Bendixson-Dulac Theorem
Theorem
Suppose there exists a continuously differentiable function $\alpha \left({x, y}\right)$ on a simply connected domain.
What is the domain? Reals, complex, or what?
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Suppose that:
- $\nabla \cdot \left({\alpha F}\right)$
is either always positive or always negative.
Then the two-dimensional autonomous system:
- $ \left({x, y}\right)' = F \left({x, y}\right)$
does not have a periodic solution.
Proof
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Source of Name
This entry was named for Ivar Otto Bendixson and Henri Claudius Rosaris Dulac.
