Consider the equations: $$\dot{x}=Cx+F(x,y), \qquad x\in \mathbb{R}^n,$$ $$\dot{y}=Sy+G(x,y), \qquad y\in \mathbb{R}^m.$$ The eigenvalues of matrix $C$ are on the imaginary axis, and those of S are having negative real part.

Center-manifold theorem says that there is a mapping $h:\mathbb{R}^n\to \mathbb{R}^m$ such that the set $W^c:=\{(x,h(x)), \,x\in\mathbb{R}^n\}$ is invariant for the above set of equations. Further, $h(0)=0$, $h'(0)=0$.

The following reduction is generally used: $$ \dot{x}=Cx+F(x,h(x)) $$

How to justify this reduction? Are there any theorems which say that the solution with any initial condition $(x_0,y_0)$ not in $W^c$ would reach $W^c$ very quickly (exponentially?) ? Please suggest some references.


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