For a real analytic system of
ordinary differential equations with an integral H
H
=λ(x2+ y2) + F(z) + G(x,y,z)
ẋ
= λy + X(x,y,z)
ẏ
= −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,⋯,μm and
iλ−1μj≠ integer (j = 1,⋯,m); the existence of a unique, local, real analytic,
two-dimensional, invariant subcenter manifold
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|λ−1|.
