2013 exam question 2.

A student has asked me about this exam question (I paraphrase):

• In part a, how is the stable manifold determined?
  
  The stable manifold of the equilibrium 0 of a linear ODE is precisely equal to the union of all (generalized) eigenspaces for eigenvalues that have negative real part. Namely, on these (generalized) eigenspaces, all solution curves converge (exponentially fast) to the equilibrium 0, whereas on the other eigenspaces solutions that start outside the equilbrium never converge to 0. These properties all follow from the explicit solutions of the linear ODE restricted to generalized eigenspaces. In the example at hand the matrix A has eigenvalues 1 and -1 and the eigenvector for -1 is (1,-1). Hence the stable manifold is the line through 0 and (1,-1). 
• In part d, why does the model answer use the Euler-Lagrange equation rather than the conservation of the Hamiltonian?
  
  Either route is possible. I will show here how we can obtain the answer using the conservation of the Hamiltonian. The Hamiltonain is given by \(H=y'f_{y'}[y]-f[y]=(2y)^2+(y')^2\). The level sets \(H=c\) are ellipses in the \(y-y'\) plane if \(c>0\). We observe that \(c\geq0\). \(H=0\) corresponds to an equilibrium. So let \(c=d^2\). Then we have \(y'=\pm\sqrt{d^2-(2y)^2}\) which can be solved by separation of variables
  \[\int dt =\int  \sqrt{d^2-(2y)^2}^{-1}.\] Then \[ t+T=\frac{1}{2}\tan^{-1}\left(\frac{2y}{d^2-4y^2}\right),\] where \(T\) is constant. After some algebraic manipulations, one can rewrite this as \(y=\pm d \cos(2(t+T))\). The boundary condition \(y(0)=0\) yields \(T=\pm\frac{\pi}{4}\)  so that \(y=\pm d \sin(2t)\) and \(y(1)= 1\) implies that \(y(t)=\frac{\sin(2t)}{\sin(2)}\), in accordance with the model answer.
  
  Clearly in this case, the calculations using the Hamiltonian appear more involved than using the Euler-Lagrange equation. Which route to the answer is most efficient will depend on the example. My advice would be when approaching such a calculation, is to try one route and if it looks tedious, quickly try the other one as well to see if it makes a difference.

Bifurcation points

A few students have asked me what they need to know about bifurcation points. Of course we only touched upon this topic superficially (and a more detailed analysis is well beyond the scope of M2AA1). If an equilibrium point is hyperbolic (no eigenvalues of the Jacobian are on the imaginary axis), then the flow near the equilibrium is determined by its linearized flow (Hartman Grobman theorem) and the flow does not essentially change under sufficiently small perturbations, so hyperbolicity is a counter indicator for (local) bifurcation. If an equilibrium is not hyperbolic, then small perturbations to the vector field may lead to substantial changes of the flow near the equilibrium, which would amount to a (local) bifurcation. We have not really discussed the precise analysis of the flow at a non-hyperbolic equilibrium point, so this is not something you need to master. However, if we have a parameter in our problem, and at one value of the parameter we have a non-hyperbolic equilibrium, we can often induce from the local behaviour near the hyperbolic equilibrium/equilibria before and/or after this parameter value, what may have happened at the bifurcation point. Nothing beyond this superficial level of analysis will be expected or required at the exam.

Phase portrait sketch in case of a Lyapunov function that is a conserved quantity

A student asked me:"I was looking at the past exam paper from 2010, and in question 03. (a) (iii) when drawing the phase plane, was wondering how we are supposed to conclude that it is a periodic orbit? I'm not completely sure how to proceed after finding the nullclines and the direction of the solution curve."
In the 2010 exam question 3(a)(iii), we have a Lyapunov function \(V\) with \(\frac{d}{dt}V=0\). This means that the level sets \(V_C:=\{x~|V(x)=C\}\) are flow-invariant and thus that every solution curve must lie inside one particular level set. As the level sets \(V_C\) with \(C>0\) in this example are closed curves that do not contain equilibria, necessarily these level sets must be periodic solutions. The level set \(V_0\) is the unique equilibrium \((x,y)=(1,1)\).
A similar situation was encountered in Test 2, question 2(ii).

Lyapunov functions and phase portraits

A student asked me: "I have a question regarding Q3 on the 2008 M2AA1 paper. In the question you are asked to sketch the phase portrait of the system for various parameters. The question gives you two pictures of the lyapunov function for these parameters. I was wondering how one might use a lyapunov function to deduce the phase portrait as the question suggests. I have checked the answers for that question which don't give much detail, however the portraits look remarkable similar to the lyapunov functions. "

Let \(V(x)\) be a Lyapunov function, ie \(\frac{d}{dt}V(x)\leq 0\). Then \(V(x(t)\) must not increase when \(t\) increases. It is also insightful to note that \(\frac{d}{dt}V(x)=\nabla V(x) \cdot\dot{x}\leq 0\). Recall that \(\nabla V(x)\) is the normal to level set \(\{\tilde{x}~|~V(\tilde{x})=V(x)\}\) at the point \(x\). Thus \( \nabla V(x)\cdot \dot{x}\leq 0\) indeed means that the vector \(\dot{x}\) does not point in the direction of higher level sets of \(V\)and thus that if \(t\geq\tau\) then \(V(x(t))\leq V(x(\tau))\).

Sketching phase portraits

A student asked me:
"With regards to Problem Sheet 7, question 2 (the first question on the sheet), after sketching the nullclines and determining the direction of flow at the nullclines, as well as the nature of the equilibrium point at (0.5,0.5) - stable equilibrium, how should I attempt to sketch the phase portrait, i.e. the diagram on the right in the answers. "

It may be instructive to go through the steps of how to try sketching a phase portrait:

Locally near equilibria:
1. Find the equilibria.  (Depending on the type of the equations, this may be easy or impossible. In most exercises this is easy.)
2. Calculate the linearization of the vector field (ie Jacobian) at these equilibria and deduce ,  where possible (ie when hyperbolic), what the phase portrait should look like near these equilibria.

More globally:
1. Where relevant or possible, determine a bounded invariant set to which all solutions are attracted (most of our examples have such a region).
2. Try drawing nullclines where the vector field (and thus the tangent to solution curves) is horizonal and vertical. These nullclines may be helpful. In simple examples, often the nullclines can be computed explicitly.
3. Where there are saddles, it may be useful to try sketching where stable and unstable manifolds may end up.
4. Determine possibilities for the \(\omega\)-limit sets, in view of the Poincare-Bendixson theory.

All-in-all, this is what you should be doing when sketching a phase portrait. It often not possible to get all the properties of the flow this way, so that there still may be some unknowns. In particular, it is often hard to rule out the existence of a periodic solution around an attracting or repelling equilibrium point (unless it lies at the border of a forward invariant set, in which case no periodic solutions can encircle it).

If at the exam, you think there is some ambiguity or unknown property of the phase portrait, you should just write that. "Sketching" a phase portrait is precisely that: provide those features which you are certain of and discuss which additional features may or may not be present, on the basis of the theory.

For the specific problem in question: it is completely reasonable to conclude that around the attracting equilibrium (0.5,0.5), the Poincare-bendixson theorem leave open the possibility of an \(\omega\)-limit set that is a periodic solution encircling this equilibrium. The argument I give in the model answer is correct, but not so easily verifiable (so you are not supposed to discover such a subtle property in an exam question, for instance).

I hope this helps.

Revision classes

Just to let you know that after consultation with the class rep, it has been decided that the scheduled revision classes of Tuesday 26 April 12:00-13:00 and Tuesday 3 May 12:00-13:00 in Clore, will take the form of problem classes. Graduate Teaching Assistants and I will be available for questions.

Guidance on past exam papers

Past summer M2AA1 exam papers can be found here (the course first featured in 2007-2008).    I provide some guidance here on the relevance of the questions on the past papers for the current exam.

2008: all questions
2009: all questions apart from Q3(c,d,e); for the model answers to Q3 (which do not seem to appear on the web), see this link
2010: all questions
2011: not relevant
2012: not relevant
2013: all questions apart from Q3.
2014: all questions apart from Q1b(ii) and Q3
2015: all questions apart from Q1(d) and Q3

I was the setter (and lecturer of the course) in 2008, 2009 and 2010. It may be no surprise that the exam questions in these years are more representative of my style, than those of the other years.

Coordinate systems for calculus of variations

Your class rep wrote me "A student asked me about calculus of variations, if we have to be able to define coordinate systems ourselves? For example in Problem sheet 8 Question 8, students might struggle to get started on it."

This question addresses a common type of anxiety that some of you may feel. It is unjustified.

Problem sheet exercises are not necessarily model exam questions. When I list question 8 as "important" this means that I think it is important you understand how this problem is solved, and that it is a good exercise. Of course, in this question a "tricky" part is how to define the coordinates. If I would use this example as the base for an exam question, I would find it most important that you can show how to apply the Euler Lagrange equation. Most likely, as getting the coordinates right is perhaps a little tricky, I would most likely suggest you to use certain coordinates, since without a reasonable choice, you would not be able to get anywhere.

As I told many of you before: it is my "problem" to set an exam that tests how well you understand the material. So I would aim to avoid exam questions to depend on "tricky" bits that are not at the core of the course material.

So in answer to the question whether "[do] we have to be able to define coordinate systems ourselves?", the answer is "I would likely avoid obstacles, so if the choice of coordinates is not obvious, I would likely provide you with a suggestion."

I hope this helps.

Lecture notes Chapter 5 and Chapter 6

Chapters 5 and 6 of the lectures notes correspond to Chapters 10 and 11 of  Differential equations, dynamical systems and an introduction to chaos  by M.W. Hirsch, S. Smale and R.L. Devaney. Please be aware that the entire book can be consulted digitally from the Imperial library website (where these chapters can also be downloaded; just search for the authors on the Library search and you find a link to the digitcal copy). Because of copy-right issues I unfortunately cannot link on this webpage directly to these chapters. I understand that the class reps also sent around instructions how this material can be downloaded and consulted from the library website.

Chapter 7 lecture notes

I have just posted typed up lecture notes of Chapter 7 on the Calculus of Variations. I apologize for the delay in finishing this. I also updated the instructions on "how to study for the exam" concerning this chapter of the notes.