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:
\(\dot{x} = x-x^3\),
\[\dot{y} = -y, (x, y) \in \mathbb{R}^2. \label{3.13}\]
- Determine 0-dimensional, 1-dimensional, and 2-dimensional invariant sets.
- Determine the attracting sets and their basins of attraction.
- Describe the heteroclinic orbits and compute analytical expressions for the heteroclinic orbits.
- Does the vector field have periodic orbits?
- Sketch the phase portrait.6