J. Chem. Phys. 91, 186 (1989).
58. -1 is a dimensionless decay constant.
59. M(x) are both positive branches of a hyperbola and since the phase flow is into the region bounded by these nullclines and there is a unique equilibrium at x=y = 0, this implies the existence of a slow manifold lying between the nullclines and terminating at the origin.
60. See M. R. Roussel, M.Sc. thesis, University of Toronto, 1990, for a discussion of the equivalence of order-by-order expansion and the functional iteration method for mechanism (M).
61. M(x;ε, η) = 0(x), gives the slope of the slow manifold for mechanism (M) at the origin.
62. See Ref. 13 for a discussion of the asymptotic expansion for mechanism (M); see also M. R. Roussel, Ph.D. thesis, University of Toronto, 1994.
63. k-1 is a dimensionless decay constant.
64. The center manifold is stable because the equilibrium is a stable node.
65. Like mechanism (M) (see Ref. 34), and mechanism (L) (see Ref. 39), mechanisms (B.s) and (B.n) can be written so that, chemically speaking, η is a quasiequilibrium constant and ε governs the probabilities for the two decay pathways of the intermediate complex.
66. T, are orthogonal.
67. Every solution of the trajectory Eq. (29) that starts from a point above the EA for mechanism (B.s) has (square-root) singularity on the EA, apart from the solution corresponding to the slow manifold itself, which is globally bounded from above by the EA nullcline. Thus Eq. (29) possesses a continuum of such movable singularities. Indeed, the trajectory equation for every mechanism discussed in this paper has this property.
68. 2) nullclines at sufficiently large x.
69. n.
70. There is no evidence of a buckling instability (see Ref. 31) in the iteration of Eq. (35) for mechanism (B.s), at least for a wide range of parameter values. This absence of instability occurs because of the small curvature of the slow manifold for mechanism (B.s).
71. With respect to the planar flows discussed here, stiffness of the ODEs does not present a problem in the convergence of the iterative form of the slow-manifold functional equation (35). The phase flow always moves into the confining region defined by the SSA and EA nullclines and the slow manifold can be found by iteration. All phase portraits appearing in figures in this paper correspond to a large value of the stiffness parameter, η= 1, rather than small values, 0<≪1, near the singular perturbation limit, η=0. Nevertheless, the phase portraits clearly indicate the existence of a slow manifold. As noted in Ref. 8 small values of the perturbation parameters are not necessary for the existence of a distinct slow manifold. However, in two-variable, kinetically controlled reactions a slow manifold need not exist, as discussed in S. J. Fraser, J. Chem. Phys. 116, 1277 (2002). In three-dimensional systems of ODEs, the corresponding phase flow may clearly show intrinsic lack of stiffness if two real negative eigenvalues on a low-dimensional invariant manifold coalesce on the negative real axis and then separate into a complex conjugate pair; an example of this behavior in inhibited enzyme systems is examined in M. R. Roussel and S. J. Fraser, J. Phys. Chem. 97, 8316 (1993).
72. With respect to the planar flows discussed here, stiffness of the ODEs does not present a problem in the convergence of the iterative form of the slow-manifold functional equation (35). The phase flow always moves into the confining region defined by the SSA and EA nullclines and the slow manifold can be found by iteration. All phase portraits appearing in figures in this paper correspond to a large value of the stiffness parameter, η= 1, rather than small values, 0<η≪1, near the singular perturbation limit, η=0. Nevertheless, the phase portraits clearly indicate the existence of a slow manifold. As noted in Ref. 8 small values of the perturbation parameters are not necessary for the existence of a distinct slow manifold. However, in two-variable, kinetically controlled reactions a slow manifold need not exist, as discussed in S. J. Fraser, J. Chem. Phys. 116, 1277 (2002). In three-dimensional systems of ODEs, the corresponding phase flow may clearly show intrinsic lack of stiffness if two real negative eigenvalues on a low-dimensional invariant manifold coalesce on the negative real axis and then separate into a complex conjugate pair; an example of this behavior in inhibited enzyme systems is examined in M. R. Roussel and S. J. Fraser, J. Phys. Chem. 97, 8316 (1993).
73. S, must be used as the iteration starting function.
74. See, e.g., G. H. Hardy, Divergent Series (AMS Chelsea, Providence, Rhode Island, 1991) regarding the theory of the summability of divergent series.
75. In principle the concentration difference ±Δ can be used in the scaling, giving ±1 in the scaled equations. For the positively signed case the system equilibrium lies at the origin, (x,y) = (0,0), whereas, for the case with a negative sign the system equilibrium is at (x,y) = (1,0). Consequently, the positively signed case leads to the simpler interpretation.
76. -(<0), which diverges to -∞ as η→0+.
77. 2.
78. The remarks made about buckling in Ref. 46 also apply to iteration of Eq. (49) for mechanism (B.n).
79. The dynamical variable x and the parameters η, and ε are included in this section to clarify the structure of the equations, expansions, etc.
80. j, at η=0.