The role of slow system dynamics in predicting the degeneracy of slow invariant manifolds: The case of vdP relaxation-oscillations

Physica D: Nonlinear Phenomena

Volumn 248, Issue 1, 2013, Pages 16-32

  Goussis, Dimitris A ^a  

^a NATIONAL TECHNICAL UNIVERSITY OF ATHENS   (Greece)

Author keywords

Dynamical systems; Model reduction; Relaxation oscillations; Singular perturbations; van der Pol

Indexed keywords

ASYMPTOTIC ANALYSIS; DYNAMICAL SYSTEMS; FORECASTING; PERTURBATION TECHNIQUES; TIME MEASUREMENT;

FAST AND SLOW DYNAMICS; HIGHER ORDER EFFECTS; MODEL REDUCTION; RELAXATION OSCILLATION; SINGULAR PERTURBATION TECHNIQUE; SINGULAR PERTURBATIONS; SLOW INVARIANT MANIFOLDS; VAN DER POL;

EXPLOSIVES;

EID: 84874508779     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2012.12.013     Document Type: Article

References (59)

1. N. Fenichel Geometric singular perturbation theory for ordinary differential equations J. Differential Equations 31 1971 53 78
2. T.J. Kaper Jr. An introduction to geometric methods and dynamical systems theory for singular perturbation problems J. Cronin, R.E. O' Malley, Analyzing Multiscale Phenomena Using Singular Perturbation Methods, Proceedings of Symposia in Appl. Mathematics 1999 American Mathematical Society Rhode Island 85 132
3. S.H. Lam, and D.A. Goussis Understanding complex chemical kinetics with computational singular perturbation Proc. Comb. Inst. 22 1988 931 941
4. S.H. Lam, and D.A. Goussis Conventional asymptotics and computational singular perturbation for simplified kinetics modeling M.O. Smooke, Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane-air Flames Springer Lecture Notes vol. 384 1991 Springer Berlin 227 242
5. D.A. Goussis, and S.H. Lam A study of homogeneous methanol oxidation kinetic using CSP Proc. Comb. Inst. 24 1992 113 120
6. M. Valorani, D.A. Goussis, F. Creta, and H.N. Najm CSP analysis of a transient flame-vortex interaction: time scales and manifolds Combust. Flame 134 2003 35 53
7. D.A. Goussis, and H.N. Najm Model reduction and physical understanding of slowly oscillating processes: the circadian cycle SIAM Multiscale Model. Simul. 5 2006 1297 1332
8. J.C. Lee, H.N. Najm, S. Lefantzi, J. Ray, M. Frenklach, M. Valorani, and D.A. Goussis A CSP and tabulation-based adaptive chemistry model Combust. Theory Model. 11 2007 73 102
9. A.N. Tikhonov Systems of differential equations containing a small parameter multiplying the derivative Mat. Sb. 31 1952 575 586
10. N. Levinson Small periodic perturbations of an autonomous system with a stable orbit Ann. of Math. 52 1950 727 738
11. A.B. Vassil'eva Asymptotic behavior of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives Russian Math. Surveys 18 1963 13 84
12. R.E. O' Malley Jr. Introduction to Asymptotics and Special Functions 1974 Academic Press New York
13. M. Hirsch, C. Pugh, and M. Shub Invariant Manifolds Lecture Notes in Mathematics vol. 583 1977 Springer-Verlag Berlin
14. K. Nipp Invariant manifolds of singularly perturbed ordinary differential equations Z. Angew. Math. Phys. 36 1985 309 320
15. C.K.R.T. Jones Geometric singular perturbation theory R. Johnson, Dynamical Systems Lecture Notes in Mathematics vol. 1609 1995 Springer Verlag Berlin 44 118
16. F. Verhulst Periodic Solutions and Slow Manifolds Int. J. of Bifurcation and Chaos 17 2007 2533 2540
17. R.E. O' Malley Jr. Singular Perturbation Methods for Ordinary Differential Equations Applied Mathematical Sciences vol. 89 1991 Springer Verlag New York
18. J. Kevorkian, and J.D. Cole Perturbation Methods in Applied Mathematics 1980 Springer Verlag New York
19. P.A. Lagerstrom Matched Asymptotic Expansions 1980 Springer Verlag New York
20. J. Grasman Asymptotic Methods for Relaxation Oscillations and Applications 1987 Springer-Verlag New York
21. V.N. Bogaevski, and A. Povzner Algebraic Methods in Nonlinear Perturbation Theory 1991 Springer Verlag New York
22. F. Verhulst Methods and Applications of Singular Perturbations 2000 Springer Verlag New York
23. L.A. Segel, and M. Slemrod The quasi-steady-state assumption: a case study in perturbation SIAM Rev. 31 1989 446 477
24. M. Hadjinicolaou, and D.A. Goussis Asymptotic solution of stiff PDEs with the CSP method: the reaction diffusion equation SIAM J. Sci. Comput. 20 1998 781 810
25. D.A. Goussis, and U. Maas Model reduction for combustion chemistry T. Echekki, E. Mastorakos, Turbulent Combustion Modeling 2011 Springer Berlin 193 220
26. D.A. Goussis Quasi steady state and partial equilibrium approximations: their relation and their validity Combust. Theory Model. 16 2012 869 926
27. M.R. Roussel, and S.J. Fraser Geometry of the steady-state approximation: perturbation and accelerated convergence methods J. Chem. Phys. 93 1990 1072 1081
28. U. Maas, and S.B. Pope Simplifying chemical kinetics - intrinsic low dimensional manifolds in composition space Combust. Flame 88 1992 239 264
29. A.N. Gorban, and I.V. Karlin Method of invariant manifold for chemical kinetics Chem. Eng. Sci. 58 2003 4751 4768
30. C.W. Gear, and I.G. Kevrekidis Constraint-defined manifolds: a legacy code approach to low-dimensional computation J. Scientific Computing 25 2005 17 28
31. M.N. Contou-Carrere, and P. Daoutidis Model reduction and control of multi-scale reaction-convection processes Chem. Eng. Sci. 63 2008 4012 4025
32. Terese Løvås Model Reduction Techniques for Chemical Mechanisms V. Patel, Chemical Kinetics 2012 InTech Croatia 79 114
33. M. Valorani, D.A. Goussis, F. Creta, and H.N. Najm Higher order corrections in the approximation of low dimensional manifolds and the construction of simplified problems with the CSP method J. Comput. Phys. 209 2005 754 786
34. A. Zagaris, H.G. Kaper, and T.J. Kaper Two perspectives on reduction of ordinary differential equations Math. Nachr. 278 2005 1629 1642
35. D.A. Goussis, and M. Valorani An efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems J. Comput. Phys. 214 2006 316 346
36. H.N. Najm, M. Valorani, D.A. Goussis, and J. Prager Analysis of methane-air edge flame Combust. Theory Model. 14 2010 257 294
37. C. Kuehn A mathematical framework for critical transitions: bifurcations, fast-slow systems and stochastic systems Physica D 240 2011 1020 1035
38. P.E. Phillipson, and P. Schuster Modeling by Nonlinear Differential Equations 2009 World Scientific Singapore
39. E.F. Mishchenko, and N.Kh. Rozov Differential Equations with Small Parameters and Relaxation Oscillations 1980 Plenum Press New York
40. B. van der Pol A theory of the amplitude of free and forced triode vibrations Radio Review 1 1920 701 710
41. J. Guckenheimer, and P. Holmes Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields 1983 Springer Verlag New York
42. A.D. MacGillivray On the leading term of the inner asymptotic expansion of Van Der Pol's equation SIAM J. Appl. Math. 43 1983 594 612
43. A.D. MacGillivray On the leading term of the outer asymptotic expansion of Van Der Pol's equation SIAM J. Appl. Math. 43 1983 1221 1239
44. M. Krupa, and P. Szmolyan Extending geometric singular perturbation theory to nonhyperbolic points - fold and canard points in two dimensions SIAM J. Math. Anal. 33 2001 286 314
45. M. Krupa, and P. Szmolyan Relaxation Oscillation and canard explosion J. Differential Equations 174 2001 312 368
46. A.H. Nayfeh, and D.T. Mook Nonlinear Oscillations 1979 Wiley New York
47. A. Zagaris, H.G. Kaper, and T.J. Kaper Fast and slow dynamics for the computational singular perturbation method Multiscale Model. Simul. 2 2004 613 638
48. D.A. Goussis, M. Valorani, F. Creta, and H.N. Najm Reactive and reactive-diffusive time scales in stiff reaction-diffusion systems Prog. Comput. Fluid Dyn. 5 2005 316 326
49. P.D. Kourdis, R. Steuer, and D.A. Goussis Physical understanding of complex multiscale biochemical models via algorithmic simplification: Glycolysis in sacharomyces cerevisiae Physica D 239 2010 1798 1817
50. F. Verhulst Nonlinear Differential Equations and Dynamical Systems Universitext 2000 Springer Verlag Berlin and New York
51. S.H. Lam Using CSP to understand complex chemical kinetics Combust. Sci. Tech. 89 1993 375 404
52. S.H. Lam, and D.A. Goussis The CSP method for simplifying kinetics International Journal of Chemical Kinetics 26 1994 461 486
53. C.M. Bender, and S.A. Orszag Advanced Mathematical Methods for Scientists and Engineers 1978 McGraw-Hill New York
54. N.N. Semenov Zur Theorie des Verbrennungsprozesses Z.fur Physik. 48 1928 571 582
55. I. Goldfarb, V. Gol'dshtein, and U. Maas Comparative analysis of two asymptotic approaches based on integral manifolds IMA J. Appl. Math. 69 2004 353 374
56. O.E. Rössler An equation for continuous chaos Phys. Lett. 57A 1976 397 398
57. L.M. Pecora, and T.L. Carroll Synchronization in chaotic systems Phys. Rev. Lett. 64 1990 821 824
58. F.A. Williams, Combustion Theory, Benjamin/Cummings, Menlo Park, 1985.
59. K.J. Laidler Chemical Kinetics 1950 McGraw-Hill New York