# TMA4165 Differential Equations and Dynamical Systems - Spring 2019

## Exercises

There are no mandatory exercises. Solutions, partial or full in some cases, will be posted as time permits. (There will be delays during the first few weeks of the term; I'll catch up as soon as I can.)

week no Exercise Solution
3 1 J&S problem 1.1(i–iv), 1.2
For 1.2, just consider the cases $\alpha>0$ and $\alpha\le0$ separately. No need to try the method of §1.7 (but feel free to try it if you wish!
4 2 J&S problem 1.4–6.
Consider the initial value problem $\dot x=x^2+e^{-|x|}$ with $x(0)=0$: Show that the problem has a unique solution, and that the maximal interval of existence has the form $(-a,a)$ with some $a\in(\frac{1}{2}\pi,\infty)$. Give a specific upper bound for $a$, as good as you can get it without working too hard at it.
Hint: Write up and estimate an integral expression for the blowup time $a$.
5 3 J&S problem 2.1 (i)–(iv), (vi); 2.3 (iii); 2.4 (i), (iii); 2.19
6 4 J&S problem 2.3 (vi), (ix); 2.47
Revisit 2.4 (i) viewing it as a Hamiltonian system (find the Hamiltonian and consider the types of its critical points: maxima, minima, saddles)
Given a system $\dot x=X(x,y)$, $\dot y=Y(x,y)$, consider the system $\dot u=-Y(u,v)$, $\dot v=X(u,v)$. Note that the trajectories for the $(u,v)$ system are orthogonal to the trajectories for the $(x,y)$ system. Assume that the origin is an equilibrium point, and that the linearisation is a centre for the $(x,y)$ system. Show that it is a node for the $(u,v)$ system.
7 5 J&S problem 2.41, 3.1, 3.3, 3.8, 3.11, 3.23
Hint for 3.23 (Dulac's test): Use Green's theorem. (One form of which is really the two-dimensional divergence theorem.)
8 6 J&S problem 11.4, 5, 15
Assume a Liénard equation (system) satisfying the conditions of Theorem 11.2 has only one closed orbit. Shows that any nonzero solution converges to that orbit. (Hint: Assume it is not so, and try to derive a contradiction.)
Find and classify all equilibrium points of the system $\quad\dot x=x-y,\quad \dot y=x^2-1\quad$. Sketch the phase diagram. Does the system have any closed orbits?
Determine whether the system $\quad\dot x=y,\quad \dot y=-x-(1+x^2)y-x^4y^3\quad$ has any non-constant periodic solutions.
Consider a population model $\quad\dot x=xF(x,y),\quad \dot y=yG(x,y)\quad$ where $\partial F/\partial x<0$ and $\partial G/\partial y<0$. Show that there are no closed orbits in the first quadrant.
9 7 J&S problem 8.1, 3, 11, 19–21, 24
10 8 J&S problem 8.14, 16, 18, 26, 27
Also: Assuming that the linear system in several variables $\dot x=A(t)x$ has fundamental matrix $\Phi(t)$, what is the flow of the system?
Edited to add: Do this first for autonomous systems ($A(t)$ independent of $t$). For the non-autonomous case, add a single coordinate to get an equivalent $n+1$-dimensional autonomous system in order to define the flow.
11 9 This week's problems revisit earlier parts of the course, and extends the index theory with a useful technique. Rough solution
12 10 J&S problems 12.1, 4, 5, 7, 9, 10, 11 Rough solution
13 11 J&S problems 12.13, 15, 17, 18
In 12.18, the text says “Show that $x=x^2+y^2$ is a solution of this system of equations.” More appropriate would be: “… is an invariant set (or manifold) for this system of equations.”
This problem set was much too hard.
Rough solution
14 12 J&S problems 12.20, 22
A Lotka–Volterra system with logistic growth for the prey can be written as $\dot x = x-x^2-xy$, $\dot y = -ay + bxy$ with positive constants $a$ and $b$. (The variables have been rescaled to rid the first equation of any coefficients.) Find and classify the equilibrium points, and show that the system undergoes a saddle-node bifurcation. Draw some representative phase diagrams.
Review of earlier material, as time permits: J&S problems 2.33, 44; 3.19, 24, 29
15 13 J&S problems 10.1 (ii, iii, iv, ix), 3, 5, 7, 15, 16, 17 as far as time permits