KrylovBogolyubov averaging of asymptotically autonomous differential equations
Authors:
Anatoliy Samoilenko, Manuel Pinto and Sergei Trofimchuk
Journal:
Proc. Amer. Math. Soc. 133 (2005), 145154
MSC (2000):
Primary 34E05
Published electronically:
June 23, 2004
MathSciNet review:
2085163
Fulltext 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 quasiperiodic coefficient matrix multiplied by a scalar factor vanishing at infinity. Next, we study the asymptotically autonomous VanderPol oscillator.
 1.
V.I. Arnold, Mathematical methods of classical mechanics, Springer, 1978. MR 57:14033b
 2.
F.V. Atkinson, `On asymptotically periodic linear systems', J. Math. Anal. Appl. 24 (1968), 646653. MR 39:3095
 3.
N.N. Bogolyubov, Yu.A. Mitropolskii, Asymptotic methods in the theory of nonlinear oscillations, (in Russian), Fourth edition, Nauka, Moscow, 1974. MR 51:10750
 4.
V. Sh. Burd and V.A. Karakulin, `Asymptotic integration of systems of linear differential equations with oscillatorily decreasing coefficients', Math. Notes 64 (1999), 571578.MR 2000a:34097
 5.
J.S. Cassell, `The asymptotic integration of some oscillatory differential equations', Quart. J. Math. Oxford Ser. (2) 33 (1982), 281296. MR 84c:34079
 6.
M.S.P. Eastham, The asymptotic solution of linear differential systems, London Mathematical Society Monographs, Clarendon Press, Oxford, 1989. MR 91d:34001
 7.
M.S.P. Eastham, `The number of resonant states in perturbed harmonic oscillation', Quart. J. Math. Oxford(2) 42 (1991), 4955. MR 92d:34075
 8.
W.A. Harris and D.A. Lutz, `A unified theory of asymptotic integration,' J. Math. Anal. Appl. 57 (1977), no. 3, 571586. MR 55:3441
 9.
W.A. Harris and Y. Sibuya, `Asymptotic behaviors of solutions of a system of linear ordinary differential equations as ', Lecture Notes in Math., 1475, Springer, 1991, pp. 210217. MR 93k:34119
 10.
N. Levinson, `The asymptotic nature of solutions of linear differential equations', Duke Math. J., 15 (1958), 111126. MR 9:509h
 11.
K. Mischaikow, H. Smith and H. Thieme, `Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions', Trans. Amer. Math. Soc., 347 (1995), 16691685. MR 95h:34063
 12.
S. Trofimchuk and M. Pinto, `perturbations of invariant subbundles for linear systems', Journal of Dynamics and Differential Equations, 14 (2002), 743761. MR 2003m:34116
 13.
A.M. Samoilenko, Elements of the mathematical theory of multifrequency oscillations, Kluwer, Dordrecht, 1991, 313 pp. MR 93d:34050
 14.
Yu. A. Samokhin 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', Siberian Math. J. 17 (1976), 926931.MR 54:10767
 15.
M.M. Skriganov, `The eigenvalues of the Schrödinger operator situated on the continuous spectrum', J. Soviet Math. 8 (1977), 464467. MR 49:887
 16.
I.Z. Shtokalo, Linear differential equations with variable coefficients, Gordon and Breach Science Publishers, Inc., New York, 1961, 100 pp.MR 27:5979
 1.
 V.I. Arnold, Mathematical methods of classical mechanics, Springer, 1978. MR 57:14033b
 2.
 F.V. Atkinson, `On asymptotically periodic linear systems', J. Math. Anal. Appl. 24 (1968), 646653. MR 39:3095
 3.
 N.N. Bogolyubov, Yu.A. Mitropolskii, Asymptotic methods in the theory of nonlinear oscillations, (in Russian), Fourth edition, Nauka, Moscow, 1974. MR 51:10750
 4.
 V. Sh. Burd and V.A. Karakulin, `Asymptotic integration of systems of linear differential equations with oscillatorily decreasing coefficients', Math. Notes 64 (1999), 571578.MR 2000a:34097
 5.
 J.S. Cassell, `The asymptotic integration of some oscillatory differential equations', Quart. J. Math. Oxford Ser. (2) 33 (1982), 281296. MR 84c:34079
 6.
 M.S.P. Eastham, The asymptotic solution of linear differential systems, London Mathematical Society Monographs, Clarendon Press, Oxford, 1989. MR 91d:34001
 7.
 M.S.P. Eastham, `The number of resonant states in perturbed harmonic oscillation', Quart. J. Math. Oxford(2) 42 (1991), 4955. MR 92d:34075
 8.
 W.A. Harris and D.A. Lutz, `A unified theory of asymptotic integration,' J. Math. Anal. Appl. 57 (1977), no. 3, 571586. MR 55:3441
 9.
 W.A. Harris and Y. Sibuya, `Asymptotic behaviors of solutions of a system of linear ordinary differential equations as ', Lecture Notes in Math., 1475, Springer, 1991, pp. 210217. MR 93k:34119
 10.
 N. Levinson, `The asymptotic nature of solutions of linear differential equations', Duke Math. J., 15 (1958), 111126. MR 9:509h
 11.
 K. Mischaikow, H. Smith and H. Thieme, `Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions', Trans. Amer. Math. Soc., 347 (1995), 16691685. MR 95h:34063
 12.
 S. Trofimchuk and M. Pinto, `perturbations of invariant subbundles for linear systems', Journal of Dynamics and Differential Equations, 14 (2002), 743761. MR 2003m:34116
 13.
 A.M. Samoilenko, Elements of the mathematical theory of multifrequency oscillations, Kluwer, Dordrecht, 1991, 313 pp. MR 93d:34050
 14.
 Yu. A. Samokhin 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', Siberian Math. J. 17 (1976), 926931.MR 54:10767
 15.
 M.M. Skriganov, `The eigenvalues of the Schrödinger operator situated on the continuous spectrum', J. Soviet Math. 8 (1977), 464467. MR 49:887
 16.
 I.Z. Shtokalo, Linear differential equations with variable coefficients, Gordon and Breach Science Publishers, Inc., New York, 1961, 100 pp.MR 27:5979
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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/S0002993904075203
PII:
S 00029939(04)075203
Keywords:
Asymptotic integration,
asymptotically autonomous equation,
Levinson theorem,
KrylovBogolyubov averaging principle,
VanderPol 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
