Poincare-Bendixon Theorem: periodic orbits in a second order autonomous system

A gentle introduction to the fundamentals of nonlinear dynamics and controls

Introduction

The poincare bendixon theorem gives conditions for the existence of periodic orbits in a second order autonomous system.

\(Lemma\): The bounded region trapped by the closed curve contains a periodic orbit if the following conditions are satisfied:

The bounded region does not contain any equilibrium points. If there exist equilibrium points, then they must be unstable. Unstable equilibrium means eigen values of jacobian matrix \(\partial f/\partial x\) have positive real parts.

The bounded region defines an invariant set.

Imaginatively,

we can say that because the bounded region defines an invariant set, the trajectory of the system cannot leave the bounded region. This means that the trajectory must be trapped in the bounded region. Since the bounded region does not contain any equilibrium points, the trajectory cannot converge to any equilibrium point. This means that the trajectory must be periodic.

Mathematically,

let the bounded region define a set \(M=\{ V(x) = C\}\) where \(V(x)\) is a scalar function and \(C\) is a constant. \(x = \begin{bmatrix}q \\ \dot{q}\end{bmatrix}\) is the state vector. We can impose that for the bounded region \(M\) to be invariant, the directional derivative of \(V(x)\) along the trajectories of the system should be less than 0.

even if we find an unstable equilibrium inside this bounded region we can always construct the set \(M\) such that the equilibrium point is not contained in it. That is, on the shorter boundary of the annular region, the directional derivative is positive and on the longer boundary of this annular region the directional derivative is negative thereby trapping the trajectory in the annular region, and thereby proving the existence of a periodic orbit.