From 225d391eb88847cc89041584bdb56269e9950beb Mon Sep 17 00:00:00 2001
From: JB <janbouwe.vanden.berg@vu.nl>
Date: Sun, 18 Aug 2024 10:57:58 +0200
Subject: [PATCH] Added Fourier series notebook

---
 src/mod3/fourier_lecture_notes.jl | 2327 +++++++++++++++++++++++++++++
 src/mod3/mod3.md                  |    9 -
 2 files changed, 2327 insertions(+), 9 deletions(-)
 create mode 100644 src/mod3/fourier_lecture_notes.jl
 delete mode 100644 src/mod3/mod3.md

diff --git a/src/mod3/fourier_lecture_notes.jl b/src/mod3/fourier_lecture_notes.jl
new file mode 100644
index 0000000..a08afdf
--- /dev/null
+++ b/src/mod3/fourier_lecture_notes.jl
@@ -0,0 +1,2327 @@
+### A Pluto.jl notebook ###
+# v0.19.45
+
+#> [frontmatter]
+#> chapter = 1
+#> order = 1
+#> title = "Lecture notes: Finite-dimensional problems"
+#> tags = ["module1"]
+#> layout = "layout.jlhtml"
+
+using Markdown
+using InteractiveUtils
+
+# ╔═╡ d0e623ee-b096-4a27-977d-dc32567d6020
+using PlutoTeachingTools, PlutoUI
+
+# ╔═╡ 2661bfc9-e398-41ed-87d9-c78f05da64cb
+using RadiiPolynomial, Plots
+
+# ╔═╡ 018ecc45-8638-4a59-b561-efb086bdc751
+using LinearAlgebra,ToeplitzMatrices
+
+# ╔═╡ 7fc40507-eda3-474d-a454-04e9173a7adb
+html"""<style>
+main {
+    max-width: 1000px;
+    margin-left: auto;
+    margin-right: auto;
+    text-align: justify;
+}
+"""
+
+# ╔═╡ d5a510a3-c518-47ed-96bc-7bb22e3b08b5
+md"""
+We continue to do CAPs to find solutions of ODEs, but rather than looking at an initial value problem, we now look for *periodic* solutions. The setting is again infinite dimensional, with some small differences:
+- we will use **Fourier series** rather than Taylor series
+- the Fourier coefficients are two-sided sequences of complex numbers
+- the convolution product is given by an infinite series
+- the problem of finding a closed orbit is global in the sense that there is no equivalent of "short time" integration
+"""
+
+# ╔═╡ f73b88cb-19e2-4d50-a45e-fbe90fb691cd
+md"""
+# Simple example: a forced, damped, nonlinear pendulum
+"""
+
+# ╔═╡ cf717e9b-26d8-449d-9f3b-d1d5d0bd2bb5
+md"""
+## Introduction and setup
+"""
+
+# ╔═╡ c33dc650-3f94-11ef-398a-8bbc4a2b69b8
+md"""
+We consider the second order ODE
+
+$\begin{align}
+u''(t) + \beta_1 u'(t) + \beta_2 u(t) - u(t)^2 = \beta_3 \cos(t)
+\end{align}$
+
+where $\beta_1, \beta_2, \beta_3$ are parameters, and we aim to find a periodic solution $u(t)$.
+"""
+
+# ╔═╡ 58892dec-c89c-4f1f-9f97-700bc27ef338
+md"""
+Since the (minimal) period of the forcing is $2\pi$, we will look for a periodic solution of period $2\pi$.
+"""
+
+# ╔═╡ 1bfcc76a-266a-4784-b76b-3871445e603e
+Foldable("""Why is this a "simple" example?""", md"""
+A more natural problem to look at is to find a periodic orbit in a system of ODEs. In such an *autonomous* system of ODEs the period of a periodic solution is unknown a priori, and determining the period $\tau$ is thus part of the problem, in addition to finding the unknown $\tau$-periodic function $u(t)$. Furthermore, when $u(t)$ is a periodic solution, so is $u(t-t_0)$ for any time shift $t_0$. In particular, periodic solutions are not isolated unless an additional phase condition is imposed. An additional complication is that for a system of equations the bookkeeping is more involved than for a single equation.
+
+Although these difficulties can be overcome, the notation can get in the way of understanding. Since we want to focus here on how to deal with periodic functions, we choose to look at a scalar, nonautonomous problem where the period is given.
+""")
+
+# ╔═╡ 3e3a3a60-f76f-401d-b265-a99e713d98c5
+md"""
+### Fourier series
+"""
+
+# ╔═╡ 8f3e96dc-0cdc-411e-8f94-e9e47488abf3
+md"""
+We substitute the Fourier series 
+
+$\begin{align}
+u(t) = \sum_{n\in\mathbb{Z}} a_n e^{int}
+\end{align}$
+
+in the ODE to find the infinite set of equations ($n \in \mathbb{Z}$)
+
+$\begin{align}
+\lambda_n a_n + (a*a)_n + c_n = 0.
+\end{align}$
+
+where
+$\lambda_n:= n^2-in \beta_1-\beta_2$, while
+$c$ represents the Fourier transform of $\beta_3 \cos(t)$,
+ given by
+
+$\begin{align}
+c_n := \begin{cases} \frac{\beta_3}{2} & n= \pm 1\\ 0 & \text{otherwise} \end{cases}
+\end{align}$
+"""
+
+# ╔═╡ 7c9ca144-02f1-4941-bb0b-becb96b4fc8a
+md"""
+### Convolution
+"""
+
+# ╔═╡ b5d97ade-bcde-4c7a-ae78-697e73c10a0c
+md"""
+The convolution product is given by 
+
+$\begin{align}
+(a*b)_n := \sum_{k\in\mathbb{Z}} a_k b_{n-k} 
+\end{align}$
+"""
+
+# ╔═╡ 47e39d6a-231e-40d1-906e-eeafe1b1f807
+Markdown.MD(Markdown.Admonition("tip", "Lemma",[md"""When $u(t) = \sum_{n\in\mathbb{Z}} a_n e^{int}$
+and $\tilde{u}(t) = \sum_{n\in\mathbb{Z}} \tilde{a}_n e^{int}$ then for their product we have $u(t)\tilde{u}(t) = \sum_{n\in\mathbb{Z}} (a*\tilde{a})_n e^{int}$.
+ """]))
+
+# ╔═╡ ac3e3cf5-e5c8-4863-8c7b-ccd5f58a3073
+md"""
+When $a_n=0$ for $|n|>N_a$ and $b_n=0$ for $|n|>N_b$ then the convolution can be computed. 
+An implementation is below.
+"""
+
+# ╔═╡ 4a8b8e9e-9562-4cb5-a029-856127e84b98
+function convolution(a, b, nab=-1)
+    # convolution for 1D Fourier; nab is size of output (default is all nonzero coefficients)
+    na = (length(a)-1)÷2
+    nb = (length(b)-1)÷2
+	if nab<0
+		nab = na + nb
+	end
+	
+    # Initialize the result array with zeros
+    result = zeros(typeof(a[1]),2*nab+1)
+    
+    # Perform the convolution
+    for ia in -na:na
+		minib=max(-nab-ia,-nb)
+		maxib=min(nab-ia,nb)
+        for ib in minib:maxib 
+            result[nab+ia+ib+1] += a[na+ia+1] * b[nb+ib+1]
+        end
+    end
+    
+    return result
+end
+
+# ╔═╡ 312d9592-1be0-4626-8773-51cc8604dd6a
+md"""
+### Norm
+"""
+
+# ╔═╡ c4d9c4f0-fd89-4a2a-96a7-17c4ba6dd747
+md"""
+We will work in the sequence space 
+
+$\begin{align}
+X=X_\nu=\{ a \in \mathbb{C}^\mathbb{Z} : \|a\|_X:= \sum_{n\in\mathbb{Z}} \nu^{|n|} |a_n| < \infty \}.
+\end{align}$
+
+We will choose the value of $\nu$ later.
+"""
+
+# ╔═╡ 766164f3-eef3-4798-bf7c-afdca52d3ba0
+Markdown.MD(Markdown.Admonition("tip", "Lemma",[md"""When $a\in X_\nu$ for some $\nu \geq 1$ then the Fourier series $\sum_{n\in\mathbb{Z}} a_n e^{int}$ converges uniformly. When $\nu>1$ then term-by-term differentiation up to any order of the Fourier series is justified."""]))
+
+# ╔═╡ bbf2e943-8c5c-469f-a2f3-5f8b758f6638
+Foldable("""Is X with this norm a Banach space?""", md"""**Exercise**: Yes""")
+
+# ╔═╡ f91abfa7-8a8f-4204-95d9-637c622dd720
+md"""
+The product and the norm play together nicely: they give the space $X$ the structure of a Banach algebra, as expressed by the following lemma.
+"""
+
+# ╔═╡ 8f6431ed-346c-4c51-83c4-ea7ec12a8d60
+Markdown.MD(Markdown.Admonition("tip", "Lemma (Banach algebra property)",[md"""For any $\nu \geq 1$ and $a,b\in X=X_\nu$ we have $\|a*b\|_X\leq \|a\|_X \|b\|_X$."""]))
+
+# ╔═╡ c5cfe80e-1728-4393-b837-187e3e7d49a9
+Foldable("""Proof""", md"""**Exercise** (rearranging series and using triangle inequality)""")
+
+# ╔═╡ 39d13194-bca9-47c1-a686-2ea5f9d62289
+md"""
+The operator norm (recall: $\|\Gamma\|_{B(X)}$ is the smallest number such that $\|\Gamma a\|_X\leq \|\Gamma\|_{B(X)} \|a\|_X$ for all $a\in X$) has a particularly nice representation for weighted $l^1$ spaces such as $X_\nu$.
+"""
+
+# ╔═╡ 7e78028c-f70b-4474-8bee-783ad7d99d56
+Markdown.MD(Markdown.Admonition("tip", "Lemma (characterisation of operator norm)",[md"""
+Let $\Gamma$ be a bounded operator on $X$, represented by $(\Gamma a)_n=\sum_{k\in\mathbb{Z}} \Gamma_{nk} a_k$. 
+Then 
+
+$\begin{align}
+  \|\Gamma\|_{B(X)} = \sup_{k\in\mathbb{Z}} \frac{1}{\nu^{|k|}} \sum_{n\in\mathbb{Z}} \nu^{|n|} |\Gamma_{nk}| .
+\end{align}$
+
+Let $e_k$ denote the basis vector given by $(e_k)_n=\delta_{kn}$ for any $k,n \in \mathbb{Z}$. Then 
+
+$\begin{align}
+  \|\Gamma\|_{B(X)} = \sup_{k\in\mathbb{Z}} \frac{\|\Gamma e_k\|_X}{\|e_k\|_X}.
+\end{align}$
+"""]))
+
+# ╔═╡ c7a10c87-a95d-4b0b-b63d-f9546e97bb36
+md"""
+Code for the (operator) norm is given below for the case of finitely represented elements/operators in/on $X$.
+"""
+
+# ╔═╡ efb766e3-e41f-4113-ae6e-c4dd7766c784
+function nunorm(B, nu)
+    # weighted l1-norm; works for both vectors and matrices B
+    n1 = (size(B,1)-1)÷2 
+    n2 = (size(B,2)-1)÷2
+
+	weights=[nu^(abs(i)) for i in -n1:n1]'
+	invweights=[nu^(-abs(i)) for i in -n2:n2]'
+
+	result=maximum((weights*abs.(B)).*invweights)
+    
+	return result
+end
+
+# ╔═╡ 0f04a415-b4c1-4b05-a35a-6d3029decb12
+maximum([1.9414153246843213 1.9290258804863343 1.4084075531504734])
+
+# ╔═╡ 31a59e3d-32af-4328-b3bc-cc434728429d
+md"""
+## Zero finding problem
+"""
+
+# ╔═╡ 8d7fb15d-32d3-4fbb-9d06-c8ec1ed7d00e
+md"""
+We define the zero finding problem $F(a)=0$ on $X$ by ($n \in \mathbb{Z}$)
+
+$\begin{align}
+F_n(a) := a_n + \lambda_n^{-1} [(a*a)+c]_n .
+\end{align}$
+"""
+
+# ╔═╡ b2defc79-da78-47c8-bb78-44cb9000ff58
+Foldable("""Why divide by lambda?""", md"""
+Dividing by $\lambda_n$ is a choice that simplifies the linear term, which is beneficial for some of the algebra. In terms of estimates it merely shifts the difficulty to the nonlinear term. Working with the alternative $\tilde{F}_n(a) := \lambda_n a_n + [(a*a)+b]_n$ is also a perfectly valid choice.
+""")
+
+# ╔═╡ 9ab6ce17-0637-4008-be29-e0bf4fda4287
+Markdown.MD(Markdown.Admonition("note", "Conjugate symmetry",[md"""
+We denote the *complex conjugate* of $z\in \mathbb{C}$ by $\overline{z}$. For real-valued functions $u(t)$ the Fourier coefficients satisfy $a_{-n} = \overline{a_n}$ for all $n\in \mathbb{Z}$. And the other way around: if $a_{-n} =\overline{a_n}$ for all $n\in \mathbb{Z}$, then the Fourier series is real-valued.
+
+For $a\in X$, we denote by $a^\dagger \in X$ the *conjugate* given by $(a^\dagger)_n = \overline{a_{-n}}$. With this notation, $a=a^\dagger$ implies the Fourier series is real-valued.
+
+We note that both $\lambda_{-n} =\overline{\lambda_n}$ and $c_{-n} =\overline{c_n}$. Additionally, $a^\dagger * b^\dagger = (a * b)^\dagger$. Hence the zero finding problem is conjugate symmetric:
+
+$\begin{align} F(a^\dagger)=F(a)^\dagger . \end{align}$
+"""]))
+
+# ╔═╡ 9002b709-ec9c-4063-a3a8-7b0c00834095
+Foldable("""Exercise""", md"""
+Prove that $a^\dagger * b^\dagger = (a * b)^\dagger$ and $F(a^\dagger)=F(a)^\dagger$.
+""")
+
+# ╔═╡ c2f4618c-dd22-477c-8fe7-324c82d3ff46
+md"""
+We now formalize that the zero finding problem leads to periodic solutions of our differential equation.
+"""
+
+# ╔═╡ 7d79ed91-7cbb-48b4-966a-d3165f2a3d89
+Markdown.MD(Markdown.Admonition("tip", "Lemma",[md"""Let $\nu>1$. When $a\in X$ with $a^\dagger=a$ solves $F(a)=0$, then the corresponding Fourier series $u(t)=\sum_{n\in\mathbb{Z}} a_n e^{int}$ is a real-valued periodic solution of the differential equation $u''(t) + \beta_1 u'(t) + \beta_2 u(t) - u(t)^2 = \beta_3 \cos(t)$."""]))
+
+# ╔═╡ 66deaecf-e201-40ec-88d3-99574195253a
+md"""
+## Finite dimensional projection
+"""
+
+# ╔═╡ c32d9839-7cb7-4018-bab1-b2211f05b71e
+md"""
+We need the projection operator on a finite number of modes ($2N+1$ coefficents) and its complement:
+
+$\begin{align}
+(\pi^{\leq N} a)_n := \begin{cases}
+a_n & |n| \leq N \\
+0 & |n| > N
+\end{cases}
+& \qquad\qquad
+(\pi^{>N} a)_n := \begin{cases}
+0 & |n| \leq N \\
+a_n & |n| > N
+\end{cases}
+\end{align}$
+
+so that $a=\pi^{\leq N}a+\pi^{>N}a$ and $\pi^{\leq N}\pi^{>N}=0=\pi^{>N}\pi^{\leq N}$.
+"""
+
+# ╔═╡ bd76555d-aa30-4313-b4c4-c3f4b9000194
+Foldable("""Why is this so complicated?""", md"""
+The strange looking superscripts $\leq N$ and $>N$ are in fact meant to indicate clearly which index values are involved.
+
+The projections provide a sleek way to do the algebra that essentially splits the problem into a finite dimensional part handled by the computer and an infinite dimensional "tail" part handled entirely by human analysis. It avoids writing down *infinite* matrices or formulas with a lot of indices.
+""")
+
+# ╔═╡ 740167dc-5bcd-48c0-b1f0-a7bf5e1b6cb7
+md"""
+The range $\pi^{\leq N}X$ is finite dimensional, and 
+the restriction of $\pi^{\leq N} F$ to $\pi^{\leq N} X$, which we denote by $F^{\leq N}$, is what we will work with in the computer. An implementation is given below"""
+
+# ╔═╡ 026aec24-4ed0-4189-af4f-28f56e6964ef
+function F(a, beta, Nf=-1)
+	# default length of output is same as input
+    Na = (length(a) - 1) ÷ 2
+	if Nf<0
+		Nf = Na
+	end
+
+	# deal with sizes
+	Naf=min(Na,Nf);
+	a1=zeros(typeof(a[1]),2*Nf+1);
+	a1[Nf+1-Naf:Nf+1+Naf]=a[Na+1-Naf:Na+1+Naf]
+
+	# the nonlinear term
+	nonlinear=convolution(a,a,Nf)
+	# add the forcing term
+	if Nf>0
+	    nonlinear[Nf] += beta[3]/2
+		nonlinear[Nf+2] += beta[3]/2
+	end
+
+	# linear terms
+    lambdainv = 1 ./ [n^2 - 1im * beta[1] * n - beta[2] for n in -Nf:Nf]
+    result = a1 + lambdainv.*nonlinear
+
+    return result
+end
+
+# ╔═╡ 06f3d467-af63-440d-b486-8da7f67c314f
+md"""It is convenient to introduce the diagonal linear operator 
+
+$\begin{align}
+(\Lambda a)_n := \lambda_n a_n
+\end{align}$
+
+When $\beta_1 \neq 0$ then the inverse $\Lambda^{-1}$ is well-defined as a bounded operator on $X$. 
+"""
+
+# ╔═╡ f5cc03ca-09eb-4419-9d51-78c956ee73d9
+md"""
+The derivative of $F(a)=a+\Lambda^{-1}[a*a+c]$ can now be written compactly as 
+
+$\begin{align}
+DF(a)b = b + 2 \Lambda^{-1} [a*b].
+\end{align}$
+"""
+
+# ╔═╡ c5706263-a08a-4742-b376-b980232d6191
+md"""
+An implementation of the Jacobian $DF^{\leq N}$ of the truncated problem is given below.
+"""
+
+# ╔═╡ e5b2628d-2e64-47ca-b594-a0e4cb8ea64a
+function DF(a, beta, Ndf=-1)
+	# Jacobian; default size of output is same as input
+    Na = (length(a) - 1) ÷ 2
+	if Ndf<0
+		Ndf = Na
+	end
+
+	# deal with nonlinear term in terms of a Toeplitz matrix
+	Naf=min(Na,2*Ndf);
+	a1c=zeros(typeof(a[1]),2*Ndf+1);
+	a1r=zeros(typeof(a[1]),2*Ndf+1);
+	a1c[1:Naf+1]=a[Na+1:Na+1+Naf]
+	a1r[1:Naf+1]=a[Na+1:-1:Na+1-Naf]
+    Ta=Toeplitz(a1c,a1r);
+
+	# linear terms
+    lambda = [n^2 - 1im * beta[1] * n - beta[2] for n in -Ndf:Ndf]
+ 	Lambdainv=Diagonal( 1 ./ lambda );
+    result = I(2*Ndf+1)+2*Lambdainv*Ta;    
+	
+    return result
+end
+
+# ╔═╡ c4363fdf-6500-4830-bc42-9777d616ac34
+md"""
+## Choosing the approximate inverse of the Jacobian
+"""
+
+# ╔═╡ 24e1a5af-569e-4ae8-a3d7-19b7962db48b
+md"""
+We also use the project to split $A$ into a finite part and a tail part:
+
+$\begin{align}A:=A^{\leq N}\pi^{\leq N} + I^{>N}\pi^{>N}, \end{align}$
+
+where $A^{\leq N}$ is a linear map (to be chosen below) on $\pi^{\leq N}X$ and $I^{>N}$ is the identity on
+$\pi^{> N}X$. Hence by construction one may also write this is block diagonal form:
+
+$\begin{align}A=\pi^{\leq N}A^{\leq N}\pi^{\leq N} + \pi^{> N}I^{>N}\pi^{>N}, \end{align}$
+"""
+
+# ╔═╡ 77b91c27-ddb0-4115-b0d0-a78f2729dadc
+Foldable("""Is A injective?""", md"""
+The operator $A$ is injective on $X$ if and only $A^{\leq N}$ is injective on $\pi^{\leq N} X$. The operator $A^{\leq N}$ has a (finite) matrix representation, hence injectivity can be proved by computer.
+
+One may also observe that $A=I+\pi^{\leq N}(A^{\leq N}-I^{\leq N})\pi^{\leq N}$, hence $A$ is of the form identity plus compacy (and thus a Fredholm operator of index $0$).
+""")
+
+# ╔═╡ 51e35c7e-77f2-4911-914c-7f1a242c3b91
+md"""
+## Numerics
+"""
+
+# ╔═╡ 4a9daa68-86ed-4327-98fc-03d436885d2e
+md"""
+We consider the finite truncation $F^{\leq N}$ and use a Newton algorithm to find a point $\bar{a} \in \pi^{\leq N} X$ such that $\pi^{\leq N} F (\bar{a}) \approx 0$. If the truncation dimensions $N$ is large enough we may hope that also 
+$\pi^{> N} F (\bar{a}) \approx 0$, since we expect the coefficients of $\bar{a}$ to decrease for large $n$.
+"""
+
+# ╔═╡ 3842ad29-ae9c-4743-b487-e76d94fb01cf
+md"""
+We choose parameter values
+"""
+
+# ╔═╡ 1688b3d8-8228-45ee-8729-22265a86484a
+# in interval form
+beta= [ I"0.1"
+	    I"4.0"
+	    I"1.0" ]
+
+# ╔═╡ aa150fa3-1729-43b1-bb55-73c741221760
+md"""
+and truncation dimension
+"""
+
+# ╔═╡ e1f6ef9b-5818-415a-8945-97b65288a433
+N=10
+
+# ╔═╡ d0858eee-5086-4754-966f-de2940183573
+md"""
+To simplify some of the expressions appearing in the bounds, we will for convenience assume that $N \geq \sqrt{\beta_3}$.
+"""
+
+# ╔═╡ a89606ee-b46f-4656-9cb9-d89850646593
+println(N >= sup(sqrt(beta[3])))
+
+# ╔═╡ 192507bd-8285-4581-8a6d-f9ff1b2c4793
+md"""
+We then run Newton iterations.
+"""
+
+# ╔═╡ 301bf3ed-b180-4c44-a92f-0f9f4c2509ed
+begin
+	initialdata = zeros(typeof(0.0im),2*N+1)   # Fourier coefficients are complex-valued
+	initialdata[N+1] += 0.0 	
+	initialdata=(initialdata+conj(reverse(initialdata)))/2
+
+	Fbeta=mid.(beta)   # floats
+	
+	a0, success = newton(a -> (F(a, Fbeta), DF(a, Fbeta)), initialdata)
+	a0=(a0+conj(reverse(a0)))/2    # symmetry
+end
+
+# ╔═╡ aa320281-42ce-4035-8c55-1f44661737e2
+md"""
+We plot the approximate periodic solution
+"""
+
+# ╔═╡ d5800e45-8809-46d6-9f41-7565e5b1cbfd
+begin
+	Na= (length(a0)-1) ÷ 2
+	Nplot=201;
+	t=LinRange(0, 2*pi, Nplot)
+	u=zeros(typeof(0.0im),Nplot)
+	for n=-Na:Na
+		u += a0[Na+1+n]*exp.(im*n*t)
+	end
+	u=real(u)
+	plot(t, u, legend=false)
+	xlabel!("t")
+	ylabel!("u")
+	xlims!(0, 2*pi)
+end
+
+# ╔═╡ 5f3d91ee-14c2-4b5e-8e2a-a607d086eb51
+Foldable("""Initializing Newton's method""", md"""Finding a good starting point for Newton's method is not always easy. Here we were a bit lucky that a simple guess suffices. For other parameter values one may need to study the solution of the ODE in more detail using numerical integration.""")
+
+# ╔═╡ 936423fd-0d78-41da-8b49-5fca3c125547
+md"""
+Next we compute a numerical inverse of the Jacobian $DF^{\leq N}(\bar{a})$ and we denote this inverse matrix by $A^{\leq N}$.
+"""
+
+# ╔═╡ e37811fa-cd67-424e-8707-25ad8534d293
+AN=inv(DF(a0,Fbeta))
+
+# ╔═╡ 0b4e2571-cbec-4a4d-aac6-dd8b32e3f474
+Markdown.MD(Markdown.Admonition("note", "Remark on truncation dimension",
+[md"""
+It is not necessary to choose the finite truncation dimensions for $\bar{a}\in \pi^{\leq N} X$ and $A^{\leq N}$ equal. 
+For purposes of exposition we do not introduce two different truncation dimension parameters here, but essentially the truncation dimension for $\bar{a}\in \pi^{\leq N} X$ controls the size of the residue, while the truncation dimension for the approximate inverse $A^{\leq N}$ controls the contractivity of the fixed point operator. These can in principle be  controled rather independently.
+"""]))
+
+# ╔═╡ 0d75d077-53c3-4f03-99d3-af796d973d66
+md"""
+## The Newton-Kantorovich proof
+"""
+
+# ╔═╡ b6cfde9d-357b-43e9-82ec-f88e8cb7ba8e
+md"""
+We choose a value for the weight $\nu$ in the Banach space norm.
+"""
+
+# ╔═╡ 671babb1-4b04-4d6a-8dba-79f807114d63
+nu=1.1
+
+# ╔═╡ 83f772ea-aeec-49a3-baa2-d2fd8771c0a7
+md"""
+Since we want to prove something we need to resort to interval arithmetic. Hence we will need to convert $\bar{a}$, $\beta$ and $A$ to interval-valued variables. 
+"""
+
+# ╔═╡ 1ace9001-93ae-46f2-98e6-7b371221beb3
+begin
+	Ia0=interval(a0)
+	Ibeta=beta
+	IAN=interval(AN)
+	Inu=interval(nu)
+	# for testing without intervals:	
+	 # Ia0=a0
+	 # Ibeta=Fbeta
+	 # IA=AN
+	 # Inu=nu
+end
+
+# ╔═╡ 9433b6f1-3460-48c4-b587-9009823327c0
+md"""
+We are now in the Newton-Kantorovich context and thus need to derive the appropriate bounds. 
+"""
+
+# ╔═╡ 9fe5120a-824a-4395-afe8-325a36ab9a07
+md"""
+### Computable expression for $Y$
+"""
+
+# ╔═╡ eb2ae1fd-5062-436b-ac50-eb3b169bca42
+md"""
+This concerns the bound on the residue $A F(\bar{a})$. Determining the residue requires only a finite computation. Indeed, since $(\bar{a}*\bar{a})_n=0$ for $|n|>2N$ (why?) we infer that $\pi^{>2N} F(\bar{a})=0$ and it can be computed explicitly. In turn this implies that $A F(\bar{a})= \pi^{\leq 2N} A \pi^{\leq 2N} F(\bar{a})$. We conclude that $\|A F(\bar{a})\|_X= \|\pi^{\leq 2N} A \pi^{\leq 2N} F(\bar{a})\|_X $ requires a finite computation only, which we can perform with interval arithmetic to get a rigorous upper bound $Y \geq \|A F(\bar{a})\|_X$.
+"""
+
+# ╔═╡ 3021e7c0-508a-4a95-9c16-098fa35b65a0
+function boundY(a, beta, A, nu)
+	# deal with sizes
+    Na = (length(a) - 1) ÷ 2
+    NA = (size(A,1)- 1) ÷ 2
+	NAfa=max(NA,2*Na)
+
+	# compute residue
+	Fa=F(a,beta,2*Na);
+	AFa=zeros(typeof(a[1]),2*NAfa+1)
+	AFa[NAfa+1-2*Na:NAfa+1+2*Na]=Fa;
+    AFa[NAfa+1-NA:NAfa+1+NA]=A*AFa[NAfa+1-NA:NAfa+1+NA];
+
+	result=nunorm(AFa,nu)
+
+    return result
+end
+
+# ╔═╡ 9bf0e55e-de5f-4ec9-92e9-5169d3d14302
+IY=boundY(Ia0,Ibeta,IAN,Inu)
+
+# ╔═╡ fc18e3ff-4a39-4aba-a2d2-94521035013e
+md"""
+### Computable expression for $Z_1$
+"""
+
+# ╔═╡ 002c7c0a-2b38-4a31-9e08-a424fa5679b8
+md"""
+This concerns the bound on the operator $I-A DF(\bar{a})$. In the analysis we will split the bound in a part which is computable by "brute force" and an estimate of the tail. The triangle inequality is often helpful for sch arguments, but in our case the characterisation of the operator norm gives an additional tool. In particular we will use that for any $M \in \mathbb{N}$ we have that 
+
+$\begin{align}
+\|\Gamma\|_{B(X)} = \max \{ \|\Gamma \pi^{\leq M}\|_{B(X)} , \|\Gamma \pi^{> M}\|_{B(X)} \}.
+\end{align}$
+
+We will make a suitable choice for $M$ later.
+"""
+
+# ╔═╡ 907f8fb5-859b-4751-b483-2006ba358d5c
+Markdown.MD(Markdown.Admonition("note", "Finite bandwidth",[md"""
+It follows from the expression for $DF(a)$ that $DF(\bar{a})$ is an finite bandwidth operator of width $N+1$.
+In particular, for any $M \in \mathbb{N}$ we have $\pi^{> M+N} DF(\bar{a}) \pi^{\leq M} =0$ and for any $M\geq N$ we have that $\pi^{\leq M-N} DF(\bar{a}) \pi^{> M} =0$.
+
+It follows from the block-diagonal structure of $A$ that this implies that 
+
+$\begin{align}
+A DF(\bar{a}) \pi^{> 2N} =  (A^{\leq N}\pi^{\leq N} + I^{>N} \pi^{>N}) DF(\bar{a}) \pi^{> 2N} =   \pi^{>N} DF(\bar{a}) \pi^{> 2N},
+\end{align}$
+
+and this expression does not involve $A^{\leq N}$.
+
+Furthermore, it also follows from the block-diagonal structure of $A$ and finite bandwidth of $DF(\bar{a})$ that 
+
+$\begin{align}
+A DF(\bar{a}) \pi^{\leq 2N} = A \pi^{\leq 3N} DF(\bar{a}) \pi^{\leq 2N} = \pi^{\leq 3N} A DF(\bar{a}) \pi^{\leq 2N},
+\end{align}$
+
+which can be represented as a matrix and requires only a finite computation to evaluate.
+"""]))
+
+# ╔═╡ 3d28b669-3294-47fb-8221-688ec3ca5491
+Foldable("""Exercise""", md"""Provide the details proving the statements in the box above.""")
+
+# ╔═╡ d35592ba-c0cf-45fe-94c1-c58c3c0bc939
+md"""
+By choosing $M=2N$ we can thus split the computation of the norm in two parts:
+
+$\begin{align}
+\|I-ADF(\bar{a})\|_{B(X)} = \max \{ \|\pi^{\leq 3N} [I-A DF(\bar{a})] \pi^{\leq 2N}\|_{B(X)} , \| [I-ADF(\bar{a})] \pi^{>2N} \|_{B(X)} \}.
+\end{align}$
+
+The latter term is computable thanks to the following lemma.
+"""
+
+# ╔═╡ d71cba6e-4f62-48d0-aa17-061e1f6bd744
+Markdown.MD(Markdown.Admonition("tip", "Lemma",[md"""
+Final lemma:
+Assume that $N\geq \sqrt{\beta_3}$. 
+Then
+
+$\begin{align}
+\| [I-ADF(\bar{a})]\pi^{>2N} \|_{B(X)} = 
+\max \left\{ \frac{\|[I-ADF(\bar{a})]e_{2N+1}\|_X}{\|e_{2N+1}\|_X} ,  
+\frac{\|[I-ADF(\bar{a})]e_{-(2N+1)}\|_X}{\|e_{-(2N+1)}\|_X} \right\}.
+\end{align}$
+"""]))
+
+# ╔═╡ ac18ecec-5d4e-4d4d-99f1-d5cefcb1355d
+Foldable("""Proof""", md"""**Exercise**. Hint: for $|k| \geq 2N+1$
+we have $[I-ADF(\bar{a})]e_k= -2\Lambda^{-1} \bar{a} * e_k$ and it follows from monotonicity of $\lambda_n$ in $|n|$ for $|n|\geq \sqrt{\beta_3}$ that $\frac{\|\Lambda^{-1} \bar{a} * e_k \|_X}{\|e_{k}\|_X}$ decreases monotonically in $|k|$ for $|k|\geq 2N+1$. """)
+
+# ╔═╡ b6ae8c03-57aa-4a45-9d6a-9886421ee934
+md"""
+Collecting and summarizing the above results, we conclude that 
+
+$\begin{align}
+\|I-ADF(\bar{a})\|_{B(X)} = \|\pi^{\leq 3N+1}[I-ADF(\bar{a})]\pi^{\leq 2N+1}\|_{B(X)}.
+\end{align}$
+
+The latter operator can be represented by a finite matrix and we can compute the operator norm $\|I-ADF(\bar{a})\|_{B(X)}$ using interval arithmetic to obtain the bound $Z_1$.
+"""
+
+# ╔═╡ a18bb200-3166-4c23-bcb9-dee71ff5035a
+Foldable("""Do we need to determine the norm of I-ADF(a̅) exactly?""", md"""No, we just need an estimate. But we need a pretty good estimate, since we can only succeed with the contraction argument if $Z_1<1$. A naive estimate like $\|I-ADF(\bar{a})\|_{B(X)} \leq \|I\|_{B(X)}+\|A\|_{B(X)}\|DF(\bar{a})\|_{B(X)}$ will certainly not work out. Indeed, we need to take advantage of the cancellations occuring in $I-ADF(\bar{a})$ due to $A$ being an approximate inverse of $DF(\bar{a})$.
+
+We are often willing to spend quite some computer power on getting a rather sharp bound.
+For the current problem we could also get away with a somewhat rougher bound,
+such as (provided $N\geq \sqrt{\beta_3}$) 
+
+$\begin{align}
+Z_1=\max\left\{ \|\pi^{\leq 3N}[I- ADF(\bar{a})]\pi^{\leq 2N}\|_{B(X)} , \frac{2  \|\bar{a}\|_X}{|\lambda_{N+1}|} \right\} .
+\end{align}$
+
+We note that sharpness of the $Z_2$-bound is less of an issue, as it plays a considerably less important role in the radii polynomial inequality. 
+""")
+
+# ╔═╡ 4a57ce24-2872-4ef1-ada1-73a64ab4f17a
+function boundZ1(a, beta, A, nu)
+    Na = (length(a) - 1) ÷ 2
+    NA = (size(A,1)- 1) ÷ 2
+
+	# deal with sizes
+	Ncol = max(NA,ceil(Integer,sup(sqrt(beta[2])))) + Na + 1
+	Nrow = Ncol + Na
+
+	# compute A*DF(a)
+	Adfa = DF(a,beta,Nrow)
+	Adfa[Nrow+1-NA:Nrow+1+NA,:] = A*Adfa[Nrow+1-NA:Nrow+1+NA,:]
+
+	B = I(2*Nrow+1)-Adfa;
+	B = B[:,Nrow+1-Ncol:Nrow+1+Ncol]
+	result=nunorm(B,nu)
+		
+    return result
+end
+
+# ╔═╡ 47ef9015-eae8-4b29-807f-d223754701d3
+IZ1=boundZ1(Ia0,Ibeta,IAN,Inu)
+
+# ╔═╡ abdca76c-832e-42aa-98fc-8a6c2c03965e
+Foldable("""Injectivity of A revisited.""", md"""**Exercise**: Prove that $\|I-A DF(\bar{a})\|_{B(X)}<1$ implies that $A$ is surjective on $X$.
+
+**Corollary**: The block-diagonal structure of $A$ implies that it follows from surjectivity of $A$ onto $X$ that $A^{\leq N}$ is surjective as an operator on $X^{\leq N}$. Since $X^{\leq N}$ is finite dimensional, this implies that $A^{\leq N}$  is injective on $X^{\leq N}$. We conclude that $A$ is also injective on $X$.
+""") 
+
+# ╔═╡ c0ce2c85-eaae-4b08-8ef7-2a9297b10e05
+md"""
+### Computable expression for $Z_2$
+"""
+
+# ╔═╡ 6f346c67-879a-4427-afd4-6acdeb17af21
+md"""
+This concerns the bound on the operator $A[DF(a)-DF(\bar{a})]$. We see that 
+$A[DF(a)-DF(\bar{a})]b =2A\Lambda^{-1}(a-\bar{a})*b$, hence by the Banach algebra property and the definition of the operator norm we arrive at
+
+$\begin{align}
+\|A[DF(a)-DF(\bar{a})]b\|_X \leq 2 \|A\Lambda^{-1}\|_{B(X)} \|a-\bar{a}\|_X \|b\|_X.
+\end{align}$
+
+We thus need to compute (or bound) $2\|A\Lambda^{-1}\|_{B(X)}$.
+"""
+
+# ╔═╡ 1c2cf614-faba-4fd4-8ceb-2a5d1373d93a
+md"""
+Since the operator $\Lambda^{-1}$ is diagonal, it has a natural restriction to $\pi^{\leq N} X$. The next lemma shows that the operator norm of $\|A\Lambda^{-1}\|_{B(X)}$ requires only a finite computation.
+"""
+
+# ╔═╡ 62cec2b5-e408-46a4-947a-ebc58745c1ec
+Markdown.MD(Markdown.Admonition("tip", "Lemma",[md"""Assume $N\geq\sqrt{\beta_3}$. Then 
+$\|A\Lambda^{-1}\|_{B(X)} = \max\{ \|A^{\leq N} \Lambda^{-1}  \|_{B(\pi^{\leq N} X)} , |\lambda_{N+1}|^{-1}  \}$
+"""]))
+
+# ╔═╡ 84478c6d-e157-40dc-987b-80c7f4d11a68
+Foldable("""Proof""", md"""**Exercise**. Hint: consider $\|A\Lambda^{-1}e_n\|_X$ for $|n|\leq N$ and for $|n|>N$ separately.""")
+
+# ╔═╡ 6688fe5d-0019-485d-9059-bb5bd2be7d6e
+function boundZ2(a, beta, A, nu)
+	NA = (size(A,1)- 1) ÷ 2
+
+	# bound on finite projection
+    lambda = [n^2 - 1im * beta[1] * n - beta[2] for n in -NA:NA]
+ 	Lambdainv=Diagonal( 1 ./ lambda );
+	normALinv=nunorm(A*Lambdainv,nu)
+
+	#tail bound
+	Nmax=max(NA+1,ceil(Integer,sup(sqrt(beta[2])))+1)
+	abslambda=[abs(n^2 - 1im * beta[1] * n - beta[2]) for n in NA+1:Nmax]
+    tailbound=1/minimum(abslambda)
+
+	result=2*max(normALinv,tailbound);
+	
+    return result
+end
+
+# ╔═╡ 0d7146b3-3015-416b-aae8-697ac0850603
+IZ2=boundZ2(Ia0,Ibeta,IAN,Inu)
+
+# ╔═╡ 5d0b6b50-0170-4f8b-ad5a-0dda212a00e1
+md"""
+## Finishing the CAP
+"""
+
+# ╔═╡ a9f597fe-4058-47a8-8a39-8d6b36594a96
+md"""
+We now evaluate the radii polynomial to finish the proof.
+"""
+
+# ╔═╡ fceb89d1-127a-41b0-b0ec-ec58763a9caa
+begin
+	# set proper bounds and absorb case of non-interval-arithmetic
+	Y=interval(sup(IY))
+	Z1=interval(sup(IZ1))
+	Z2=interval(sup(IZ2))
+end
+
+# ╔═╡ 60b43603-9937-4869-b5fc-f74c1c18e710
+r=interval_of_existence(Y, Z1, Z2, Inf)
+
+# ╔═╡ ee6195ff-70b0-46f9-9911-c17bac67aa82
+md"""
+## Interval arithmetic in Julia 
+"""
+
+# ╔═╡ a16e9d49-7fd2-4594-8b81-f8c758b7efd1
+md"""
+This does not look great:
+"""
+
+# ╔═╡ 64a95b2a-f747-4de1-aca1-c54e43119f64
+begin
+	[isguaranteed(IY)
+	 isguaranteed(IZ1)
+	 isguaranteed(IZ2)]
+end
+
+# ╔═╡ b0612642-f710-496b-be28-c12d5f867e20
+md"""
+## Symmetry
+"""
+
+# ╔═╡ eb1f2c71-60fc-4ee5-9716-f18dd5eb0d2d
+md"""
+We made sure that the numerical approximation was conjugate symmetric: $\bar{a}^\dagger=\bar{a}$.
+"""
+
+# ╔═╡ 6df91aa3-ad36-4b9e-ba4a-4dfb46eb1d27
+	println(conj(reverse(a0))==a0)
+
+# ╔═╡ 49eae38b-786f-45fc-b331-3b01c32363a1
+md"""
+Hence the following lemma finishes our proof.
+"""
+
+# ╔═╡ 35df9eed-5f22-4e26-b119-ea7354d2c762
+Markdown.MD(Markdown.Admonition("tip", "Lemma (symmetry of the solution)",
+[md"""
+Assume that $\bar{a}^\dagger=\bar{a}$ and that the asummptions of the Newton-Kantorovich theorem were satisfied for some $r=r_0>0$. Then the zero $a^*$ of $F$ such that $\|a^*-\bar{a}\|_X \leq r_0$ satisfies $(a^*)^\dagger=a^*$.
+"""]))
+
+# ╔═╡ bae570fe-8219-4cf2-b763-fcd9f0f02735
+md"""
+*Proof*: First, based on the symmetry $F(a^\dagger)=F(a)^\dagger$ of the zero finding problem, we see that $F(a^*)=0$ implies that $F((a^*)^\dagger)=(F(a^*))^\dagger=0$, hence $(a^*)^\dagger$ is a zero of $F$.  Second, the norm on $X$ is such that $\|a^\dagger\|_X=\|a\|_X$  for all $a\in X$. We find that 
+
+$\begin{align} 
+\|(a^*)^\dagger-\bar{a}\|_X=\|(a^*)^\dagger-\bar{a}^\dagger\|_X 
+=\|(a^*-\bar{a})^\dagger\|_X = \|a^*-\bar{a}\|_X \leq r_0.
+\end{align}$
+
+Hence $(a^*)^\dagger$ is a zero of $F$ inside the ball of radius $r_0$ around $\bar{a}$, and since this zero is unique by the Newton-Kantorovich theorem, we conclude that $(a^*)^\dagger=a^*$.
+"""
+
+# ╔═╡ 09922158-6ac9-410e-90ac-5240a9056086
+md"""
+## Conclusion
+"""
+
+# ╔═╡ 3cc0fc2c-9512-41d6-9798-4e8cbec94c9e
+md"""This means that we have proven that there is a periodic solution of the differential equation within distance"""
+
+# ╔═╡ 670dd539-75b4-45b5-b9f3-de4354945648
+r₀=inf(r)
+
+# ╔═╡ f4f432f8-ea80-4169-94ad-ee3289e1a93f
+md"""
+of the approximation depicted in the figure below (with distance measured in the max-norm on $[0,2\pi]$). 
+"""
+
+# ╔═╡ 26d06cfe-24cc-41ec-87d2-8e489865b5b8
+begin
+	plot(t, u, legend=false, width=2)
+	xlabel!("t")
+	ylabel!("u")
+	xlims!(0, 2*pi)
+	middle=(maximum(u)+minimum(u))/2
+	annotate!(pi,middle, Plots.text("Capified",50, :magenta, rotation = 30 ))
+end
+
+# ╔═╡ e40d36d5-93a6-4b76-bccd-c629f621b22b
+Foldable("""Error bound""", md"""
+**Exercise**: Prove that the error between the solution $u^*(t)=\sum_{n\in \mathbb{Z}} a^*_n e^{int}$ and the approximation $\bar{u}(t)=\sum_{|n| \leq N} \bar{a}_n e^{int}$ is indeed controled by
+
+$\begin{align}
+\max_{t \in [0,2\pi]} | u^*(t) - \bar{u}(t)| \leq r_0
+\end{align}$
+""")
+
+# ╔═╡ 194e7df2-a234-45c7-8760-8c49a0d6e651
+
+
+# ╔═╡ 00000000-0000-0000-0000-000000000001
+PLUTO_PROJECT_TOML_CONTENTS = """
+[deps]
+LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
+PlutoTeachingTools = "661c6b06-c737-4d37-b85c-46df65de6f69"
+PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
+RadiiPolynomial = "f2081a94-c849-46b6-8dc9-07bb90ed72a9"
+ToeplitzMatrices = "c751599d-da0a-543b-9d20-d0a503d91d24"
+
+[compat]
+Plots = "~1.40.5"
+PlutoTeachingTools = "~0.2.15"
+PlutoUI = "~0.7.59"
+RadiiPolynomial = "~0.8.12"
+ToeplitzMatrices = "~0.8.4"
+"""
+
+# ╔═╡ 00000000-0000-0000-0000-000000000002
+PLUTO_MANIFEST_TOML_CONTENTS = """
+# This file is machine-generated - editing it directly is not advised
+
+julia_version = "1.10.4"
+manifest_format = "2.0"
+project_hash = "db3d94c0ad35dd4b82a364910d1cea9158e423fd"
+
+[[deps.AbstractFFTs]]
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+version = "1.5.0"
+
+    [deps.AbstractFFTs.extensions]
+    AbstractFFTsChainRulesCoreExt = "ChainRulesCore"
+    AbstractFFTsTestExt = "Test"
+
+    [deps.AbstractFFTs.weakdeps]
+    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+    Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
+
+[[deps.AbstractPlutoDingetjes]]
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+
+[[deps.ArgTools]]
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+
+[[deps.Cairo_jll]]
+deps = ["Artifacts", "Bzip2_jll", "CompilerSupportLibraries_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "JLLWrappers", "LZO_jll", "Libdl", "Pixman_jll", "Xorg_libXext_jll", "Xorg_libXrender_jll", "Zlib_jll", "libpng_jll"]
+git-tree-sha1 = "a2f1c8c668c8e3cb4cca4e57a8efdb09067bb3fd"
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+version = "1.18.0+2"
+
+[[deps.CodeTracking]]
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+
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+uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
+version = "2.0.2+0"
+
+[[deps.libinput_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"]
+git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce"
+uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
+version = "1.18.0+0"
+
+[[deps.libpng_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
+git-tree-sha1 = "d7015d2e18a5fd9a4f47de711837e980519781a4"
+uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
+version = "1.6.43+1"
+
+[[deps.libvorbis_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
+git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
+uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
+version = "1.3.7+1"
+
+[[deps.mtdev_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11"
+uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
+version = "1.1.6+0"
+
+[[deps.nghttp2_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
+version = "1.52.0+1"
+
+[[deps.oneTBB_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "7d0ea0f4895ef2f5cb83645fa689e52cb55cf493"
+uuid = "1317d2d5-d96f-522e-a858-c73665f53c3e"
+version = "2021.12.0+0"
+
+[[deps.p7zip_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
+version = "17.4.0+2"
+
+[[deps.x264_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
+uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
+version = "2021.5.5+0"
+
+[[deps.x265_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
+git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
+uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
+version = "3.5.0+0"
+
+[[deps.xkbcommon_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
+git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
+uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
+version = "1.4.1+1"
+"""
+
+# ╔═╡ Cell order:
+# ╟─7fc40507-eda3-474d-a454-04e9173a7adb
+# ╠═d0e623ee-b096-4a27-977d-dc32567d6020
+# ╠═2661bfc9-e398-41ed-87d9-c78f05da64cb
+# ╠═018ecc45-8638-4a59-b561-efb086bdc751
+# ╟─d5a510a3-c518-47ed-96bc-7bb22e3b08b5
+# ╟─f73b88cb-19e2-4d50-a45e-fbe90fb691cd
+# ╟─cf717e9b-26d8-449d-9f3b-d1d5d0bd2bb5
+# ╟─c33dc650-3f94-11ef-398a-8bbc4a2b69b8
+# ╟─58892dec-c89c-4f1f-9f97-700bc27ef338
+# ╟─1bfcc76a-266a-4784-b76b-3871445e603e
+# ╟─3e3a3a60-f76f-401d-b265-a99e713d98c5
+# ╟─8f3e96dc-0cdc-411e-8f94-e9e47488abf3
+# ╟─7c9ca144-02f1-4941-bb0b-becb96b4fc8a
+# ╟─b5d97ade-bcde-4c7a-ae78-697e73c10a0c
+# ╟─47e39d6a-231e-40d1-906e-eeafe1b1f807
+# ╟─ac3e3cf5-e5c8-4863-8c7b-ccd5f58a3073
+# ╠═4a8b8e9e-9562-4cb5-a029-856127e84b98
+# ╟─312d9592-1be0-4626-8773-51cc8604dd6a
+# ╟─c4d9c4f0-fd89-4a2a-96a7-17c4ba6dd747
+# ╟─766164f3-eef3-4798-bf7c-afdca52d3ba0
+# ╟─bbf2e943-8c5c-469f-a2f3-5f8b758f6638
+# ╟─f91abfa7-8a8f-4204-95d9-637c622dd720
+# ╟─8f6431ed-346c-4c51-83c4-ea7ec12a8d60
+# ╟─c5cfe80e-1728-4393-b837-187e3e7d49a9
+# ╟─39d13194-bca9-47c1-a686-2ea5f9d62289
+# ╟─7e78028c-f70b-4474-8bee-783ad7d99d56
+# ╟─c7a10c87-a95d-4b0b-b63d-f9546e97bb36
+# ╠═efb766e3-e41f-4113-ae6e-c4dd7766c784
+# ╠═0f04a415-b4c1-4b05-a35a-6d3029decb12
+# ╟─31a59e3d-32af-4328-b3bc-cc434728429d
+# ╟─8d7fb15d-32d3-4fbb-9d06-c8ec1ed7d00e
+# ╟─b2defc79-da78-47c8-bb78-44cb9000ff58
+# ╟─9ab6ce17-0637-4008-be29-e0bf4fda4287
+# ╟─9002b709-ec9c-4063-a3a8-7b0c00834095
+# ╟─c2f4618c-dd22-477c-8fe7-324c82d3ff46
+# ╟─7d79ed91-7cbb-48b4-966a-d3165f2a3d89
+# ╟─66deaecf-e201-40ec-88d3-99574195253a
+# ╟─c32d9839-7cb7-4018-bab1-b2211f05b71e
+# ╟─bd76555d-aa30-4313-b4c4-c3f4b9000194
+# ╟─740167dc-5bcd-48c0-b1f0-a7bf5e1b6cb7
+# ╠═026aec24-4ed0-4189-af4f-28f56e6964ef
+# ╟─06f3d467-af63-440d-b486-8da7f67c314f
+# ╟─f5cc03ca-09eb-4419-9d51-78c956ee73d9
+# ╟─c5706263-a08a-4742-b376-b980232d6191
+# ╠═e5b2628d-2e64-47ca-b594-a0e4cb8ea64a
+# ╟─c4363fdf-6500-4830-bc42-9777d616ac34
+# ╟─24e1a5af-569e-4ae8-a3d7-19b7962db48b
+# ╟─77b91c27-ddb0-4115-b0d0-a78f2729dadc
+# ╟─51e35c7e-77f2-4911-914c-7f1a242c3b91
+# ╟─4a9daa68-86ed-4327-98fc-03d436885d2e
+# ╟─3842ad29-ae9c-4743-b487-e76d94fb01cf
+# ╠═1688b3d8-8228-45ee-8729-22265a86484a
+# ╟─aa150fa3-1729-43b1-bb55-73c741221760
+# ╠═e1f6ef9b-5818-415a-8945-97b65288a433
+# ╟─d0858eee-5086-4754-966f-de2940183573
+# ╠═a89606ee-b46f-4656-9cb9-d89850646593
+# ╟─192507bd-8285-4581-8a6d-f9ff1b2c4793
+# ╠═301bf3ed-b180-4c44-a92f-0f9f4c2509ed
+# ╟─aa320281-42ce-4035-8c55-1f44661737e2
+# ╠═d5800e45-8809-46d6-9f41-7565e5b1cbfd
+# ╟─5f3d91ee-14c2-4b5e-8e2a-a607d086eb51
+# ╟─936423fd-0d78-41da-8b49-5fca3c125547
+# ╠═e37811fa-cd67-424e-8707-25ad8534d293
+# ╟─0b4e2571-cbec-4a4d-aac6-dd8b32e3f474
+# ╟─0d75d077-53c3-4f03-99d3-af796d973d66
+# ╟─b6cfde9d-357b-43e9-82ec-f88e8cb7ba8e
+# ╠═671babb1-4b04-4d6a-8dba-79f807114d63
+# ╟─83f772ea-aeec-49a3-baa2-d2fd8771c0a7
+# ╠═1ace9001-93ae-46f2-98e6-7b371221beb3
+# ╟─9433b6f1-3460-48c4-b587-9009823327c0
+# ╟─9fe5120a-824a-4395-afe8-325a36ab9a07
+# ╟─eb2ae1fd-5062-436b-ac50-eb3b169bca42
+# ╠═3021e7c0-508a-4a95-9c16-098fa35b65a0
+# ╠═9bf0e55e-de5f-4ec9-92e9-5169d3d14302
+# ╟─fc18e3ff-4a39-4aba-a2d2-94521035013e
+# ╟─002c7c0a-2b38-4a31-9e08-a424fa5679b8
+# ╟─907f8fb5-859b-4751-b483-2006ba358d5c
+# ╟─3d28b669-3294-47fb-8221-688ec3ca5491
+# ╟─d35592ba-c0cf-45fe-94c1-c58c3c0bc939
+# ╟─d71cba6e-4f62-48d0-aa17-061e1f6bd744
+# ╟─ac18ecec-5d4e-4d4d-99f1-d5cefcb1355d
+# ╟─b6ae8c03-57aa-4a45-9d6a-9886421ee934
+# ╟─a18bb200-3166-4c23-bcb9-dee71ff5035a
+# ╠═4a57ce24-2872-4ef1-ada1-73a64ab4f17a
+# ╠═47ef9015-eae8-4b29-807f-d223754701d3
+# ╟─abdca76c-832e-42aa-98fc-8a6c2c03965e
+# ╟─c0ce2c85-eaae-4b08-8ef7-2a9297b10e05
+# ╟─6f346c67-879a-4427-afd4-6acdeb17af21
+# ╟─1c2cf614-faba-4fd4-8ceb-2a5d1373d93a
+# ╟─62cec2b5-e408-46a4-947a-ebc58745c1ec
+# ╟─84478c6d-e157-40dc-987b-80c7f4d11a68
+# ╠═6688fe5d-0019-485d-9059-bb5bd2be7d6e
+# ╠═0d7146b3-3015-416b-aae8-697ac0850603
+# ╟─5d0b6b50-0170-4f8b-ad5a-0dda212a00e1
+# ╟─a9f597fe-4058-47a8-8a39-8d6b36594a96
+# ╠═fceb89d1-127a-41b0-b0ec-ec58763a9caa
+# ╠═60b43603-9937-4869-b5fc-f74c1c18e710
+# ╟─ee6195ff-70b0-46f9-9911-c17bac67aa82
+# ╟─a16e9d49-7fd2-4594-8b81-f8c758b7efd1
+# ╠═64a95b2a-f747-4de1-aca1-c54e43119f64
+# ╟─b0612642-f710-496b-be28-c12d5f867e20
+# ╟─eb1f2c71-60fc-4ee5-9716-f18dd5eb0d2d
+# ╠═6df91aa3-ad36-4b9e-ba4a-4dfb46eb1d27
+# ╟─49eae38b-786f-45fc-b331-3b01c32363a1
+# ╟─35df9eed-5f22-4e26-b119-ea7354d2c762
+# ╟─bae570fe-8219-4cf2-b763-fcd9f0f02735
+# ╟─09922158-6ac9-410e-90ac-5240a9056086
+# ╟─3cc0fc2c-9512-41d6-9798-4e8cbec94c9e
+# ╠═670dd539-75b4-45b5-b9f3-de4354945648
+# ╟─f4f432f8-ea80-4169-94ad-ee3289e1a93f
+# ╟─26d06cfe-24cc-41ec-87d2-8e489865b5b8
+# ╟─e40d36d5-93a6-4b76-bccd-c629f621b22b
+# ╠═194e7df2-a234-45c7-8760-8c49a0d6e651
+# ╟─00000000-0000-0000-0000-000000000001
+# ╟─00000000-0000-0000-0000-000000000002
diff --git a/src/mod3/mod3.md b/src/mod3/mod3.md
deleted file mode 100644
index 12344b2..0000000
--- a/src/mod3/mod3.md
+++ /dev/null
@@ -1,9 +0,0 @@
----
-title: "Periodic orbits"
-order: 1
-chapter: 4
-layout: "md.jlmd"
-tags: ["module3"]
----
-
-## Jan Bouwe's section