ODEs that do not give rise to a flow: non-unique solutions.
Recurrent and non-wandering points.
Readings
Section 7.1.
Homework #2
Due date: Feb 5
Find the general solution of the ODE $\dot x=\frac{1}{1+x^2}$, $x\in\mathbb R$, and show that:
the function $\frac{1}{1+x^2}$ is Lipschitz in $\mathbb R$;
this ODE has no equilibria;
this ODE gives rise to a flow on $\mathbb R$.
Find the general solution of the ODE $\dot x=\sqrt[3]{x}$, $x\in\mathbb R$, and show that:
the function $\sqrt[3]{x}$ is not Lipschitz in $\mathbb R$;
there are solutions that fall in finite time on the equilibrium solution $x(t)=0$;
this ODE gives rise to a semiflow but not to a flow on $\mathbb R$.
Consider the "constant speed" flow on the 2-torus $F^t(x,y)=(x+at,y+bt)$, where $a,b\in\mathbb R$ and the coordinates $(x,y)$ are meant mod 1
(we already talked briefly about this flow in class last semester but see the wikipedia page
for more info about it).
Show that, independently on $a$ and $b$, every point is recurrent under this flow.
Recall that a point $x$ is recurrent under a flow $F$ if, for every $\varepsilon>0$ and $\tau>0$, there is a time $T\geq\tau$ such that $d(F^T(x),x)<\varepsilon$.
Consider the flow sketched in the figure below. Points A,B,C,D are fixed while all blue points move counterclockwise so that, for instance, every blue point between A and B asymptote to B forward in time and to A backward in time. The center of the disc is fixed (red point) while every other point in the interior of the disc moves on a outward spiral that asymptotes to the boundary circle (in orange it is shown one of these spirals). Prove that no blue point is recurrent and that each of them is non-wandering. Is any orange point recurrent? is any non-wandering?
Recall that a point $x$ is non-wandering under a flow $F$ if, for every $\varepsilon>0$ and $\tau>0$, there is a point $y$ such that $d(y,x)<\varepsilon$ and a time $T\geq\tau$ such that $d(F^T(y),x)<\varepsilon$.
The ODE $\begin{cases}\dot x &= \phantom{-}y\\ \dot y&= -x\end{cases}$ and its relations with Physics (harmonic oscillator) and Geometry (Euclidean rotations and their Lie Algebra).
The ODE $\begin{cases}\dot x &= y\\ \dot y&= x\end{cases}$ and its relations with Physics (Special Relativity) and Geometry (Minowski rotations and their Lie Algebra).
The ODE $\begin{cases}\dot x &= \phantom{-}ax+by\\ \dot y&= -bx+ay\end{cases}$.
Readings
Section 7.3.
Extra Readings
Learn more about the harmonic oscillator from this lecture by Richard Feynman.
Learn a bit more about Lie groups and Algebras from this chapter of "Geometric Methods and Applications" by Jean Gallier (UPenn). It contains all I told you in class and much more.
Read elementary facts and see nice picts about Special Relativiy from this page by Izaak Neutelings (U. of Zurich, Swisse)
Let $A,B$ be two $n\times n$ matrices. Prove that, if $AB=BA$, then $e^{At}e^{Bt}=e^{(A+B)t}$.
Find the general solution of the system $\begin{cases}\dot x &= 3x-4y\\ \dot y&= 4x+3y\end{cases}$ and sketch, in the $(x,y)$ plane, the orbits of the system passing through $(1,0)$ and $(0,1)$.
Find the general solution of the system $\begin{cases}\dot x &= 3x+5y\\ \dot y&= x-y\end{cases}$ and write the solution passing through $(1,1)$.
Use the definition of exponential to find the exponential of the matrix $\begin{pmatrix}1&1&0\\ 0&1&1\\ 0&0&1\\\end{pmatrix}$.
Then, use this result to write the general solution of the system
$\begin{cases}\dot x &= x+y\\ \dot y&=y+z\\\dot z&=z\\\end{cases}$.
Classification of linear ODEs: the case of complex eigenvalues.
Symplectic geometry.
Stability of equilibria.
Readings
Section 7.5.
Extra Readings
I like a lot the lecture notes on ODEs by Kim Feldman (U. of Utah).
This first set is about how an ODE induces a flow.
This second set is about how linear ODEs.
This third set is about how limit sets and stability of equilibria.
Notice that these notes say much more than we did in class, you do not need to learn the extra stuff but certainly go over it if you like it.
Let $(x^1,\dots,x^n,p_1,\dots,p_n)$ be linear coordinates on $\mathbb R^{2n}$.
Given a function $H=H(x^1,\dots,x^n,p_1,\dots,p_n)$ (called Hamiltonian function), consider the ODE system
$$
\begin{cases}
\dot x^1&=\phantom{-}\frac{\partial H}{\partial p_1}\\
\vdots\\
\dot x^n&=\phantom{-}\frac{\partial H}{\partial p_n}\\
\dot p_1&=-\frac{\partial H}{\partial x^1}\\
\vdots\\
\dot p_n&=-\frac{\partial H}{\partial x^n}\\
\end{cases}
$$
Prove that $H$ is conserved along the solutions of the ODE.
Given another smooth function $f=f(x^1,\dots,x^n,p_1,\dots,p_n)$,
define their Poisson bracket as
$\{H,f\}=\frac{\partial H}{\partial x^1}\frac{\partial f}{\partial p_1}+\dots\frac{\partial H}{\partial x^n}\frac{\partial f}{\partial p_n}
-\frac{\partial f}{\partial x^1}\frac{\partial H}{\partial p_1}-\dots-\frac{\partial f}{\partial x^n}\frac{\partial H}{\partial p_n}$.
Show that $f$ is conserved along the ODE above if and only if $\{H,f\}=0$.
Use the point above to show that $p_k$ is conserved along the solutions of the ODE above if and only if $\frac{\partial H}{\partial x^k}=0$.
Show that the ODE above can be re-written as follows:
$$
\begin{cases}
\dot x^1&=\{x^1,H\}\\
\vdots\\
\dot x^n&=\{x^n,H\}\\
\dot p_1&=\{p_1,H\}\\
\vdots\\
\dot p_n&=\{p_n,H\}\\
\end{cases}
$$
Evaluate the following Poisson brackets: $\{x^i,x^j\}$, $\{p_i,p_j\}$, $\{x^i,p_j\}$.
Consider the Hamiltonian $H(x,p)=\frac{1}{2}p^2+V(x)$ in $\mathbb R^2$ and show that the ODE above gives the equations of motion of a particle in a potential field we saw in Section 7.5.
Prove that the origin is an asymptotically stable equilibrium of the ODE
$$
\begin{cases}
\dot x&=-x^3+xy\\
\dot y&=-y^3-x^2\\
\end{cases}
$$
and find its basin of attraction.
Show that $L(x,p)=\frac{1}{2} p^2+1-\cos x$ is a Lyapunov function for the equilibria $(x_0,v_0)=(2k,0)$, $k\in\mathbb Z$, of the ODE
$\ddot x=-\sin x$. Hint: first write the ODE as a first order system.
Sketch the phase portrait for the linear ODE
$$
\begin{cases}
\dot x&=-2x+3y\\
\dot y&=\phantom{-}7x-6y\\
\end{cases}
$$
Crises and the right endpoint of the period-3 window.
Readings
Section 10.1.
Homework #5
Due date: Apr 7
Explain why $x(t)=\sin t$ cannot be a solution of a differential eqation on the line $x'=f(x)$.
Find the $\omega$-limit sets of the orbits of the ODE
$$
\begin{cases}
\dot r &= r(r-1)(r-3)\\
\dot \theta &= 1\\
\end{cases}
$$
with initial conditions $(r,\theta)=(0,0),(1/2,\pi),(1,\pi/2),(2,0)$.
Show that a planar vector field whose flow has a periodic orbit has necessarily at least a zero.
Let $X$ be a planar Lipschitz vector field, namely there is a $L>0$ such that $||X(p)-X(q)||\leq L d(p,q)$ for every pair $p,q$ of points in the plane.
Prove that the period of each (if any) periodic orbit of the flow of $\dot v=X(v)$ is not smaller than $2\pi/L$.
Find out what happens in the Lorenz system to the orbits of points htat start on the $z$ axis.
Prove that the stable and unstable manifolds $S,U$ of a diffeomorphism $f$ are invariant under $f$, namely $f(S)=S$ and $f(U)=U$.
Let $f(x,y)=(x/2, 2y-7x^2) $. Verify that, for every $x\in\mathbb R$, $(x,4x^2)$ belongs to the stable manifold of the fixed point $(0,0)$.
Show that this implies that the stable manifold of the origin is the parabola $y=4x^2$.
Let $F$ be a discrete flow. Use the Stable Manifold Theorem to show that the stable and unstable manifolds of any saddle of $F$
have no endpoints and that each of them does not cross itself.
Explore numerically the Ikeda map and visualize the attractor and the period-5 periodic orbit that collides with it at about $a=7.24$.
Use the code to produce pictures to verify that the orbit does not belong to the attractor for $a\lessapprox 7.24$ and does belong to it
for $a\gtrapprox 7.24$.
Produce pictures for at least four different values of the parameter $a$ and use enough iterations to make the attractor decently visible.
Write code to visualize the basins of attraction of the two fixed point attractors under the Lorenz system
at $\rho=17.5,\sigma=10,\beta=8/3$ in the planes $z=5,10,15,20,27,50$ in the range $(x,y)\in[-100,100]\times[-100,100]$.
Bifurcations in area-contracting maps over the plane.
Readings
Sections 11.4 and 11.5.
Homework #7
Due date: Apr 28
Write code to visualize the bifurcation diagram of the family $f_a(x) = a\sin(x)$, $(a,x)\in[-10,10]\times[-10,10]$.
Then find a saddle-node and a period-doubling bifurcation for this family.
You can double-check your diagram with the one in
this recent article, page 18.
Find all saddle-node and period-doubling bifurcations of fixed-points for the map
$f_a(x)=a+x-e^x$.
Find the saddle-node and period-doubling bifurcations of fixed points for the Henon map
$f_a(x,y) = (a-x^2+0.3y,x)$.
Find a family $f_a(x)$ such that there is a fixed point $(\bar a,\bar x)$ such that $f'_{\bar a}(\bar x)=-1$ but there is no period-doubling bifurcation at $(\bar a,\bar x)$.
