Course page for Differential Equations
1st March (Lecture 14): We covered local existence and uniqueness of solutions. Sections 8.1-8.3 in Hirsch-Smale.
Homework: Problems 1-4 on p.177 in Hirsch-Smale. In addition, attempt Homework 1.
4th March, 8th March (Lecture 15, 16): We finished the proof that a Lipschitz continuous vector field generates a continuous flow. The flow is well-defined on an open interval of the product of the real line (time co-ordinate) and the phase space. This is proved in the rest of Chapter 8 of Hirsch-Smale, but the approach in class was different.
Homework: Problems 6,7, 9 in Hirsch-Smale, p.177. In addition, attempt Homework 2. We will discuss homework on Tuesday 15th March.
11th March (Lecture 17): We discussed how flow and vector fields transform under diffeomorphism of phase space. We showed that in the neighbourhood of a point in the phase space that is not an equilibrium point, there is a diffeomorphism that transforms the flow to a standard form (called a flow box). Refer to Chapter 11, Section 2 in Hirch-Smale. Next we started Chapter 9 in Hirsch-Smale: we defined stability and asymptotic stability of equilibria, and stated a sufficient condition for asymptotic stability.
15th March (Lecture 18): We proved the Theorem in Section 9.1 of Hirsch-Smale.
Homework: Problems 2, 3, 4 on p.185, Problems 2, 3 on p. 191 of Hirsch-Smale.
18th March (Lecture 19): We discussed problems in Homework 2. We also talked about left and right-invariant vector fields on GL(n,R). The vector field in Problem 2 is a time-dependent right invariant vector field on GL(n,R).
22nd March (Lecture 20): I assigned a homework problem Homework 3 that is to be submitted. Aim to submit by 29th March, however 1st April is the hard deadline.
In class today, we discussed the assigned homework problem. Next we defined Liapunov functions (see Section 9.3 of Hirsch-Smale). We showed that for a linear system all of whose eigenvalues have negative real part, we can construct Liapunov functions - it is just the norm square function for a suitable choice of inner product on the vector space.
29th March (Lecture 21): Homework 4 is due latest by Friday April 8th. You are encouraged to aim to finish by Tuesday April 5, so that you can ask me questions.
In class on 29th March, we talked about topological classification of linear systems.
1st and 5th April (Lectures 22 and 23): Discussed theorems 1 and 2 of Section 9.3 about Liapunov stability. Here is a note filling in the gap in the proof of Theorem 2 in Section 9.3. Electrical ciruits were introduced - we covered Sections 10.1 and 10.2.
8th April (Lecture 24): Section 9.4 on gradient flow. Homework 5 is due on 19th April Tuesday. This is the last homework that has to be submitted. Out of the 5 grades coming out of 2 quizzes and 3 homeworks, the lowest one will be dropped.