Dear all,
I am planning to investigate laminar-turbulent phase transition in fluid dynamics via studying the bifurcation characteristics of the Navier-Stokes equation. I would like to know if there is any theory which details the variation of an index at a bifurcations points during the phase transition.
Thank you.
Wang Zhe
Hi,
I quoted the following arguments from T. B. Benjamin, Proc. R. Soc. Lond. A. 359. 1-26, 1978. (attached)
"In the neighbourhood of a bifurcation point, the sum of the indices of the solutions entailed in the bifurcation does not change as the parameter (Re) is varied.
However, for static bifurcations
1. saddle-node: before bifurcation, the flow is ordinary so that the index is zero; after which, a saddle (-1) and a node(+1) are born and the sum of the indices is zero.
2. transcritical: fixed points interchange their stabilities with as the parameter is varied, so that the sum of the indices remains unchanged.
3. pitch fork: before bifurcation, a fixed point with index 1; after which, 2 fixed points with index 1 each and a saddle with index -1. The sum of indices 2-1=1 remains unchanged.
For dynamic bifurcations:
Hopf: before bifurcation, a fixed point with index 1; after which a fixed point with index 1 and a limit cycle is born with index 1. The sum of indices 1+1=2 is increased by 1.
Therefore, according to Benjamin's theory, it seems that the Hopf bifurcation is ruled out from the possible bifurcation in the flow, which, to me, ridiculous. I am pretty sure that I have somehow misunderstood what Benjamin implies, or I miscalculate the indices before and after the Hopf bifurcation.
Thanks for any comments, ideas, info...
Wang Zhe
You have two different equations: an algebraic one for computing fixpoints (in the framework what you call "static bifurcation") and a boundary value problem for computing periodic solutions (your "dynamic bifurcation"). Considering Poincare maps, the latter may be turned into an algebraic equation, too. Both algebraic equations are quite different and each yields its own index value as the sign of its own Jacobian determinant. Putting the index values of both equations together is like comparing apples and oranges.