9. Let be a vector field on , be the flow induced by on . That is, . Then we take a curve , and consider the solutions . There are two ways to take derivative:
(2) , which induces the tangent flow of .
Combine these two derivatives together:
This gives rise to an equation
Formally, one can consider the differential equation along a solution :
, . Then is called the linear Poincare map along . Suppose . Then determines if the periodic orbit is hyperbolic or elliptic. Note that the path , contains more information than the above characterization.