The center problem on a center manifold in

Stability and periodic oscillations in the Moon–Rand systems

“…A proof of the following theorem can be found in [8]. The equivalence of statements (a) and (b) is called the Lyapunov Center Theorem; it is proved in many places, including [4].…”

“…When Ψ has the form (2.4) the coefficient g k 1 k 2 k 3 of x k 1 y k 2 z k 3 in ZΨ can be calculated explicitly (see [8]). Except when (k 1 , k 2 , k 3 ) = (K, K, 0) for K ∈ N, the equation g k 1 k 2 k 3 = 0 can be solved uniquely for v k 1 k 2 k 3 in terms of the known quantities v αβγ .…”

“…It is known ( [8]) that it is necessary by proving that if for one choice of the v KK0 at least one focus quantity is non-zero, then the same is true for every other choice of the v KK0 . To shorten the notation we let (a, b, c) stand for the coefficient string Theorem 2.2.…”

“…For a proof see [8]. Recall that the set of common zeros of a collection of polynomials that generate an ideal I is the variety V(I) of I. Theorem 2.3.…”

“…Instead we avoid the center manifold entirely using the ideas developed in [8] (and reviewed in Section 2): complexify (1.1) and compute a sufficiently long initial string of an infinite sequence of polynomials g k in the ring C[λ, c 20 , c 11 , c 02 ], the focus quantities of family (1.1), whose variety specifies those systems in (1.1) with a center at the origin on the center manifold (which is unique in such circumstances). By this means we are able to solve the center problem for the flow of (1.1) on the center manifold.…”