4. Orbits, Invariant Sets, and their Stability

Question 1

Definition 4.1
Orbit of a point.
Forward and backward orbit.
Fixed point.
Periodic point.
Periodic orbit.
Period of periodic orbit.

Question 2

Exercise 4.2
The period of a periodic orbit is what?

Question 3

Definition 4.3
Homoclinic and heteroclinic points.
Homoclinic and heteroclinic orbits.
Heteroclinic loop.

Question 4

Definition 4.4
Invariant subset: positively and negatively.

Question 5

Proposition 4.5
If A \subset X is invariant then…

Proof.

Question 6

Definition
Set of fixed points.

Question 7

Proposition 4.6
If I \subset R \ Fix(f) is bounded then…
The orbit is…

Proof.

Question 8

Corollary 4.7
When X = R, the orbits consist of…

Question 9

Definition 4.8
Invariant set: Lyapunov stable, unstable.

Question 10

Definition 4.9
Attracting Invariant Set.
Its basin of attraction.
Asymptotically Stable Invariant Set.

Question 11

Definition 4.10
Omega- and alpha-limit sets.

Question 12

Proposition 4.11
a) If x* is a fixed point, then w(x) = {…} = …
b) If x is periodic then w(x*) = …

Proof.

Question 13

Exercise
If x is a homo- or heteroclinic point, then w(x) is one point.

Question 14

Proposition 4.12
w(x) is what for every x \in X.

Proof.

Question 15

Lemma 4.13
If x \in X and O+(x) is bounded then…

Proof.

Question 16

Lemma 4.14
Let x \in X, O+(x) bounded, if w(x) = A \cup B s.t. … then…

Proof.

Question 17

Definition 4.14
A Lyapunov function for a flow.
M_E = {…}

Question 18

Theorem 4.16 (LaSalle Invariance Principle)

Proof.

Question 19

Remark 4.17
If L < 0 on large parts of V, then M_E what?