Bendixson–Dulac theorem

The Bendixson–Dulac theorem rules out periodic orbits in a simply connected planar region when a suitable weighted divergence has a strict sign throughout it.

Divergence: A scalar measure of the net outward flow of a vector field from an infinitesimal region. The theorem tests whether the vector field’s weighted divergence keeps one strict sign.
Planar dynamical system: An autonomous differential equation whose state consists of two real variables. The theorem concerns trajectories in a two-dimensional phase plane.
Lotka–Volterra equations: A pair of differential equations modeling interactions between a predator population and its prey. Dulac functions can rule out cycles in variants of this ecological model.
Poincaré–Bendixson theorem: A theorem describing the possible limit sets of bounded planar trajectories when equilibria are absent from the limit set. It can support cycle-existence arguments, while Bendixson–Dulac supplies a cycle-exclusion test.
Ivar Bendixson: A Swedish mathematician known for work on differential equations and the nonexistence of periodic solutions. His work supplied the original divergence-based exclusion criterion.
Dulac function: A differentiable scalar function used to weight a planar vector field in the Bendixson–Dulac criterion. Choosing this function can make the weighted divergence sign-definite.
Autonomous system: A differential equation whose governing rule does not depend explicitly on time. The theorem applies to time-independent planar vector fields.
Competitive Lotka–Volterra equations: A system modeling two species whose populations suppress each other’s growth. A suitable weighting can exclude periodic trajectories in its positive quadrant.
Hopf bifurcation: A local bifurcation in which an equilibrium changes stability as a conjugate pair of eigenvalues crosses the imaginary axis. It explains how cycles can arise as parameters vary, a conclusion the criterion does not provide.
Henri Dulac: A French mathematician who developed criteria for excluding periodic solutions of differential equations. His contribution generalized the criterion through a suitable scalar weighting.

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