Interpretation:
Assuming the hypotheses of Dulac’s criteria and R is topologically equivalent to annulus (exactly one hole in it).By using Green’s theorem, showthat there exists at mostone closed orbit in R.
Concept Introduction:
Dulac’s Criterion: For
The hypotheses of the Dulac’s criteria are
If the limit cycle doesn’t enclose a hole in the region, then the cycle is not possible.
If the limit cycle encloses a hole in the region, then there is at least one limit cycle.
Green’s theorem: If C is positively oriented, simple curve, piecewise smooth, and D be the region enclosed by the curve. If P and Q have continuous first order partial derivatives on D then
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Nonlinear Dynamics and Chaos
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