The Bendixson–Dulac theorem is a result in the field of dynamical systems and differential equations. It provides criteria to determine the existence of periodic orbits in a planar autonomous system of ordinary differential equations. Specifically, the theorem helps to identify non-existence of periodic orbits in a given planar system, which can be useful for analyzing the behavior of the system.
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If is simply connected and a function makes have one sign and not vanish identically on any open subset of , then the planar system has no periodic orbit lying in .