3. Stability of Fixed Points for Non-linear ODEs

Question 1

Definition
Fixed point of autonomous ODE.

Question 2

Definition
Flow.

Question 3

Definition 3.2
Fixed points: Lyapunov stable, asymptotically stable, and unstable.

Question 4

Definition 3.3
Fixed points: hyperbolic, sink, source, and saddle.

Question 5

Theorem 3.4 (Linear Stability Test)

Proof.

Question 6

Corollary 3.5
A hyperbolic fixed point is asymptotically stable iff.

Question 7

Lemma 3.7 (Variation of Constants)

Proof.

Question 8

Corollary 3.8
Sources are what?

Proof.

Question 9

Proposition 3.9
If N : [0, \infty) -> [0, \infty) and N(t) >= LN(t), L > 0, then…

Proof.

Question 10

Definition 3.13
Lyapunov function.

Question 11

Theorem 3.14 (Lyapunov’s Stability Theorem)

Proof.

Question 12

Definition 3.15
Positively invariant set.

Question 13

Definition 3.16
Basin of attraction for asymptotically stable fixed point.

Question 14

Lemma 3.17
The basin of attraction is … for every …

Proof.

Question 15

Theorem 3.18
Suppose L : Z -> [0, \infty) is a Lyapunov function for a fixed point x*. Let a > 0 and set V = {…} and K = {…}.

If K is compact then…
If also L is strict then…

Proof.