View source for Center manifold

==An Example== 

We first look at a simple example.  Consider
:<math>\label{eq1}
x' =ax^3 \,, \qquad y' =-y + y^2
</math>

where <math>a</math> is a constant.  Since the equations are uncoupled, we see that the stationary  solution <math> x=y =0</math> of \eqref{eq1} is asymptotically [[stable]] if and only if <math>a < 0\ .</math>  Suppose now that
:<math>\label{eq2}
x' =ax^3 + xy - xy^2\,, \qquad y' =-y + bx^2 +x^2 y 
</math>

Since the equations are coupled we cannot immediately decide if the
stationary  solution <math> x=y =0</math>  of \eqref{eq2} is asymptotically stable.  The key is an abstraction of the idea of uncoupled equations.

A curve <math>y =h(x)\ ,</math> defined for <math>|x|</math> small, is said to be an '''invariant manifold''' for the system 
:<math>\label{eq3}
x' =f(x,y)\,, \qquad y' = g(x,y)
</math>

if the solution of \eqref{eq3} with <math>x(0) =x_0\ ,</math> <math>y(0) = h(x_0)</math> lies on the curve <math>y =h(x)</math> as long as <math>x(t)</math> remains small.  For the system \eqref{eq1}, <math>y=0</math> is an invariant manifold.  Note that in deciding upon the stability of the stationary  solution of \eqref{eq1}, the only important equation is <math>x' = ax^3\ ,</math> that is, we need only study a first order equation on a particular invariant manifold.

Center manifold theory tells us that \eqref{eq2} has an invariant manifold 
<math>y =h(x) = \mbox{O}(x^2)</math> for small <math>x\ .</math>  Furthermore, the local behaviour of solutions of the two dimensional system  \eqref{eq2} can be determined by studying the scalar equation
:<math>\label{eq4}
u' = au^3 + uh(u) -uh^2(u)
</math>

The theory also tells us how to compute approximations to the invariant manifold <math>y = h(x)\ .</math>  For \eqref{eq2} we have that <math>h(x) = bx^2 + \mbox{O}(x^4)</math> and using this information in \eqref{eq4} gives 
:<math>\label{eq5}
u' =(a+b)u^3 + \mbox{O}(u^5)
</math>

Hence the stationary  solution of  \eqref{eq2} is asymptotically stable if <math>a+b < 0</math> and unstable if <math>a+b>0\ .</math>  If <math>a+b = 0</math> we need a better approximation to the invariant manifold in order to decide on the stability.