Vol. 29, No. 2, 1969

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ISSN: 0030-8730
Analytic two-dimensional subcenter manifolds for systems with an integral

Al (Allen Frederick) Kelley, Jr.

Vol. 29 (1969), No. 2, 335–350
Abstract

For a real analytic system of ordinary differential equations with an integral H

H = 1
2λ(x2 + y2) + F(z) + G(x,y,z)
= λy + X(x,y,z)
= λx + Y (x,y,z)
ż = Bz + Z(x,y,z)
where x and y are scalars; z is an m-vector; X, Y , Z are power series with no constant or linear terms; B is a constant matrix with eigenvalues μ1,m and 1μj integer (j = 1,,m); the existence of a unique, local, real analytic, two-dimensional, invariant subcenter manifold
  λ
M   = {(x,y,z) | |x|+ |y| < δ,z = w(x,y)}

is proved, where w is a real analytic function with no constant or linear terms in its expansion about the origin. The manifold Mλ is composed of a nested, one-parameter (ρ 0) family of periodic orbits, and as ρ 0 the corresponding periodic orbit goes to the origin and its period goes to 2π|λ1|.

Mathematical Subject Classification
Primary: 34.45
Milestones
Received: 10 May 1968
Published: 1 May 1969
Authors
Al (Allen Frederick) Kelley, Jr.