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November 1, 2004, 01:29 
pointcare index

#1 
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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 = Dphi/2*pi, i dong't know how to get the phi? who can give me detailed intepretation? thanks 

November 6, 2004, 23:57 
Re: Poincare index

#2 
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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).


November 8, 2004, 14:32 
Re: pointcare index

#3 
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For a general surface, the sum(N)sum(S)=22*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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