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MAT 574 Theory of Differential Equations

Chetaevs instability theorem

When trying to adapt Lyapunov methods for stability to an instability theorem, a first thought mightbe to keep V positive definite and argue with indefinite V. But unlike in stability theorems, one doesneed to consider all trajectories. Hence it suffices to identify a suitable region in which both V and Vhave the same sign. The following statement and arguments use the delicate intersection of a closed set

Fwith the open set V1(0,).

Theorem [adapted from Chetaev 1934]. Suppose E Rn is open, 0 E, f: E Rn and V: E Rare continuously differentiable, f(0) = 0, V(0) = 0. Suppose Uan open neighborhood of 0 with compactclosure F = UEsuch that the restriction ofV = (DV)f to F V1(0,) is strictly positive. If forevery open neighborhoodW Rn of 0 the setWV1(0,) is nonempty, then the origin is an unstableequilibrium of x= f(x).

Proof.

1. Assume above hypotheses.

2. Let Wbe an open neighborhood of 0 Rn.

3. There existsz W V1(0,). Define a= V(z)> 0.

4. The set K=F V1[a,) is compact, and it is nonempty since z K.

5. Using the continuity ofV and V there exist m= minxK

V(x) and M= maxxK

V(x).

6. SinceK=F V1[a,) F V1(0,) it follows that m >0.

7. LetI[0,) be the maximal interval of existence for : I Esuch that(0) =z , and =f.

8. Define the setT={t I: for all s [0, t], (s) K}.

9. For allt T, ddt

(V )(t) = (V )(t) m.

10. Hence for allt TV((t)) =V((0) +

t

0

V((s)) ds a + mt.

11. SinceV is bounded above by M, T is also bounded above. Let t0 the the least upper bound ofT.

12. SinceKE, I[0, t0] and (t0) is well-defined. Moreover t0 lies in the interior ofI.

13. Since for all t [0, t0), (t) K, and for every >0 there exist t I , t0 < t < t0+ such that

(t)Kit follows that (t0) lies on the boundary of the set K.

14. The boundary ofKis contained in the union KV1(a) (F\ U).

15. SinceV((t0)) a + mt0 > a it follows that (t0)V1(a).

16. Therefore(t0) (F\ U), and, in particular, (t0)U.

17. SinceWwas arbitrary, this shows that the origin is an unstable equilibrium point off.

All rights reserved, Matthias Kawski Version 0.9: October 17, 2007.http://math.asu.edu/kawski