Centre manifold

Consider an autonomous system of ordinary differential equations

$$ \tag{a1 } {\dot{x} } = f ( x ) , \quad x \in \mathbf R ^ {n} , $$

where $ f : {\mathbf R ^ {n} } \rightarrow {\mathbf R ^ {n} } $ is sufficiently smooth, $ f ( 0 ) = 0 $. Let the eigenvalues of the Jacobi matrix $ A $ evaluated at the equilibrium position $ x _ {0} = 0 $ be $ \lambda _ {1} \dots \lambda _ {n} $. Suppose the equilibrium is non-hyperbolic, i.e. has eigenvalues with zero real part. Assume also that there are $ n _ {u} $ eigenvalues (counting multiplicities) with $ { \mathop{\rm Re} } \lambda _ {j} > 0 $, $ n _ {c} $ eigenvalues with $ { \mathop{\rm Re} } \lambda _ {j} = 0 $, and $ n _ {s} $ eigenvalues with $ { \mathop{\rm Re} } \lambda _ {j} < 0 $. Let $ T ^ {c} $ denote the linear (generalized) eigenspace of $ A $ corresponding to the union of the $ n _ {c} $ eigenvalues on the imaginary axis. The eigenvalues with $ { \mathop{\rm Re} } \lambda _ {j} = 0 $ are often called critical, as is the eigenspace $ T ^ {c} $. Let $ \varphi ^ {t} $ 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 $ n _ {c} $- dimensional invariant manifold $ W ^ {c} ( 0 ) $ of $ \varphi ^ {t} $ that is tangent to $ T ^ {c} $ at $ x = 0 $.

The manifold $ W ^ {c} ( 0 ) $ is called the centre manifold. The centre manifold $ W ^ {c} ( 0 ) $ need not be unique. If $ f \in C ^ {k} $ with finite $ k $, $ W ^ {c} ( 0 ) $ is a $ C ^ {k} $- manifold in some neighbourhood $ U $ of $ x _ {0} $. However, as $ k \rightarrow \infty $ the neighbourhood $ U $ may shrink, thus resulting in the non-existence of a $ C ^ \infty $- manifold $ W ^ {c} ( 0 ) $ for certain $ C ^ \infty $ systems.

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

$$ \tag{a2 } \left \{ \begin{array}{l} { {\dot{u} } = Bu + g ( u,v ) , \ } \\ { {\dot{v} } = Cv + h ( u,v ) , \ } \end{array} \right . $$

where $ u \in \mathbf R ^ {n _ {c} } $, $ v \in \mathbf R ^ {n _ {u} + n _ {s} } $, $ B $ is an $ ( n _ {c} \times n _ {c} ) $- matrix with all its $ n _ {c} $ eigenvalues on the imaginary axis, while $ C $ is an $ ( ( n _ {u} + n _ {s} ) \times ( n _ {u} + n _ {s} ) ) $- matrix with no eigenvalue on the imaginary axis; $ g,h = O ( \| {( u,v ) } \| ^ {2} ) $. A centre manifold $ W ^ {c} $ of (a2) can be locally represented as the graph of a smooth function $ V : {\mathbf R ^ {n _ {c} } } \rightarrow {\mathbf R ^ {n _ {u} + n _ {s} } } $, $ V ( u ) = O ( \| u \| ^ {2} ) $:

$$ W ^ {c} = \left \{ {( u,v ) } : {v = V ( u ) , \left \| u \right \| < \varepsilon } \right \} . $$

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

$$ \tag{a3 } \left \{ \begin{array}{l} { {\dot{u} } = Bu + g ( u,V ( u ) ) , \ } \\ { {\dot{v} } = Cv. \ } \end{array} \right . $$

The equations for $ u $ and $ v $ 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 $ u = 0 $ is the asymptotically stable equilibrium of the restriction and the matrix $ C $ has no eigenvalue with positive real part, then $ ( u,v ) = ( 0,0 ) $ is the asymptotically stable equilibrium of (a2). If there is more than one centre manifold, then all the resulting systems (a3) with different $ V $ are locally topologically equivalent (actually, the $ V $ differ only by flat functions).

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

$$ \tag{a4 } \left \{ \begin{array}{l} { {\dot{v} } = - v \ } \\ { {\dot{w} } = w, \ } \end{array} \right . $$

with $ ( v,w ) \in \mathbf R ^ {n _ {s} } \times \mathbf R ^ {n _ {u} } $. 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:

$$ \tag{a5 } {\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R ^ {n} , \alpha \in \mathbf R ^ {m} . $$

Suppose that at $ \alpha = 0 $ the system has a non-hyperbolic equilibrium $ x = 0 $ with $ n _ {c} $ eigenvalues on the imaginary axis and $ ( n - n _ {c} ) $ eigenvalues with non-zero real part. Let $ n _ {s} $ of them have negative real part, while $ n _ {u} $ have positive real part. Applying the centre manifold theorem to the following extended system:

$$ \tag{a6 } \left \{ \begin{array}{l} { {\dot \alpha } = 0, \ } \\ { {\dot{x} } = f ( x, \alpha ) , \ } \end{array} \right . $$

one can prove the existence of a parameter-dependent local invariant manifold $ {\mathcal M} _ \alpha $ in (a5). The manifold has dimension $ n _ {c} $ and $ {\mathcal M} _ {0} $ coincides with a centre manifold $ W ^ {c} ( 0 ) $ of the (a5) at $ \alpha = 0 $. Often, the manifold $ {\mathcal M} _ \alpha $ is called the centre manifold for all $ \alpha $. For each small $ | \alpha | $ one can restrict system (a5) to $ {\mathcal M} _ \alpha $. Introducing a coordinate system on $ {\mathcal M} _ \alpha $ with $ u \in \mathbf R ^ {n _ {c} } $ as the coordinate, this restriction will be represented by a smooth system:

$$ \tag{a7 } {\dot{u} } = \Phi ( u, \alpha ) . $$

At $ \alpha = 0 $, the system (a7) is equivalent to the restriction of (a5) to its centre manifold $ W ^ {c} ( 0 ) $. 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) Zbl 0791.00009
[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) MR1345150 Zbl 0826.34002
[a5]	J. Hale, S.M. Verduyn Lunel, "Introduction to functional differential equations" , Springer (1993) MR1243878 Zbl 0787.34002
[a6]	D. Henry, "Geometric theory of semilinear parabolic equations" , Springer (1981) MR0610244 Zbl 0456.35001
[a7]	A. Kelley, "The stable, center stable, center, center unstable and unstable manifolds" J. Diff. Eq. , 3 (1967) pp. 546–570 MR0221044
[a8]	Yu.A. Kuznetsov, "Elements of applied bifurcation theory" , Springer (1995) MR1344214 Zbl 0829.58029
[a9]	J. Marsden, M. McCracken, "Hopf bifurcation and its applications" , Springer (1976) MR0494309 Zbl 0346.58007
[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 MR1000977 Zbl 0677.58001