Topics
- Introduction to the course.
- What is a Dynamical System: discrete-time and continuous-time flows on a topological (metric) space.
- Example 1: exponential growth of population.
- Example 2: flow of an Ordinary Differential Equation.
Readings
Homework #1a
Due date: Aug 28
- Solve the ODE $\dot x = x^2, x(t_0)=x_0$, $x\in\Bbb R$, and show that every solution reaches infinity in finite time.
- Solve the ODE $\dot x = \sqrt{x}, x(t_0)=0$, $x\in[0,+\infty)$, and show that it has infinitely many solutions.
Topics
- Discrete-time and continuous-time flows on a topological space
- The main questions in the field of Dynamical Systems:
- what are the invariant sets of the system?
- what are the possible asymptotics of the orbits?
- Fixed points of a flow.
- Cobweb plots.
- Stability of fixed points.
- Example: exponential growth of population.
Readings
- Sections 1.2-1.4 of textbook.
Topics
- One-dimensional maps.
- The logistic map family $\ell_k(x)=kx(1-x)$ for $0\leq k\leq 3$.
Readings
Topics
- The logistic map family $\ell_k(x)=kx(1-x)$ for $0\leq k\leq 3$.
Readings
Homework #2
Due date: Sep 11
- Find all period-1 (i.e. fixed points) and period-2 orbits for the logistic map with $k=0.5,1.5,2.5,3.5,4$ and discuss their stability.
Remark: in order to find period-2 orbits you will need to solve a 4th order polynomial equation. Feel free to use any help you need to solve this equation:
python code, matlab, wolfram alpha, chatgpt, you name it -- in fact, you can easily solve the equation numerically by tweaking a bit the code below for the "Graph of iterates"!
In your solution, though, make sure to explain all your steps you took.
- I mentioned in class that a period-2 orbit $\{x_1,x_2\}$ (namely $f(x_1)=x_2$ and $f(x_2)=x_1$, with $x_1\neq x_2$), is attracting if $|(f^2)'(x_1)|<1$.
The same in fact holds for any period-$k$ orbit. Show that this criterion makes sense, namely that we get the same result no matter which point of the periodic orbit
we choose to do the test. In fact, show that $(f^2)'(x_1)=(f^2)'(x_2)$. Then show that the same is valid for every period-$k$ orbit.
Hint: it is an immediate consequence of the chain rule.
- Set the following parameters in the Bifurcation Diagram code below: imgx=imgy=1000, ka=3.82, kb=3.87, maxit=20000.
Run the code, look at the picture and describe at the best of your understanding what is happening in that range of parameter values of the logistic map.
Identify an attracting period-3 orbit in the picture and find numerically (as in the problem above) the coordinates of the three points of the orbit for $k=3.835$.
How many period-3 orbits are there at $k=3.835$? Use the derivative criterion (page 16 of the textbook) to check which period-3 orbit is stable (i.e. attacting, i.e. a sink)
and which is unstable (i.e. repelling, i.e. a source).
Bifurcation diagram of the logistic map family
Graph of iterates of a logistic map
Topics
- Bifurcation cascade in the logistic map family.
- The logistic map for $k=4$.
- Sensitivity to initial conditions.
Readings
- Sections 1.5-1.7 of textbook.
Topics
- A chaotic discrete-time dynamical system on the circle: the map $f(x) = 10x (mod 1)$.
- Itineraries.
Readings
- Sections 1.7-1.8 of textbook.
Homework #3
Due date: Sep 18
Let me remark once again that, in order to solve any part of problems, and especially the numerics, you are allowed to use any tool, whether is coding in python, wolfram alpha, chatgpt and so on.
However you got your result, your duty is 1. making sure it is correct, and explain clearly why you believe it is; 2. write nicely your solution, a bit as if it were part of a research article.
- Let $f(x) = 10x (mod 1)$ be the map on the circle discussed in the last lecture.
Provide examples of a a fixed point, a period-2 orbit and a period-3 orbit of the map $f$.
Then show that $f$ has periodic orbits of all periods and that the periodic orbits of $f$ are dense in the circle (seen as the segment $[0,1]$ with the identification of 0 and 1).
- Find all period-3 and period-4 orbits of the logistic map $\ell_4$ (namely $\ell_4(x)=4x(1-x)$).
- Show the map $\ell_4^n$ has exactly $2^n$ fixed points.
Hint: first show that, for any $n$, every local max of $\ell_4^n$ is equal to 1 and every local min to 0. Using the fact that $\ell_4$ is surjective, evaluate the number of maxima
of $\ell_4^n$ (start with $n=1$ and show that, at every step, the number of maxima doubles). After that the result should be obvious.
- Let $p>2$ be a prime number. Find the expression of the number of period-$p$ orbits of $\ell_4$ as function of $p$ and use it to prove
Fermat's little theorem.
- Find a point that lies in the subinterval RRL.
- Let $x_0$ be a point in the subinterval RLLRRRLRLR. Is $x_0$ less than, equal to or greater than $1/2$? How about $\ell_4^6(x_0)$?
Sensitive dependance from the initial conditions
Topics
- Cantor sets.
- Invariant sets of the logistic map.
- Discrete-time flows in higher dimension: Poincaré return maps.
Readings
Extra Readings
Topics
- Recurrent points.
- An example of non-periodic recurrent points: motion with constant speed on the 2-torus.
- Invariant sets of recurrent points in the logistic map.
- Full bifurcation diagram of the logistic map.
- Full bifurcation map of a Poincaré section of the Lorenz system.
- Solution of some problem in 1D dynamics.
Readings
Extra Readings
Topics
- The tent map -- a piecewise linear version of the logistic map.
- The circle map $f(x)=2x (mod 1)$.
- The Henon map -- a 2D version of the logistic map.
- Phase space portrait of the physical pendulum.
Readings
Topics
- Phase space portrait of the physical pendulum with friction.
- The forced damped pendulum.
- Dynamics of the Newton's root-finding method.
- Classification of fixed points in dimension 2 and more.
Readings
- Sections 2.1 and 2.2 of textbook.
Topics
- Real Newton method in dimension 2.
- Classification of linear maps.
- Sources, saddles and sinks; rotations, spiraling inwards and outwards.
Readings
- Sections 2.2, 2.3 and 2.4 of textbook.
Topics
- Dynamics of linear maps on the plane.
- Nonlinear maps.
- Jacobian.
- Example: the Henon map.
- Stable and unstable manifolds.
Readings
Topics
- Example: the Henon map.
- Stable and unstable manifolds.
Readings
- Sections 2.5 and 2.6 of textbook.
Homework #4
Due date: Oct 10
Solve the following problems from textbook: T2.6 (p. 73), T2.7 (p. 74), 2.1, 2.2, 2.3 (p. 98)