Previous |  Up |  Next

Article

Keywords:
singular perturbations; invariant measures; slow and fast motions
Summary:
The limit behaviour of solutions of a singularly perturbed system is examined in the case where the fast flow need not converge to a stationary point. The topological convergence as well as information about the distribution of the values of the solutions can be determined in the case that the support of the limit invariant measure of the fast flow is an asymptotically stable attractor.
References:
[1] Z. Artstein: Stability in the presence of singular perturbations. Nonlinear Analysis 34 (1998), 817–827. MR 1636588 | Zbl 0948.34029
[2] Z. Artstein: Singularly perturbed ordinary differential equations with nonautonomous fast dynamics. J. Dynamics Differential Equations 11 (1999), 297–318. MR 1695247 | Zbl 0937.34044
[3] Z. Artstein, M. Slemrod: The singular perturbation limit of an elastic structure in a rapidly flowing invicid fluid. Q. Appl. Math. 59 (2000), 543–555. MR 1848534
[4] Z. Artstein, M. Slemrod: On singularly perturbed retarded functional differential equations. J. Differential Equations 171 (2001), 88–109. MR 1816795
[5] Z. Artstein, A. Vigodner: Singularly perturbed ordinary differential equations with dynamic limits. Proceedings of the Royal Society of Edinburgh 126A (1996), 541–569. MR 1396278
[6] P. Billingsley: Convergence of Probability Measures. Wiley, New York, 1968. MR 0233396 | Zbl 0172.21201
[7] W. E. Boyce, R. C. Diprima: Elementary Differential Equations and Boundary Value Problems (2nd Edition). Wiley, New York, 1969. MR 0179403
[8] N. Kriloff, N. Bogoliuboff: La théorie générale de la mesure dans son application à l’etude des systèmes dynamiques de la mécanique non linéaire. Ann. Math. 38 (1937), 65–113.
[9] R. E. O’Malley, Jr.: Singular Perturbation Methods for Ordinary Differential Equations. Springer, New York, 1991. MR 1123483
[10] A. N. Tikhonov: Systems of differential equations containing small parameters in the derivative. Mat. Sbornik N. S. 31 (1952), 575–586. MR 0055515
[11] A. N. Tikhonov, A. B. Vasileva, A. G. Sveshnikov: Differential Equations. Springer, Berlin, 1985. MR 0788499
[12] T. Ura: On the flow outside a closed invariant set; stability, relative stability and saddle sets. Contrib. Differential Equations 3 (1964), 249–294. MR 0163018 | Zbl 0161.28803
[13] W. Wasow: Asymptotic Expansions for Ordinary Differential Equations. Wiley Interscience, New York, 1965. MR 0203188 | Zbl 0133.35301
[14] T. Yoshizawa: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. Springer, New York, 1975. MR 0466797 | Zbl 0304.34051
Partner of
EuDML logo