1. Center manifold and submanifold

Hi all,

In the weakly nonlinear stability analysis, the center manifold reduction can be used to study the first linear bifurcation. This lead to the Ginzburg-Landau equation
$\displaystyle \frac{\partial A}{\partial t}=a_1A + a_3 A^*A^2 + a_5 A^{*2}A^3 + ......$

I have several questions.

Is the center manifold corresponding to the space $\displaystyle a_1=0$? I feel this because at the linear bifurcation, the growth rate of the disturbance is zero, which implies that $\displaystyle a_1=0$ in the above equation.

Then does there exist a submanifold corresponding to $\displaystyle a_3=0$?

Thanks a lot.

2. Re: Center manifold and submanifold

Hey Mengqi.

For those of us not acquainted with specifics (like me) could you provide a couple of definitions and some context for what you are studying?

3. Re: Center manifold and submanifold

Hi chiro,

Thanks for reminding me about this. My major is fluid dynamics. The equations are the Navier-Stokes (NS) equations.

In the hydrodynamic stability theory, the NS equations are usually studied by expansion method $\displaystyle q=Q+\epsilon q_1+\epsilon^2 q_2 +\epsilon^3 q_3 +$ ..... where $\displaystyle Q$ is the base flow upon which the expansion is done and $\displaystyle \epsilon$ is a small parameter.

When plugging this expansion into NS equation and equating the coefficients of the different $\displaystyle \epsilon^i$, we are speaking of the linear stability theory or weakly nonlinear stability theory. In the latter, the solvability condition should be applied and during this procedure, the Ginzburg-Landau (GL) equation results.

In GL equation, the first coefficient $\displaystyle a_1$ is obviously the linear growth rate (one can solve $\displaystyle \frac{\partial A}{\partial t}=a_1 A$). The other coefficients are for the nonlinear terms. The coefficient of the lowest nonlinear term $\displaystyle a_3$ indicates the subcritical/supercritical bifurcation.

So my question is can we find in the solution a submanifold for $\displaystyle a_3=0$? Thanks.