3.1: Problem Set

EXERCISE \(\PageIndex{1}\)

Consider the following autonomous vector field on R:

\[\dot{x} = x-x^3, x \in \mathbb{R}. \label{3.11}\]

• Compute all equilibria and determine their stability, i.e., are they Lyapunov stable, asymptotically stable, or unstable?
• Compute the flow generated by (3.11) and verify the stability results for the equilibria directly from the flow.

EXERCISE \(\PageIndex{2}\)

Consider an autonomous vector field on \(\mathbb{R}^n\):

\[\dot{x} = f(x), x \in \mathbb{R}^n. \label{3.12}\]

Suppose \(M \subset \mathbb{R}^n\) is a bounded, invariant set for (3.12). Let \(\phi_{t}(\cdot)\) denote the flow generated by (3.12). Suppose \(p \in \mathbb{R}^n, p \notin M\). Is it possible for

\(\phi_{t}(p) \in M\),

for some finite t?

EXERCISE \(\PageIndex{3}\)

Consider the following vector field on the plane:

\(\dot{x} = x-x^3\),
\[\dot{y} = -y, (x, y) \in \mathbb{R}^2. \label{3.13}\]

1. Determine 0-dimensional, 1-dimensional, and 2-dimensional invariant sets.
2. Determine the attracting sets and their basins of attraction.
3. Describe the heteroclinic orbits and compute analytical expressions for the heteroclinic orbits.
4. Does the vector field have periodic orbits?
5. Sketch the phase portrait.6