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POINCARÉ-BENDIXSON THEORY

The results of Worked Example 1 can be formalised in the following theorem.

Theorem - Poincaré- Bendixson

Let D be a closed bounded region of the x-y plane and

be a dynamical system in which f and g are continuously differentiable. If a trajectory of the dynamical system is such that it remains in D for all then the trajectory must

> > (i) a closed orbit,

> > (ii) approach a closed orbit or

> > (iii) approach an equilibrium point as .

The implication of this theorem is that if we can find a trapping region for a dynamical system which does not contain an equilibrium point then there must be at least one limit cycle within the region. The technique of transforming to polar coordinates although often employed does not always work. The following examples demonstrates other methods of obtaining trapping regions.

- Worked Example 2

In the biochemical process of glycolis, living cells obtain energy by breaking down sugar molecules. Often this process turns out to be oscillatory in nature. A simple model of the process is described by the equations

where x and y are concentrations of reactants and a and b are positive parameters.

Maple Solution

The diagram below shows the graphs of the expressions

and

on which and respectively. The region within the dotted lines is a trapping region for the system.

However we cannot conlude that there is a limit cycle within region since the equilibrium point lies within the region.

The Jacobian matrix of the linearzation is

Now and

If we choose the parameters a and b so that then the equilibrium point will be a repeller. In this case suuround the equilibrium point by a small circular region as shown below.

Since the equilubrium point is a repeller, trajectories crossing the boundary must point into the region defined by the dotted lines. In this case the new region is a trapping region which does not include an equilibrium point. Thus the Poincaré-Bendixson theorem guarantees that the region must contain at least one limit cycle.

We can verify this result using Maple to obtain the phase portrait. Suitable parameter values are

Limit Cycle

- Worked Example 3

By considering the rate of change with respect to time of the function

on the trajectories of the dynamical system

show that the dynamical system has a stable elliptical limit cycle.

Maple Solution

It is easily seen that the only equilibrium point of the system occurs at the origin.

Now

Thus we see that V decreases on the trajectories except at the origin where it clearly has a relative maximum, and on the ellipse where it is identically zero.

Consider the plot of V shown below

Outside the ellipse the function is decreasing towards the ellipse. Inside the ellipse the function is decreasing towards the ellipse again. Thus the trajectories of the system must move towards the ellipse from points outside and inside for increasing t.

Now consider a contour plot of V

We can construct a trapping region by taking any contour inside the ellipse as the inner boundary of an annular region together with any contour outside the ellipse as the outer boundary. Hence by the Poincaré-Bendixson theorem since there are no equilibrium points within the annular region there must be a stable limit cycle within the region.

Note also that as the contours can be taken as close to the ellipse as we like the limit cycle must be the ellipse.