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Averaging and integral manifolds (II)

Published online by Cambridge University Press:  17 April 2009

K. J. Palmer
K. J. Palmer
Affiliation:
Australian National University, Canberra, ACT.
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Abstract

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In the first part of this paper (written jointly with W.A. Coppel) the existence and properties of an integral manifold were established for the system

x′ = f(t, x, y)

y′ = A(t)y + g(t, x, y)

where f and g are “integrally small”. In this second part of the paper the stability properties of the integral manifold are investigated. Solutions are found which are bounded on the positive half of the real line and it is shown that these solutions approach the manifold exponentially and, moreover, that they are asymptotic to particular solutions on the manifold.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 2 , Issue 3 , June 1970 , pp. 369 - 399
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Coppel, W.A. and Palmer, K.J., “Averaging and integral manifolds”, Bull. Austral. Math. Soc. 2 (1970), 197–222.CrossRefGoogle Scholar
[2]Bogolyubov, N.N. and Mitropolskii, Yu.A., Asymptotic methods in the theory of nonlinear oscillations (Russian), 3rd ed. (Gos. Izdat. Fiz.-Mat. Lit., Moscow, 1963). English transl. 2nd ed. (Gordon & Breach, New York, 1962).Google Scholar
[3]Hale, Jack K., “Integral manifolds of perturbed differential systems”, Ann. of Math. (2) 73 (1961), 496–531.CrossRefGoogle Scholar
[4]Hale, Jack K., Oscillations in nonlinear systems (McGraw-Hill, New York, Toronto, London, 1963).Google Scholar
[5]Gihman, I.I., “Concerning a theorem of N.N. Bogolyubov”, (Russian), Ukrain. Mat. Ž. 4 (1952), 215–219.Google Scholar
[6]Coppel, W.A., “Dichotomies and reducibility (II)”, J. Differential Equations 4 (1968), 386–398.CrossRefGoogle Scholar
[7]Levinson, Norman, “Small periodic perturbations of an autonomous system with a stable orbit”, Ann. of Math. (2) 52 (1950), 729–738.CrossRefGoogle Scholar
[8]Diliberto, S.P. and Hufford, G., “Perturbation theorems for non-linear ordinary differential equations”, Contributions to the theory of nonlinear oscillations, 3, 207–236. (Annals of Mathematics Studies, 36, Princeton University Press, Princeton, New Jersey, 1956).Google Scholar
[9]Diliberto, Stephen P., “Perturbation theorems for periodic surfaces I. Definitions and main theorems”, Rend. Circ. Mat. Palermo (2) 9 (1960), 265–299.CrossRefGoogle Scholar
[10]Diliberto, Stephen P., “Perturbation theorems for periodic surfaces. II”, Rend. Cir. Mat. Palermo (2) 10 (1961), 111–161.Google Scholar
[11]Sacker, Robert J., “A new approach to the perturbation theory of invariant surfaces”, Comm. Pure Appl. Math. 18 (1965), 717–732.CrossRefGoogle Scholar
[12]Sacker, Robert J., “A perturbation theorem for invariant manifolds and Hölder continuity”, J. Math. Mech. 18 (1969), 705–762.Google Scholar
[13]Kurzweil, Jaroslav, “Exponentially stable integral manifolds, averaging principle and continuous dependence on a parameter”, Czechoslovak Math. J. 16 (91) (1966), 380–423.CrossRefGoogle Scholar
[14]Kurzweil, Jaroslav, “Invariant manifolds for flows”, Differential Equations and Dynamical Systems (Proc. Internat. Sympos., Mayaguez, P.R., 1965, 431–468. Academic Press, New York, 1967).Google Scholar
[15]Kurzweil, Jaroslav, “Invariant manifolds for differential systems”, (Russian), Differencial'nye Vravnenija 4 (1968), 785–797.Google Scholar