Results 1 to 3 of 3

Thread: Center manifold and submanifold

  1. #1
    Junior Member
    Joined
    Oct 2013
    From
    China
    Posts
    28

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    6,608
    Thanks
    1714

    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?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Oct 2013
    From
    China
    Posts
    28

    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. (n - 1) dimensional submanifold of the manifold R^n
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Dec 19th 2012, 08:24 AM
  2. Orientation of submanifold
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Feb 9th 2011, 04:21 AM
  3. Submanifold
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: Jan 2nd 2011, 07:59 AM
  4. Submanifold Problem
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 19th 2010, 07:48 AM
  5. Manifold is a manifold with boundary?
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: Mar 1st 2010, 12:38 PM

Search Tags


/mathhelpforum @mathhelpforum