Centre manifold

Consider an autonomous system of ordinary differential equations

(a1)

where is sufficiently smooth, . Let the eigenvalues of the Jacobi matrix evaluated at the equilibrium position be . Suppose the equilibrium is non-hyperbolic, i.e. has eigenvalues with zero real part. Assume also that there are eigenvalues (counting multiplicities) with , eigenvalues with , and eigenvalues with . Let denote the linear (generalized) eigenspace of corresponding to the union of the eigenvalues on the imaginary axis. The eigenvalues with are often called critical, as is the eigenspace . Let denote the flow (continuous-time dynamical system) associated with (a1). Under the assumptions stated above, the following centre manifold theorem holds [a7], [a9], [a3], [a11]: There is a locally defined smooth -dimensional invariant manifold of that is tangent to at .

The manifold is called the centre manifold. The centre manifold need not be unique. If with finite , is a -manifold in some neighbourhood of . However, as the neighbourhood may shrink, thus resulting in the non-existence of a -manifold for certain systems.

In a basis formed by all (generalized) eigenvectors of (or their linear combinations if the corresponding eigenvalues are complex), the system (a1) can be written as

(a2)

where , , is an -matrix with all its eigenvalues on the imaginary axis, while is an -matrix with no eigenvalue on the imaginary axis; . A centre manifold of (a2) can be locally represented as the graph of a smooth function , :

The following reduction principle is valid (see [a1], [a8]): The system (a2) is locally topologically equivalent (cf. Equivalence of dynamical systems) near the origin to the system

(a3)

The equations for and are uncoupled in (a3). The first equation is the restriction of (a2) to its centre manifold. Thus, the dynamics of (a2) near a non-hyperbolic equilibrium are determined by this restriction, since the second equation in (a3) is linear and has exponentially decaying/growing solutions. For example, if is the asymptotically stable equilibrium of the restriction and the matrix has no eigenvalue with positive real part, then is the asymptotically stable equilibrium of (a2). If there is more than one centre manifold, then all the resulting systems (a3) with different are locally topologically equivalent (actually, the differ only by flat functions).

The second equation in (a3) can be replaced by the standard saddle:

(a4)

with . In other words, near a non-hyperbolic equilibrium the system is locally topologically equivalent to the suspension of its restriction to the centre manifold by the standard saddle.

Consider now a system that depends smoothly on parameters:

(a5)

Suppose that at the system has a non-hyperbolic equilibrium with eigenvalues on the imaginary axis and eigenvalues with non-zero real part. Let of them have negative real part, while have positive real part. Applying the centre manifold theorem to the following extended system:

(a6)

one can prove the existence of a parameter-dependent local invariant manifold in (a5). The manifold has dimension and coincides with a centre manifold of the (a5) at . Often, the manifold is called the centre manifold for all . For each small one can restrict system (a5) to . Introducing a coordinate system on with as the coordinate, this restriction will be represented by a smooth system:

(a7)

At , the system (a7) is equivalent to the restriction of (a5) to its centre manifold . The following results are known as the Šošitaišvili theorem [a10] (see also [a1], [a2], [a8]): The system (a5) is locally topologically equivalent near the origin to the suspension of (a7) by the standard saddle (a4). Moreover, (a7) can be replaced by any locally topologically equivalent system.

This theorem reduces the study of bifurcations of non-hyperbolic equilibria (cf. also Bifurcation) to those on the corresponding centre manifold of dimension equal to the number of critical eigenvalues. There are analogues of the reduction principle and Šošitaišvili's theorem for discrete-time dynamical systems defined by iterations of diffeomorphisms (see, for example, [a1], [a8]). Existence of centre manifolds has also been proved for certain infinite-dimensional dynamical systems defined by partial differential equations [a9], [a3], [a6] and delay differential equations [a5], [a4].

References

[a1]	V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , Grundlehren math. Wiss. , 250 , Springer (1983) (In Russian)
[a2]	V.I. Arnol'd, V.S. Afraimovich, Yu.S. Il'yashenko, L.P. Shil'nikov, "Bifurcation theory" V.I. Arnol'd (ed.) , Dynamical Systems V , Encycl. Math. Sci. , Springer (1994) (In Russian)
[a3]	J. Carr, "Applications of center manifold theory" , Springer (1981)
[a4]	O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, H.-O. Walther, "Delay equations" , Springer (1995)
[a5]	J. Hale, S.M. Verduyn Lunel, "Introduction to functional differential equations" , Springer (1993)
[a6]	D. Henry, "Geometric theory of semilinear parabolic equations" , Springer (1981)
[a7]	A. Kelley, "The stable, center stable, center, center unstable and unstable manifolds" J. Diff. Eq. , 3 (1967) pp. 546–570
[a8]	Yu.A. Kuznetsov, "Elements of applied bifurcation theory" , Springer (1995)
[a9]	J. Marsden, M. McCracken, "Hopf bifurcation and its applications" , Springer (1976)
[a10]	A.N. Šošitaišvili, "Bifurcations of topological type of a vector field near a singular point" , Proc. Petrovskii Sem. , 1 , Moscow Univ. (1975) pp. 279–309 (In Russian)
[a11]	A. Vanderbauwhede, "Centre manifolds, normal forms and elementary bifurcations" Dynamics Reported , 2 (1989) pp. 89–169