Limit Cycles and the Poincare-Bendixson Theorem

Limit cycles are isolated periodic orbits that attract or repel nearby trajectories. Their existence is guaranteed under certain conditions by the Poincare-Bendixson theorem.

Limit Cycles

Definition6.5Limit cycle

A limit cycle is an isolated closed trajectory $\gamma$ in the phase plane. It is stable (attracting) if nearby trajectories spiral toward $\gamma$ , unstable (repelling) if they spiral away, and semi-stable if attraction occurs from one side and repulsion from the other.

ExampleA system with a limit cycle

The system $r' = r(1-r)$ , $\theta' = 1$ (in polar coordinates) has the unit circle $r = 1$ as a stable limit cycle. For $r < 1$ , $r' > 0$ (trajectories spiral outward); for $r > 1$ , $r' < 0$ (trajectories spiral inward).

The Poincare-Bendixson Theorem

Theorem6.3Poincare-Bendixson theorem

Let $R$ be a closed, bounded region in $\mathbb{R}^2$ that contains no equilibrium points. If a trajectory of an autonomous system enters $R$ and remains inside for all $t > t_0$ , then the trajectory either is a periodic orbit or spirals toward a periodic orbit (limit cycle) in $R$ .

RemarkSignificance

The Poincare-Bendixson theorem is specific to $\mathbb{R}^2$ and fails in $\mathbb{R}^3$ (where chaotic behavior is possible). It provides the main tool for proving existence of limit cycles by constructing a trapping region.

ExampleProving existence of a limit cycle

For the Van der Pol equation $x'' - \mu(1-x^2)x' + x = 0$ with $\mu > 0$ : the origin is an unstable spiral (trajectories leave any small disk). The function $V = x^2 + y^2$ satisfies $\dot{V} < 0$ on a sufficiently large circle (trajectories enter the disk). Therefore, the annular region between the small and large circles is a trapping region with no equilibria, and a limit cycle exists by Poincare-Bendixson.

Dulac's Criterion (Ruling Out Limit Cycles)

Theorem6.4Bendixson-Dulac criterion

If there exists a $C^1$ function $B(x,y)$ such that $\frac{\partial(Bf)}{\partial x} + \frac{\partial(Bg)}{\partial y}$ is of one sign (and not identically zero) on a simply connected domain $D$ , then the system $x' = f$ , $y' = g$ has no periodic orbits in $D$ .

ExampleNo limit cycles for a gradient system

For $x' = -\partial V/\partial x$ , $y' = -\partial V/\partial y$ (gradient system), take $B = 1$ : $\frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} = -V_{xx} - V_{yy} = -\Delta V$ . If $V$ is strictly convex ( $\Delta V > 0$ ), the criterion applies and no limit cycles exist.

RemarkIndex theory

The index of a closed curve $\gamma$ with respect to the vector field $(f, g)$ counts the net rotation of the field around $\gamma$ . Key results: the index of a node, focus, or center is $+1$ ; the index of a saddle is $-1$ ; any closed orbit must enclose equilibria whose indices sum to $+1$ . This constrains where limit cycles can exist.