Topics Covered

Oct 8th

- Grobman-Hartman theorem
- Stable and unstable manifolds
- Stable Manifold Theorem

Oct 10th

Recommended Exercises

Read Lynch, Section 3.4 and 3.5 and do problems in Lynch, Section 3.7 -- 4 (b), (c), (e), (f), 5, 6, 7, 8.
Consider the nonlinear system (see examples discussed in the notes)
$\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} =\begin{pmatrix} -x \\ y+x^2 \end{pmatrix}.$

(a) Given the initial condition $(x_0,y_0)$ , what is the solution to the system?

(b) Hence write down an explicit formula for the flow of the system $\phi_t : \mathbb{R}^2 \to \mathbb{R}^2$ .

(c) Show that the set $S=\{(x,y)\in \mathbb{R}^2|3y+x^2=0\}$ is invariant under the flow $\phi_t$ in part (b).

(d) Determine the behaviour of the solutions with initial conditions on $S$ , as $t \to \infty$ .
(e) Show that the origin is the only critical point of the system. Is it stable? If so, is it asymptotically stable?
(f) What is the linearization of the nonlinear system at the origin?

(g) Is the origin a sink, a source or a saddle?

(h) Find stable/unstable manifolds corresponding to the origin.
(i) Stable manifold theorem tells us that in this case the function whose graph gives us the stable manifold has a power series expansion. Use this fact to determine stable manifold.
(See Example 3 in the notes Non-linear systems II. By uniqueness of stable manifold your answer should agree with the one you obtained previously).
(j) Draw the phase portrait of the system near the origin.