\DocumentMetadata{ % Compile with lualatex not pdflatex for an accessible PDF 2.0 doc.
	lang        = en, % or or en-US
	tagging     = on,
	tagging-setup = {math/setup={mathml-SE,mathml-AF}}, % optional but recommended to get both MathML mechanisms
	pdfstandard = {ua-2,a-4f}, % Use both
	uncompress,
	check-tagging-status % From https://www.tug.org/TUGboat/tb46-3/tb144ltnews42.pdf
	% The line above adds, to the log file, a summary of the packages and classes
	% compatibility with the new tagging framework, using the package/class status
	% information at https://latex3.github.io/tagging-project/tagging-status/
}

%% NOTE:  Please add, to all tables and figures, suitable \alt={...} tags and otherwise 
%%		  ensure that your document meets the required accessiblity standards.

\documentclass[12pt]{report}

% Need help with Math equations?  Here's an excellent reference to get you started!
%  https://tug.ctan.org/info/short-math-guide/short-math-guide.pdf
%  Also available, along with other resources, at https://pauljhurtado.com/latex/

%% PACKAGES:
%% Many packages are still not fully compatible with LuaLaTeX tagging. 
%% Before loading packages, see https://latex3.github.io/tagging-project/tagging-status/

% These packages help with equations, math symbols, etc.
\usepackage{amsmath} % add amssymb, amsthm, etc. here if needed.
\usepackage{unicode-math}              % <- to help with ML tagging of equations
\setmathfont{Latin Modern Math}        % or another OpenType math font

%% Remove ligatures (e.g. ``ff") to allow text copying computer code from the PDF
\setmonofont{Latin Modern Mono}[ % Above works for the whole  PDF, but below is better w/ lualatex
Ligatures = TeXOff
]

% The following gives us better margins and spacing
\usepackage[margin=1in]{geometry} % replaces the 'fullpage' package, which can be buggy!
\usepackage{parskip}
\usepackage{setspace}


%% The next few lines setup list formatting.
%% For more compact numbered lists, and setting label formats using the new lualatex built-in
%% replacement for customizing lists mimicing the enumitem package. It's STILL IN DEVELOPMENT,
%% so this list formatting setup might be replaced with something closer to the old enumitem
%% setup in future updates to the lualatex.
%\renewcommand{\labelenumii}{\alph{enumii})} % These two lines might be needed with slightly
%\renewcommand{\labelenumiii}{\roman{enumiii})} % older lualatex & blocks package (before fall 2026)
\setlist[enumerate,2]{label=\alph*)}  % These should work w/ the latest lualatex & blocks package
\setlist[enumerate,3]{label=\roman*)} % 
\setlist[enumerate,1]{label=\arabic*.,wide=0pt} % This removes top level list item indentation.

% Shaded boxes for solutions
\usepackage[most]{tcolorbox}

% To include images using \includegraphics
\usepackage{graphicx}

% In case you want to link anything...
\usepackage[hidelinks]{hyperref}
\hypersetup{ % Set for PDF accessibility checkers, which require a title.
	pdftitle   = {Homework}, % MATH XXX Homework \#X by NAMEHERE
	pdfauthor  = {} 
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\vspace*{-1.75cm}
\textbf{\large MATH XXX (Fall 2026) -- Homework \#?? \hfill NAME HERE}  % Add your name.
\hrule

% Configure some details for knitr
<<setup, include=FALSE>>=
library(knitr)
# smaller font size for chunks
opts_chunk$set(size = 'footnotesize', concordance=TRUE)
@

\textbf{Instructions:} Please complete the following exercises, etc. \\

\textbf{Exercises:}
\begin{enumerate}
\item (Adapted from Wiggins, \S18.7 exercise 4) For each of the following systems, do a center manifold reduction at the origin (with the bifurcation occurring at $\epsilon=0$) and use the equation for the center manifold dynamics to classify the bifurcation type. 

\begin{enumerate}	
	\item Suppose $x$ is on $S$ (the circle, $0$ to $2\pi$) and $v\in\mathbb{R}$. 
	\begin{align*}\frac{dx}{dt} =& - x + \epsilon v + v^2\\ \frac{dv}{dt} =& -\sin(x)\end{align*}
	
	\item Suppose $x$ and $y$ are real, and 
	\begin{align*} \frac{dx}{dt} =& \frac{x}2+y+x^2 y \\ \frac{dy}{dt} =& x + 2y + \epsilon y + y^2 \end{align*}
\end{enumerate}

\begin{tcolorbox}[breakable, enhanced]
\textbf{Solution:} 
\begin{enumerate}[wide=0pt] % or set left-margin=1.5em or something similar.	
\item 
\item 
\end{enumerate} \end{tcolorbox}

\bigskip

\item Recall that $\displaystyle \frac{d}{dx}\tan(x)=\sec^2(x)$. For $a,b\in(-\pi/2,\pi/2)$, find \[\int_a^b\sec^2(u)\,du.\] 

\begin{tcolorbox}[breakable, enhanced]
	\textbf{Solution:} 
\end{tcolorbox} % End Solutions

\bigskip

\item You can embed snippets of R code (or  python code, etc.) into a LaTeX document using the knitr package in R to parse the code and format the code and output in a LaTeX document. For more, see \url{https://pauljhurtado.com/latex/#quickrknitr}. If you're familiar with R Markdown code chunks, these LaTeX code chunks follow the same general format with only a few minor differences! :)

<<r, echo=TRUE, fig.alt="Graph of the function y equals the negative of the natural log of, open paren, one minus x squared, close paren. The graph has domain form -1 to 1, with vertical asymptotes at each endpoint of that interval with function values diverging towards infinity at each end of the interval. The function is concave up, has even symmetry, and a minimum at the origin. The graph overall is U shaped.">>=	
## R code to draw the graph below
curve(-log(1-x^2),-1,1,n=250, ylim=c(-0.2,2.5), lwd=3,
ylab="y = - ln(1-x^2)", xlim=c(-1.2,1.2))
points(0,0,pch=19)
abline(h=0,v=0)
abline(v=1,lty=2)
abline(v=-1,lty=2)
## Example R code and output
1+1
diff(c(1,2,5,10))
@

\end{enumerate}
\end{document}