Krylov-Bogolyubov averaging of asymptotically autonomous differential equations

Authors:
Anatoliy Samoilenko, Manuel Pinto and Sergei Trofimchuk

Journal:
Proc. Amer. Math. Soc. **133** (2005), 145-154

MSC (2000):
Primary 34E05

Published electronically:
June 23, 2004

MathSciNet review:
2085163

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We apply the Krylov and Bogolyubov asymptotic integration procedure to asymptotically autonomous systems. First, we consider linear systems with quasi-periodic coefficient matrix multiplied by a scalar factor vanishing at infinity. Next, we study the asymptotically autonomous Van-der-Pol oscillator.

**1.**V. I. Arnold,*Mathematical methods of classical mechanics*, Springer-Verlag, New York-Heidelberg, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein; Graduate Texts in Mathematics, 60. MR**0690288****2.**F. V. Atkinson,*On asymptotically periodic linear systems*, J. Math. Anal. Appl.**24**(1968), 646–653. MR**0241756****3.**N. N. Bogolyubov and Ju. A. Mitropol′skiĭ,*Asimptoticheskie metody v teorii nelineinykh kolebanii*, Izdat. “Nauka”, Moscow, 1974 (Russian). Fourth edition, revised and augmented. MR**0374550****4.**V. Sh. Burd and V. A. Karakulin,*Asymptotic integration of systems of linear differential equations with oscillatorily decreasing coefficients*, Mat. Zametki**64**(1998), no. 5, 658–666 (Russian, with Russian summary); English transl., Math. Notes**64**(1998), no. 5-6, 571–578 (1999). MR**1691208**, 10.1007/BF02316281**5.**J. S. Cassell,*The asymptotic integration of some oscillatory differential equations*, Quart. J. Math. Oxford Ser. (2)**33**(1982), no. 131, 281–296. MR**668174**, 10.1093/qmath/33.3.281**6.**M. S. P. Eastham,*The asymptotic solution of linear differential systems*, London Mathematical Society Monographs. New Series, vol. 4, The Clarendon Press, Oxford University Press, New York, 1989. Applications of the Levinson theorem; Oxford Science Publications. MR**1006434****7.**M. S. P. Eastham,*The number of resonant states in perturbed harmonic oscillation*, Quart. J. Math. Oxford Ser. (2)**42**(1991), no. 165, 49–55. MR**1094341**, 10.1093/qmath/42.1.49**8.**W. A. Harris Jr. and D. A. Lutz,*A unified theory of asymptotic integration*, J. Math. Anal. Appl.**57**(1977), no. 3, 571–586. MR**0430436****9.**William A. Harris Jr. and Yasutaka Sibuya,*Asymptotic behaviors of solutions of a system of linear ordinary differential equations as 𝑡→∞*, Delay differential equations and dynamical systems (Claremont, CA, 1990), Lecture Notes in Math., vol. 1475, Springer, Berlin, 1991, pp. 210–217. MR**1132032**, 10.1007/BFb0083493**10.**Norman Levinson,*The asymptotic nature of solutions of linear systems of differential equations*, Duke Math. J.**15**(1948), 111–126. MR**0024538****11.**Konstantin Mischaikow, Hal Smith, and Horst R. Thieme,*Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions*, Trans. Amer. Math. Soc.**347**(1995), no. 5, 1669–1685. MR**1290727**, 10.1090/S0002-9947-1995-1290727-7**12.**Sergei Trofimchuk and Manuel Pinto,*𝐿_{𝑝}-perturbations of invariant subbundles for linear systems*, J. Dynam. Differential Equations**14**(2002), no. 4, 743–761. MR**1940101**, 10.1023/A:1020804409250**13.**A. M. Samoĭlenko,*Elements of the mathematical theory of multi-frequency oscillations*, Mathematics and its Applications (Soviet Series), vol. 71, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the 1987 Russian original by Yuri Chapovsky. MR**1172117****14.**Ju. A. Samohin and V. N. Fomin,*A method for studying the stability of the oscillations of linear systems that are subject to the action of parametric loads with continuous spectrum*, Sibirsk. Mat. Ž.**17**(1976), no. 4, 926–931 (Russian). MR**0422781****15.**M. M. Skriganov,*The eigenvalues of the Schrödinger operator that are located on the continuous spectrum*, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)**38**(1973), 149–152 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 7. MR**0336111****16.**I. Z. Shtokalo,*Linear differential equations with variable coefficients. (Criteria of stability and unstability of their solutions.)*, Translated from Russian, Hindustan Publishing Corpn., Delhi, 1961; Gordon and Breach Science Publishers, Inc., New York, 1961. MR**0156046**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
34E05

Retrieve articles in all journals with MSC (2000): 34E05

Additional Information

**Anatoliy Samoilenko**

Affiliation:
Institute of Mathematics, National Academy of Sciences, Tereshchenkyvs’ka str., 3, Kiev, 252601, Ukraine

Email:
sam@imath.kiev.ua

**Manuel Pinto**

Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile

Email:
pintoj@uchile.cl

**Sergei Trofimchuk**

Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile

Address at time of publication:
Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile

Email:
trofimch@uchile.cl

DOI:
http://dx.doi.org/10.1090/S0002-9939-04-07520-3

Keywords:
Asymptotic integration,
asymptotically autonomous equation,
Levinson theorem,
Krylov-Bogolyubov averaging principle,
Van-der-Pol oscillator,
adiabatic oscillator

Received by editor(s):
May 7, 2002

Received by editor(s) in revised form:
September 9, 2003

Published electronically:
June 23, 2004

Additional Notes:
The first author was supported in part by FONDECYT (Chile), project 7960723

The second and third authors were supported in part by FONDECYT (Chile), project 8990013

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2004
American Mathematical Society