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autofly November 1, 2004 01:29

pointcare index
 
hi I meet difficulty in understanding the definition of pointcare index. when study the topotical structure on surface. the sum of saddle and nodal points must meet certain relationship. for a common case.sum( N) -sum(S) = 2 but for special case sum(N) + 1/2 sum(N')- sum(S) - 1/2 sum(S')= I

I is the pointcare index. how to get it?

in a formula I = D-phi/2*pi, i dong't know how to get the phi? who can give me detailed intepretation? thanks

versi November 6, 2004 23:57

Re: Poincare index
 
Your first formula is right , but your second formula about relation of Poincare indices in 2D should be sum(N) + 1/2 sum(N')- sum(S) - 1/2 sum(S')= -1 , where N is a full node or focus point or center point, S is a full saddle point, N' or S' are half node (saddle) point on the boundary of the 2D domain. The index of a full node = 1 while the index of a saddle =-1, the index of a node (saddle) on the boundary is counted as 1/2 (-1/2). The meaning of Poincare Index can be found in many text books about topology and dynamic theory. If you draw a closed loop around a critical point and monitor the direction change of the unit field vetor along this loop counterclockwise, if change of the direction of the unit field vector is counterclock +360 (or -360) deg , then Poincare index =+1 (or -1).


Nicola November 8, 2004 14:32

Re: pointcare index
 
For a general surface, the sum(N)-sum(S)=2-2*g=I where g is the number of holes in the surface. So, for the flow over a sphere I=2, for the flow over a thorus I=0.

Hope it helps,

Nicola


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