The center manifold theorem

When a system loses stability, the number of eigenvalues and eigenvectors which are associated with this change is typically small. Hence bifurcation problems usually involve systems where the linearization has a very large, and possibly infinite dimensional, stable part and a small number of ``critical" modes which change from stable to unstable as the bifurcation parameter exceeds a threshold. The central idea of bifurcation theory is that the dynamics of the system near the onset of instability is governed by the evolution of these critical modes, while the stable modes follow in a passive fashion, they are ``enslaved." The center manifold theorem is the rigorous formulation of this idea; it allows us to reduce a large problem to a small and manageable one.

We shall state the theorem for systems of ordinary differential equations; of course the real physical applications are partial differential equations. We let $x\in{\rm I\!R}^m$ and $y\in{\rm I\!R}^n$, and we consider the system

   $$\begin{eqnarray} \dot x=A(\lambda)x+f(x,y,\lambda),\nonumber\ \dot y=B(\lambda)y+g(x,y,\lambda).\end{eqnarray}$$

We make the following assumptions:

1. f and g are smooth ($C^\infty$) functions, and they are quadratic at the origin, i.e.

$$\Vert f(x,y,\lambda)\Vert+\Vert g(x,y,\lambda)\Vert\le C(\Vert x\Vert^2+\Vert y\Vert^2)$$ (84)

for small $\Vert x\Vert$ and $\Vert y\Vert$.

2. A and B depend smoothly on $\lambda$. All eigenvalues of A(0) are purely imaginary, while all eigenvalues of B(0) have negative real parts.

Theorem 6

If assumptions 1 and 2 hold, then, in some neighborhood of x=0, $\lambda=0$,there exists a $C^\infty$-smooth manifold of the form $y=\phi(x,\lambda)$, called the center manifold, with the following properties.
A. $\phi(0,\lambda)=0$ and $\partial\phi/\partial x_i(0,0)=0$ for i=1,...,m.
B. Every solution of (3.12) which starts on the center manifold remains on it, i.e. if $y(0)=\phi(x(0),\lambda)$, then $y(t)=\phi(x(t),\lambda)$ for all t.
C. Every solution of (3.12) which remains small for positive time approaches the center manifold. That is, there exists an $\epsilon\gt$ such that, if $\vert\lambda\vert< \epsilon$ and $\Vert x(t)\Vert+\Vert y(t)\Vert<\epsilon$ for all t>0, then $\Vert y(t)-\phi(x(t),\lambda)\Vert\to 0$ as $t\to\infty$. Every solution which satisfies $\Vert x(t)\Vert+\Vert y(t)\Vert<\epsilon$ for all t (positive and negative) is on the center manifold. Moreover, the stability of such a solution is determined by its stability within the center manifold.

Property C implies that if we are looking for steady or periodic solutions bifurcating from the trivial solution x=0, y=0, then we can confine our search to the center manifold. This is a substantial simplification, since in applications m is typically very small (in the situations considered below, m ranges from 1 to 4), while n is large or even infinite.

There is no claim of uniqueness in the theorem above. Indeed, center manifolds are usually not unique. Their Taylor approximations, however, are unique, that is if $y=\phi_1(x,\lambda)$ and $y=\phi_2(x,\lambda)$ are two center manifolds, then the difference $\Vert\phi_1(x,\lambda)-\phi_2(x,\lambda)\Vert$ is $o(\Vert x\Vert^N)$for every N as $x\to 0$. For the simplest bifurcation problems, only very few terms in the Taylor approximation of the center manifold are needed to gain a qualitative understanding of the bifurcation; often it suffices to get the quadratic terms. The formal Taylor series of the center manifold is usually not convergent, but in many cases the Taylor series for the bifurcating solutions do converge (assuming of course, that the nonlinear terms f and g have convergent Taylor expansions). We noted above that bifurcating steady or periodic solutions must lie on ``the" center manifold. If the center manifold is not unique, this means, of course, that they must lie on every center manifold.

A proof of the center manifold theorem is well beyond the scope of this course. The following example should give an idea, however, how we can go about computing the center manifold in a concrete situation.

Example 14

Let us consider the system

   $$\begin{eqnarray} \dot x&=&\lambda x+2\lambda y+x^2+y^2,\nonumber\ \dot y&=&-y+\lambda x+x^2+xy.\end{eqnarray}$$

By the center manifold theorem, there should be an invariant manifold of the form  

$$y=\phi(x,\lambda)=a(\lambda)x+b(\lambda)x^2+c(\lambda)x^3+O(\vert x\vert^4).$$ (85)

We shall now use the invariance of the manifold (property B in the theorem) to compute the coefficients $a(\lambda)$ and $b(\lambda)$. For this, we simply differentiate the equation (3.15). We find  

$$\dot y=a(\lambda)\dot x+2b(\lambda)x\dot x+3c(\lambda)x^2\dot x +O(\vert x\vert^3)\dot x.$$ (86)

Now we use the differential equation (3.14) to replace $\dot y$ and $\dot x$. This yields

$$-y+\lambda x+x^2+xy=(a(\lambda)+2b(\lambda)x+3c(\lambda)x^2+O(\vert x\vert^3)) (\lambda x+2\lambda y+x^2+y^2).$$ (87)

Next, we use (3.15) to substitute for y, and we compare terms of order x, x² and x³. At order x, we find  

$$-a(\lambda)+\lambda=a(\lambda)\lambda+2\lambda a(\lambda)^2.$$ (88)

For small $\lambda$, this has a unique solution: $a(\lambda)=\lambda+ O(\lambda^2)$. Note that the second solution of the quadratic equation for $a(\lambda)$ becomes infinite for $\lambda\to 0$ and is therefore inadmissible. At order x² we find  

$$-b(\lambda)+1+a(\lambda)=2b(\lambda)\lambda+6\lambda a(\lambda)b(\lambda)+a(\lambda)+ a(\lambda)^3.$$ (89)

This allows us to solve for $b(\lambda)$: $b(\lambda)=1+O(\lambda)$. At order x³, we find  

$$-c(\lambda)+b(\lambda)=8\lambda a(\lambda)c(\lambda)+4a(\lam... ...\lambda) +4\lambda b(\lambda)^2+2b(\lambda)+3c(\lambda)\lambda.$$ (90)

This determines $c(\lambda)$. We can continue the procedure and compute the Taylor approximation of the center manifold up to any order we desire.

Finally, we can insert (3.15) back into the first equation of (3.14), and obtain

   $$\begin{eqnarray} \dot x&=&(\lambda+2\lambda a(\lambda))x+(2\lambda b(\lambda)+1+... ...number\ &&+(2\lambda c(\lambda)+2a(\lambda)b(\lambda))x^3+O(x^4).\end{eqnarray}$$

This is the equation which governs the evolution of solutions on the center manifold. We can find stationary solutions by setting the right hand side equal to zero. After division by x this yields

   $$\begin{eqnarray} 0=f(x,\lambda)&:=&\lambda+2\lambda a(\lambda)+ (2\lambda b(\lam... ...number\ &&+(2\lambda c(\lambda)+2a(\lambda)b(\lambda))x^2+O(x^3).\end{eqnarray}$$

We note that  

$${\partial f\over\partial x}(0,0)=1,\quad {\partial f\over\partial\lambda}(0,0) =1.$$ (91)

Both of these are nonzero, so that, according to the implicit function theorem, we can solve uniquely for either variable in a neighborhood of (0,0): $x=g(\lambda)=-\lambda+O(\lambda^2)$ or $\lambda=h(x)=-x+O(x^2)$. We have thus established the existence of a bifurcating branch of steady solutions.