Interpretation:
Check the validity of theorem for various types of closed orbits on a torus, cylinder, and sphere.
Concept Introduction:
Index of a closed curve C is an integer that measures the windings of the
To find the index of the curve, draw a closed contour C on the phase portrait. Then on each point “x” on the contour C, vector field
The index of closed curve C with respect to vector field is given as
Here
Properties of Indices
If contour C continuously deformed into C’ without passing through a fixed point then
If contour C doesn’t enclose any fixed points then
If the directions of all arrows in the vector field are reversed by changing
If the closed curved C is a trajectory for the system then
Theorem: Any closed orbit in the phase plane mustenclose fixed points whose indices sum to
Theorem: If a close curve C surrounds n isolated fixed point
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Nonlinear Dynamics and Chaos
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