Vol. 25, No. 2, 1968

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Invariance for linear systems of ordinary differential equations

Al (Allen Frederick) Kelley, Jr.

Vol. 25 (1968), No. 2, 289–304

In studying the existence and smoothness of invariant manifolds arising from nonlinear, perturbed systems of ordinary differential equations, one encounters the study of certain linear (in x), perturbation problems of the type

𝜃 = a + 𝜖b(𝜃,𝜖)
= (A + 𝜖B(𝜃,𝜖))x
where 𝜃 and x are vectors, A and B are matrices, b and B are multiply periodic in 𝜃, and 𝜖 is a perturbation parameter. Assuming A is a constant matrix consisting of square sub-matrices on the diagonal,
A = diag(A11,⋅⋅⋅ ,Ann ),

with the maximum of the real parts of the eigenvalues of Ajj less than the minimum of the real parts of the eigenvalues of Akk for 1 j < k n; we construct a change of variables which reduces B to similar diagonal form.

Mathematical Subject Classification
Primary: 34.53
Received: 14 April 1967
Published: 1 May 1968
Al (Allen Frederick) Kelley, Jr.