The first subsystem in (5.16) is the n c -dimensional center manifold reduction to
the original n-dimensional system _
x = f(x), near a particular point of bifurcation. It
describes the non-hyperbolic part of the system response in terms of the n c critical
variables x
_ ðtÞ: The center manifold reduction retains the bifurcational behavior of
the full system, even though its dimension n c may be far less than n.
The center manifold theorem merely states that subsystems of the form (5.16)
exist, giving no hints as how to compute them. We next consider a systematic
procedure for computing the center manifold reduction.
5.5.3 Computing the Center Manifold Reduction
In applications we usually study bifurcations of some stable state of a system. Thus,
one can safely assume that there are no Jacobian eigenvalues with a positive real
part (n u = 0). We further assume that the system has been Taylor-expanded around
the bifurcation point in question. Finally we assume that a modal transformation has
been applied to the original system, so that its linear part is in block diagonal form
(that is, in Jordan canonical form, cf. Sect. 3.4.5). By these assumptions the system
under consideration can readily be split into the following subsystems:
_
x ¼ Bx þ fðx; yÞ; xðtÞ 2 R
n c ;
_
y ¼ Cy þ gðx; yÞ; yðtÞ 2 R
n s ;
ð5:17Þ
where B and C are n c  n c and n s  n s matrices whose eigenvalues have,
respectively, zero and negative real parts, and where the nonlinear functions f and
g vanish along with their first derivatives at the origin.
The first subsystem is the interesting one, describing motions on the center
manifold in terms of the critical variables x(t). However, the non-critical variables
y(t) also appear in this subsystem, through the nonlinear function f(x,y). To compute the center manifold reduction we need to eliminate y from f.
Considering the second subsystem, one could be tempted to believe that
y(t) would tend to zero exponentially fast, since all eigenvalues of the matrix
C have negative real parts. The center manifold reduction would then readily be
given as _
x = Bx + f(x, 0). However, the argument that y ! 0 due to C having
negative real parts assumes g to be independent of x. If g is independent of x, then
one can let y = 0 to obtain the center manifold reduction. On the other hand, when
g depends on x one cannot be sure that y will approach zero, and a more systematic
procedure for eliminating y is in need.
For this purpose we note that, at the origin, the center manifold W
c is tangential
to E
c (the y = 0 space). The center manifold can therefore be represented by a local
graph h(x):
5.5 Center Manifold Reduction
287
the original n-dimensional system _
x = f(x), near a particular point of bifurcation. It
describes the non-hyperbolic part of the system response in terms of the n c critical
variables x
_ ðtÞ: The center manifold reduction retains the bifurcational behavior of
the full system, even though its dimension n c may be far less than n.
The center manifold theorem merely states that subsystems of the form (5.16)
exist, giving no hints as how to compute them. We next consider a systematic
procedure for computing the center manifold reduction.
5.5.3 Computing the Center Manifold Reduction
In applications we usually study bifurcations of some stable state of a system. Thus,
one can safely assume that there are no Jacobian eigenvalues with a positive real
part (n u = 0). We further assume that the system has been Taylor-expanded around
the bifurcation point in question. Finally we assume that a modal transformation has
been applied to the original system, so that its linear part is in block diagonal form
(that is, in Jordan canonical form, cf. Sect. 3.4.5). By these assumptions the system
under consideration can readily be split into the following subsystems:
_
x ¼ Bx þ fðx; yÞ; xðtÞ 2 R
n c ;
_
y ¼ Cy þ gðx; yÞ; yðtÞ 2 R
n s ;
ð5:17Þ
where B and C are n c  n c and n s  n s matrices whose eigenvalues have,
respectively, zero and negative real parts, and where the nonlinear functions f and
g vanish along with their first derivatives at the origin.
The first subsystem is the interesting one, describing motions on the center
manifold in terms of the critical variables x(t). However, the non-critical variables
y(t) also appear in this subsystem, through the nonlinear function f(x,y). To compute the center manifold reduction we need to eliminate y from f.
Considering the second subsystem, one could be tempted to believe that
y(t) would tend to zero exponentially fast, since all eigenvalues of the matrix
C have negative real parts. The center manifold reduction would then readily be
given as _
x = Bx + f(x, 0). However, the argument that y ! 0 due to C having
negative real parts assumes g to be independent of x. If g is independent of x, then
one can let y = 0 to obtain the center manifold reduction. On the other hand, when
g depends on x one cannot be sure that y will approach zero, and a more systematic
procedure for eliminating y is in need.
For this purpose we note that, at the origin, the center manifold W
c is tangential
to E
c (the y = 0 space). The center manifold can therefore be represented by a local
graph h(x):
5.5 Center Manifold Reduction
287