Young measure approach to computing slowly advancing fast oscillations

Multiscale Modeling and Simulation

Volumn 6, Issue 4, 2007, Pages 1085-1097

  Artstein, Zvi ^a   Linshiz, Jasmine ^a   Titi, Edriss S ^a,b  

^a WEIZMANN INSTITUTE OF SCIENCE   (Israel)

^b University of California at Irvine   (United States)

Author keywords

Limit cycles; Multiscale computation; Singular perturbations; Young measures

Indexed keywords

PERTURBATION TECHNIQUES;

FAST DYNAMICS; FAST OSCILLATIONS; LIMIT CYCLES; MULTISCALE; MULTISCALE COMPUTATION; NUMERICAL EXAMPLES; SINGULAR PERTURBATIONS; SINGULARLY PERTURBED SYSTEMS; SLOW DYNAMICS; YOUNG MEASURES;

DYNAMICS;

EID: 55149104186     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/070687219     Document Type: Article

References (29)

1. E.L. ALLGOWER and K. GEORG, Introduction to Numerical Continuation Methods, SIAM, Philadelphia, 2003.
2. Z. ARTSTEIN, On singularly perturbed ordinary differential equations with measure-valued limits, Math. Bohem., 127 (2002), pp. 139-152.
3. Z. ARTSTEIN and M. SLEMROD, The singular perturbation limit of an elastic structure in a rapidly flowing inviscid fluid, Quart. Appl. Math., 59 (2001), pp. 543-555.
4. Z. ARTSTEIN and M. SLEMROD, On singularly perturbed retarded functional differential equations, J. Differential Equations, 171 (2001), pp. 88-109.
5. Z. ARTSTEIN and A. VIGODNER, Singularly perturbed ordinary differential equations with dynamic limits, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), pp. 541-569.
6. R. ASHINO, M. NAGASE, and R. VAILLANCOURT, Behind and beyond the MATLAB ODE suite, Comput. Math. Appl., 40 (2000), pp. 491-512.
7. P. BILLINGSLEY, Convergence of Probability Measures, Wiley, New York, 1968.
8. N.N. BOGOLIUBOV and Y.A. MITROPOLSKY, Asymptotic Methods in the Theory of Non-Linear Oscillations, Hindustan, Delhi, 1961.
9. R.N. BRACEWELL, The Fourier Transform and Its Applications, 3rd ed., McGraw-Hill, Boston, 1986.
10. A. DHOOGE, W. GOVAERTS, and YU.A. KUZNETSOV, MATHCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2003), pp. 141-164.
11. A. DHOOGE, W. GOVAERTS, and YU.A. KUZNETSOV, Numerical continuation of fold bifurcation of limit cycles in MATCONT, in Computational Science-ICCS 2003. Part I, Lecture Notes in Comput. Sci. 2657, Springer, Berlin, 2003, pp. 701-710.
12. E.J. DOEDEL, A.R. CHAMPNEYS, T.F. FAIRGRIEVE, YU. A. KUZNETSOV, B. SANDSTEDE, and X.J. WANG, AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations, Concordia University, Montreal QC, Canada, 1997-2000.
13. E.H. DOWELL and M. ILGAMOV, Studies in Nonlinear Aeroelasticity, Springer-Verlag, New York, 1988.
14. W.J.F. GOVAERTS, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, 2000.
15. W.D. IWAN and R.D. BELVINS, A model for vortex induced oscillation of structure, J. Appl. Mech., 41 (1974), pp. 581-586.
16. YU.A. KUZNETSOV, Elements of Applied Bifurcation Theory, 3rd ed., Springer-Verlag, New York, 2004.
17. YU.A. KUZNETSOV and V.V. LEVITIN, Content: Integrated Environment for Analysis of Dynamical Systems, CWI, Amsterdam, 1997.
18. W.L. MIRANKER, Numerical Methods for Stiff Equations and Singular Perturbation Problems, D. Reidel, Dordrecht, Boston, 1981.
19. R.E. O'MALLEY, JR., Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991.
20. A.V. OPPENHEIM, R.W. SCHAFER, and J. R. BUCK, Discrete-Time Signal Processing, 2nd ed., Prentice-Hall, Upper Saddle River, NJ, 1999.
21. P. PEDREGAL, Optimization, relaxation and Young measures, Bull. Amer. Math. Soc. (N.S.), 36 (1999), pp. 27-58.
22. L.S. PONTRYAGIN and L.V. RODYGIN, Approximate solution of a system of ordinary differential equations involving a small parameter in the derivatives, Dokl. Akad. Nauk SSSR, 131 (1960), pp. 255-258 (in Russian); Soviet Math. Dokl., 1 (1960), pp. 237-240 (in English).
23. J.G. PROAKIS and D.G. MANOLAKIS, Digital Signal Processing: Principles, Algorithms and Applications, 3rd ed., Prentice-Hall, Upper Saddle River, NJ, 1996.
24. J.A. SANDERS and F. VERHULST, Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York, 1985.
25. L.F. SHAMPINE and M.K. GORDON, Computer Solutions of Ordinary Differential Equations: The Initial Value Problem, W.H. Freeman, San Francisco, 1975.
26. L.F. SHAMPINE and M.W. REICHELT, The MATLAB ODE suite, SIAM J. Sci. Comput., 18 (1997), pp. 1-22.
27. D.R. SMITH, Singular-Perturbation Theory, Cambridge University Press, Cambridge, UK, 1985.
28. A.N. TIKHONOV, A.B. VASILÉVA, and A.G. SVESHNIKOV, Differential Equations, Springer-Verlag, Berlin, 1985.
29. E. VANDEN-EIJNDEN, Numerical techniques for multi-scale dynamical systems with stochastic effects, Commun. Math. Sci., 1 (2003), pp. 385-391.