3. fixed points; and in R C closed orbit · Poincar´e- Bendixson Theorem I Poincar´e-Bendixson Theorem: Suppose 1. R is a closed, bounded subset of the plane; 2. x˙ = f(x) is

� Poincare- Bendixson Theorem

I Poincare-Bendixson Theorem: Suppose

1. R is a closed, bounded subset of the plane;

2. x = f(x) is C1 on an open set containing R.

3. R does not contain any fixed points; and

4. There exists a trajectory confined in R.

Then either C is a closed orbit,or it spirals toward a closed orbit.

I Need to construct a trapping region R.

� Existence of Closed Orbit

I Considerr = r(1− r2) + µr cos θ, θ = 1.

Show: closed orbit exists for small µ > 0.

I Construct trapping region (0 < rmin ≤ r ≤ rmax):

• rmin <√

1− µ.

•√

1 + µ < rmax.

� Limit Cycle in Glycolysis

I Consider

x = −x + ay + x2y, y = b− ay − x2y.

where x, y are concentrations of ADP and F6P. a, b > 0.

I Construct trapping region.

� Limit Cycle in Glycolysis

I Find limit cycle for

x = −x + ay + x2y, y = b− ay − x2y.

I Fixed point:

(x∗, y∗) = (b,b

a + b2)

I Unstable ifb2 = 1

2(1− 2a±√

1− 8a).

� No Chaos in Phase Plane

I Limit possibilities in phase plane:If a trajectory is confined to a closed, bounded regionwith no fixed points. then it must approach a closed orbit.

I For n ≥ 3, trajectories may wander in a bounded region withoutsettle down to a fixed point or a closed orbit.

I Trajectories may be attracted to a strange attractor. Motionis chaotic and sensitive to tiny changes in the initial conditions.