Equivalent system in Centre manifold theory

Question

I was studying the centre manifold theory. It says (see Kuznetsov page 155, theorem 5.2) that the system on the left side of the picture is topologically equivalent to the one on the right.

$ \begin{equation} \label{} \left\lbrace \begin{array}{lcr} \dot{u} \: & = & \: Bu + g(u,v)\\ \dot{v} \: & = & \: Cv + h(u,v) \end{array}\right. \Rightarrow \left\lbrace \begin{array}{lcl} \dot{u} \: & = & \: Bu + g(u,V(u))\\ \dot{v} \: & = & \: Cv \end{array}\right. \end{equation} $

Here $v=V(u)$ is the center manifold.

I'm interested in understanding things from an heuristic point of view but i can't figure out why in the second system $h(u,v)=0$ and what this means.

Other books don't tell anything about the second equation of the second system. They mainly say that the most important information are embedded in the first equation of the second system. What if i want to graph the second system? I also searched Carr's book for a demonstration but i did not find anything useful.

Answer 1

I) The question formulation (v4) leaves out some important implicit assumptions$^1$ of Theorem 5.2 in Ref.1. These are, among other things, the following four items.

1. The word topologically equivalent should be replaced by locally topologically equivalent, i.e. in some local neighborhood.

2. The (vector-valued) functions $g(w)$ and $h(w)$ are $o(w)$ for $w\to 0$, where $w=(u,v)$. Here $o(w)$ refers to the little-o notation.

3. The eigenvalues of the $B$ matrix lie on the imaginary axis.

4. The eigenvalues of the $C$ matrix do not lie on the imaginary axis.

II) Let us first take a step back and put it in context. Where did all of this come from? Well, we wanted to analyze the stability of the dynamical system

$$ \dot{w}~=~Aw + f(w) $$

at the point $w=0$. Here $A$ is a constant(=$w$-independent) quadratic matrix$^2$, and the (vector-valued) function $f$ is of higher-order,

$$f(w)~=~o(w) \qquad \text{for} \qquad w~\to~ 0.$$

We then partially diagonalize the $A$ matrix into sub-blocks $B$ and $C$ with the aforementioned properties (3 and 4). After the diagonalization, the $w$-variables are split into $w=(u,v)$, and similarly for the (vector-valued) function $f=(g,h)$.

III) We could furthermore partially diagonalize $C$ into sub-blocks $C_{+}$ and $C_{-}$ depending of whether the eigenvalues have positive or negative real part, respectively. These will correspond to unstable or stable directions, respectively, independently of what the higher-order terms $g$ and $h$ are. (In a Renormalization group jargon, they correspond to so-called relevant or irrelevant deformations, respectively. Similarly, the $B$ sub-block corresponds to so-called marginal deformations.)

IV) The tangent plane $\{w|v=0\}$ at $w=0$ corresponds to marginal directions. We need the higher-order information $f$ to decide if they are stable or unstable directions. If $h\equiv 0$, we can choose the tangent plane $\{w|v=0\}$ as the center manifold. For general $h$, the center manifold $C:=\{w|v=V(u)\}$ is a deformation of the tangent plane $\{w|v=0\}$. The tangent plane $\{w|v=0\}$ is only a precise substitute for the center manifold $C$ when $u=0$. Intuitively, the center manifold $C$ by construction becomes curved by encoding the higher-order information $f$ in such a way that points $w\notin C$ in a neighborhood of $0$ are attracted to (repelled from) $C$, for time $t$ going to the future (coming from the past), in accordance with just the linear predictions of the $Cw$ term, respectively. In particular, the effects of the $h$ terms in the second equation have already been taking into account in the definition of the curved center manifold $C$, so that one should now only use the linear equation

$$\dot{v} ~ = ~ Cv$$

without the $h$-term. The first equation

$$\dot{u} ~ =~ Bu + g(u,V(u))$$

is used to determine the stability for points $w\in C$ on the center manifold $C$ itself in a small neighborhood of $0$.

References:

1. Yu.A. Kuztnetsov, Elements of applied Bifurcation Theory, 2nd Edition, 1998.

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$^1$ Some of the implicit assumptions are explained in the Scholarpedia link.

$^2$ A finite-dimensional quadratic matrix $A$ is not necessarily diagonalizable. Eigenvalues, eigenvectors and eigenspaces should then be understood in the sense of generalized eigenvalues, eigenvectors and eigenspaces.