� 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.