The Bendixson–Dulac theorem is a mathematical principle in the study of dynamical systems, specifically regarding the qualitative behavior of autonomous ordinary differential equations in two dimensions. The theorem provides a criterion for the non-existence of periodic orbits, such as limit cycles, within a given simply connected region of the phase plane.
Historical Background
The theorem is named after the Swedish mathematician Ivar Otto Bendixson and the French mathematician Henri Dulac. Bendixson published an initial version of the criterion in 1901, and Dulac later refined and extended the method in 1933. It serves as a "negative criterion," meaning it is utilized to rule out the possibility of closed trajectories, complementing the Poincaré–Bendixson theorem, which is often used to establish the existence of such orbits.
Formal Statement
Consider a system of two autonomous first-order ordinary differential equations: $$\frac{dx}{dt} = f(x, y)$$ $$\frac{dy}{dt} = g(x, y)$$ where the functions $f$ and $g$ are continuously differentiable in a simply connected region $D \subset \mathbb{R}^2$. If there exists a continuously differentiable, real-valued function $\phi(x, y)$, known as a Dulac function, such that the expression $$\frac{\partial (\phi f)}{\partial x} + \frac{\partial (\phi g)}{\partial y}$$ maintains a constant sign (either consistently positive or consistently negative) almost everywhere in $D$, then the dynamical system has no periodic orbits lying entirely within the region $D$.
Mathematical Mechanism
The proof of the theorem typically relies on Green's theorem. If a closed orbit (a simple closed curve) were to exist in the region $D$, the line integral of the vector field across that curve would be zero. However, by applying Green's theorem, the area integral of the divergence of the scaled vector field $(\phi f, \phi g)$ over the region enclosed by the orbit must also be zero. If the divergence has a constant sign throughout the region, the integral cannot be zero, thereby creating a contradiction that proves the orbit cannot exist.
Application and Limitations
The Bendixson–Dulac theorem is a fundamental tool in planar dynamics, used to simplify the analysis of global phase portraits by excluding complex oscillatory behaviors in specific areas of the state space. It is frequently applied in fields such as mathematical biology, chemistry, and physics to verify the stability of equilibrium points and ensure that systems do not settle into unintended periodic oscillations.
The primary challenge in applying the theorem is that there is no general systematic method for determining an appropriate Dulac function $\phi(x, y)$ for any given system. Analysts often test common functional forms, such as constants (which reduces the theorem to the original Bendixson criterion), power functions, or exponential functions, to satisfy the sign condition.