chap3-systems.tex

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%		Chapter 3 - System Models
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\begin{topic}[Models of Systems]



\vfil

\begin{center}
\begin{minipage}{300pt}
	\includegraphics*[width=300pt]{images/chap3-xkcd.png}

	\hfill {\footnotesize (image from \href{https://www.xkcd.com/2063/}{xkcd - comic \#2063})}
\end{minipage}
\end{center}


\end{topic}






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%  MODULE - Modelling Two Interconnected Quantities
%
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\begin{module}{Modelling Two Quantities}
	\label{sys:model}

	\input{modules/module16-sys-model.tex}
	\input{modules/module16-sys-model-exercises.tex}
\end{module}



\begin{lesson}
	\Title{Modelling Two Quantities I}

\Heading{Textbook}
	\begin{itemize}
		\item Module 15
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item Model with two interconnected quantities
		\item Follow the step-by-step procedure to create a model
	\end{itemize}
	
\Heading{Motivation} 

Again, this is the main goal of this course: learn how to create models.

In this class, the goal is to study how to model quantities that affect each other (and are affected by external factors). 

After the students create a model, they should get used to making sure the model is ``good'', i.e., using some software look at some approximation of solutions to see the consequences of the model.



\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook
	\item Read the core exercise \ref{sys:competition} and solve steps 1 (define problem) and 2 (create a mind map)
\end{itemize}



\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:pursuit}: \pursuittitle
	\item Project \ref{proj:epidemic}: \epidemictitle
	\item Project \ref{proj:lotkavolterra}: \lotkavolterratitle
\end{itemize}



\end{lesson}



\begin{annotation}
\begin{goals}
	\Goal{Pre-class exercise.} 
	Students bring their mind map and the definition of the functions they need to find.
\end{goals}	
\end{annotation}

\question \label{sys:competition}
	We want to model two competing populations, like cheetahs and lions: they don't hunt each other, but they hunt the same prey.
\begin{annotation}
	\begin{goals}
		Stress that students should follow the step-by-step approach from chapter 1.
		
		\emph{.4} only if there is time. Tell the students to ``go nuts'' and include everything that relates.
	\end{goals}
\end{annotation}
\begin{parts}
	\item Create a model for these two populations.
	\item Using Desmos or WolframAlpha, create a slope field in the plane where the horizontal axis is one population and the vertical one is the other.
	\item Using the slope field, deduce some properties of your model and discuss how closely it matches what you expect from these populations.

	\item Extend the model to include a population of antelopes.	
\end{parts}





\bookonlynewpage

\begin{lesson}
	\Title{Modelling Two Quantities II}

\Heading{Textbook}
	\begin{itemize}
		\item Module 15
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item Model with two interconnected quantities
		\item Follow the step-by-step procedure to create a model
	\end{itemize}
	
\Heading{Motivation} 

This class students work on a model that involves a little more calculus (for the cheetah's movement) and also is more \textbf{open ended} on how the cheetah will follow the antelope, but mainly on how the antelope will try to escape the cheetah. 

The instructor should encourage different solutions from students. 



\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook
	\item Read the core exercise \ref{sys:chase} and solve steps 1 (define problem) and 2 (create a mind map)
\end{itemize}


\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:pursuit}: \pursuittitle
	\item Project \ref{proj:epidemic}: \epidemictitle
	\item Project \ref{proj:lotkavolterra}: \lotkavolterratitle
	\item Project \ref{proj:arms}: \armstitle
\end{itemize}



\end{lesson}


\begin{annotation}
\begin{goals}
	\Goal{Pre-class exercise.} 
	Students bring their mind map and the definition of the functions they need to find.
\end{goals}	
\begin{goals}
		This exercise is not required to do in lecture. \\
		
		Be careful with assumptions! A very general model will be very hard to study. \\
		
		Allow some brainstorming and try to create a structure for this problem:
		\begin{itemize}
			\item Positions seen from above ($xy-$plane).
			\item Only need $x_a(t), y_a(t)$ and $x_c(t), y_c(t)$
			\item Focus on the cheetah: where is she heading to?
			\item For the antelope, students need to come up with an escape strategy
			\item Model will be nonlinear!
		\end{itemize}
	\end{goals}\end{annotation}
\question \label{sys:chase}
	A cheetah is chasing an antelope. We want a model of their positions as they run.
	
	




\standardonlynewpage

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%
%  MODULE - Solving Systems
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\begin{module}{Systems of two linear ODEs with constant coefficients}
	\label{sys:solve}

	\input{modules/module17-sys-solving.tex}
	\input{modules/module17-sys-solving-exercises.tex}
\end{module}



\begin{lesson}
	\Title{Systems of two linear ODEs with constant coefficients I}

\Heading{Textbook}
	\begin{itemize}
		\item Module 16
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item Know how to find eigenvalues and eigenvectors of a matrix
		\item Know how to find a solution from an eigenvalue and an eigenvector
		\item Know how to write the general solution to a system of ODEs
		\item Know how to find the constants given an initial condition
	\end{itemize}
	
\Heading{Motivation} 

Linear Algebra strikes again!!! It will be very important to know how to solve a linear system of equations (although we'll focus on $2\times 2$ systems only) and how to find eigenvalues and eigenvectors. 

This is the more computational part of the course. 

Even though it's not the focus of the course, students need to know how solutions are found. This will give some insight for the analysis of a system of ODEs later. 

We start with problems involving two real distinct eigenvalues.

\Heading{Preparation for Class}
\begin{itemize}
	\item Review Linear Algebra (Appendix 7.1)
	\item Read textbook: Two real distinct eigenvalues
	\item Watch video
	\item Solve the core exercise \ref{sys:2real}.1.
\end{itemize}


\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:arms}: \armstitle
\end{itemize}


\end{lesson}




\question \label{sys:2real}
Consider a cheetah-lion inspired problem:
$$
\frac{d \,\vec{r}}{dt} = \begin{bmatrix} 3 & -2 \\ -1 & 4\end{bmatrix} \vec{r}.
$$
\begin{annotation}
\begin{goals}
	\Goal{Superposition Principle.}
	After .5, can introduce the principle of superposition:
	\begin{itemize}
		\item If $\vec{r}_1$ and $\vec{r}_2$ are solutions of a linear homogeneous (system of) ODE(s), then $\vec{r} = c_1 \vec{r}_1 + c_2\vec{r}_2$ is also a solution for any constants $c_1,c_2$.
	\end{itemize}
\end{goals}	
\end{annotation}
\begin{parts}
	\item Find the two solutions $\vec{r}_1, \vec{r}_2$.
	\item Is $\vec{r}_1(t) + \vec{r}_2(t)$ a solution?
	\item Is $\vec{r}_1(t) - \vec{r}_2(t)$ a solution?
	\item Is $2\vec{r}_1(t) + 3\vec{r}_2(t)$ a solution?
	\item What is the general solution?
	\item Find the solution that satisfies $\vec{r}(0) = \begin{bmatrix} 6 \\ 7\end{bmatrix}$?
\end{parts}
\begin{annotation}
\begin{goals}
	\Goal{If there is more time.} 
	Start working on the core exercise for the next lesson.
\end{goals}	
\end{annotation}


\bookonlynewpage
\begin{lesson}
	\Title{Systems of two linear ODEs with constant coefficients II}

\Heading{Textbook}
	\begin{itemize}
		\item Module 16
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item Know how to find complex eigenvalues and eigenvectors of a matrix
		\item Know how to find a complex solution from an eigenvalue and an eigenvector
		\item Know how to re-write the solution using Euler's Formula and only real numbers
	\end{itemize}
	
\Heading{Motivation} 

We now add some complex numbers into the mix! Although students only need a very (Very) superficial knowledge of complex numbers. 

We study problems involving two complex eigenvalues or one repeated real eigenvalue. 

The third case: One repeated real eigenvalue will not be worked on in lecture. Students should learn it by themselves and practice it.

\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook: Two complex eigenvalues
	\item Watch corresponding video
	\item Solve the core exercise \ref{sys:corecomplex}.
\end{itemize}

\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:arms}: \armstitle
\end{itemize}


\end{lesson}




\begin{annotation}
\begin{goals}
	\Goal{Skip .2 and start working on next core exercise}
\end{goals}
\end{annotation}
\question \label{sys:corecomplex}
Consider a problem:
$$
\frac{d \,\vec{r}}{dt} = \begin{bmatrix} 2 & -5 \\ 1 & -2\end{bmatrix} \vec{r}.
$$
	
\begin{parts}
	\item Find the general solution.
	\item Find the solution that satisfies $\vec{r}(0) = \begin{bmatrix} 6 \\ 7\end{bmatrix}$?
\end{parts}



\bookonlynewpage



\begin{annotation}
	\begin{goals}
		\Goal{Non-Homogeneous Problem}
%		Introduce this core exercise as a non-homogeneous problem. \\
		
		Students don't know how to solve it yet:
		\begin{enumerate}
			\item Equilibrium solution ($\begin{bmatrix}2\\-1\end{bmatrix}$)
			\item Show phase portrait using WolframAlpha \url{https://uoft.me/modelling-sys-nonhom}
			\item Ask students about properties of the phase portrait (\emph{Goal}: solutions revolve around the equilibrium point)
			\item Redefine centre: $\vec{r} = \vec{\rm eq} + \vec{p}$. What system foes $\vec{p}$ solve?
			\item $\cdots$
		\end{enumerate}
		
		There are practice problems about this.
	\end{goals}
\end{annotation}
\question
Consider a problem:
$$
\frac{d \,\vec{r}}{dt} = \begin{bmatrix} 2 & -5 \\ 1 & -2\end{bmatrix} \vec{r} - \begin{bmatrix} 9 \\ 4 \end{bmatrix}.
$$
%[2,-1]
\begin{parts}
	\item Find the equilibrium solution.
	\item Find the general solution.
	\item Find the solution that satisfies $\vec{r}(0) = \begin{bmatrix} 8 \\ 6 \end{bmatrix}$?
\end{parts}


	





\standardonlynewpage

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%  MODULE - Phase Portraits
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\begin{module}{Phase Portraits}
	\label{sys:phase}

	\input{modules/module18-sys-phase.tex}
	\input{modules/module18-sys-phase-exercises.tex}
\end{module}



\begin{lesson}
	\Title{Phase Portraits I}

\Heading{Textbook}
	\begin{itemize}
		\item Module 17
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item Interpret a phase portrait
		\item Sketch a phase portrait
	\end{itemize}
	

\Heading{Motivation} 


After learning how to create a model involving a system of ODEs and how to find the solution, we turn our attention to how to represent the solutions.

The phase portrait is a compact way to represent all possible solutions of a system of two ODEs. 

Through sketching phase portraits, we are also learning the different possible behaviours of solutions of systems of two ODEs. 

We start with systems of two ODEs with two real distinct eigenvalues.
It usually takes a while to sketch the first phase portrait, so the first lesson is dedicated to sketching only one.

\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook
	\item Watch first video
	\item Solve the core exercise \ref{sys:phaseportrait-real}.
\end{itemize}


\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:lotkavolterra}: \lotkavolterratitle
	\item Project \ref{proj:arms}: \armstitle
\end{itemize}


\end{lesson}




\begin{annotation}
	\begin{goals}
	\Goal{Pre-class exercise}	
	\end{goals}
	\begin{goals}
	\Goal{Unstable Saddle Point}
	At the end, let the students know that the equilibrium is called \emph{saddle point} and it is \emph{unstable}, because solutions go away from it.
	
	For the interpretation question, when one population hits zero, it is extinct, so the graph doesn't make sense. 
	
	We can interpret that if a population becomes extinct, then the other will behave as it would without competitors: grow exponentially fast!
	\end{goals}
\end{annotation}
\question \label{sys:phaseportrait-real}
	Consider the following model for cheetah's and lions, where
	$$ \vec{p}(t) = \begin{bmatrix} \ell(t) = \text{population of lions} \\ c(t) = \text{population of cheetahs} \end{bmatrix} $$
	which satisfies
	$$
	\frac{d\,\vec{p}}{dt} = \begin{bmatrix}
 		1 & -1 \\
 		-3 & 1
 	\end{bmatrix}
	$$
	
	The general solution is:
	$$
	\vec{p}(t) = c_1 \begin{bmatrix} 1 \\ \sqrt{3} \end{bmatrix} e^{(1-\sqrt{3})t} + c_2 \begin{bmatrix} -1 \\ \sqrt{3} \end{bmatrix} e^{(1+\sqrt{3})t}.
	$$
\begin{parts}
	\item Without computing them, what are the eigenvalues and eigenvectors of the matrix?
	\item Sketch the graph of the solution with $c_1=\pm 1$ and $c_2=0$.
	\item Sketch the graph of the solution with $c_1=0$ and $c_2=\pm 1$.
	\item When one constant is set to 0, what is the shape of the graph? Is it always like that? Can you prove it?
	\item Sketch the graph of the solution with $c_1=\pm 1$ and $c_2=\pm 1$.
	\item Provide an interpretation of the different types of solutions.
\end{parts}






\bookonlynewpage

\begin{lesson}
	\Title{Phase Portraits II}

\Heading{Textbook}
	\begin{itemize}
		\item Module 17
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item Interpret a phase portrait
		\item Sketch a phase portrait
	\end{itemize}
	
\Heading{Motivation} 

There are a few more different behaviours, so we study some more cases. 

We start with a non-homogeneous case and then study some other cases of phase portraits with two real distinct eigenvalues.

\Heading{Preparation for Class}
\begin{itemize}
	\item Solve the core exercise \ref{sys:phaseportrait-real-equil}.
\end{itemize}


\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:lotkavolterra}: \lotkavolterratitle
	\item Project \ref{proj:arms}: \armstitle
\end{itemize}


\end{lesson}

\begin{annotation}
	\begin{goals}
		\Goal{Pre-class exercise}
		\begin{itemize}
			\item Solve quickly in class.
			\item Get students to compare their results with the previous core exercise.
			\item How does the interpretation of the population dynamics change?
		\end{itemize}
%		Some students might try to sketch everything from scratch.
%		Remind them that the solutions look very similar and they only have to adapt the phase portrait they had before.\\		
	\end{goals}
\end{annotation}
\question \label{sys:phaseportrait-real-equil}
	Let us expand the model from the previous exercise to:
	$$ \vec{p}(t) = \begin{bmatrix} \ell(t) = \text{population of lions} \\ c(t) = \text{population of cheetahs} \end{bmatrix} $$
	which satisfies
	$$
	\frac{d\,\vec{p}}{dt} = \begin{bmatrix}
 		1 & -1 \\
 		-3 & 1
 	\end{bmatrix} \vec{p}
 	+ \begin{bmatrix}
 		- 10 \\ 50
	 \end{bmatrix}.
	$$
	
	The extra term corresponds to the effect of harvesting 10 lions and bringing in 50 cheetahs every year to the reserve. \\
	
	The general solution is:
	$$
	\vec{p}(t) = \begin{bmatrix} 20 \\ 10 \end{bmatrix} +
		c_1 \begin{bmatrix} 1 \\ \sqrt{3} \end{bmatrix} e^{(1-\sqrt{3})t} + c_2 \begin{bmatrix} -1 \\ \sqrt{3} \end{bmatrix} e^{(1+\sqrt{3})t}.
	$$
\begin{parts}
	\item Sketch the phase portrait.
	\item Provide an interpretation of the different types of solutions.
\end{parts}



\bookonlynewpage

\begin{annotation}
	\begin{goals}
	At the end, let the students know what these equilibria are called:
	\begin{itemize}
		\item \emph{source} and it is \emph{unstable}, because solutions go away from it.
		\item \emph{sink} and it is \emph{asymptotically stable}, because solutions converge to it.
	\end{itemize}
	
	If there is time, students can think about:
	\begin{itemize}
		\item Given a matrix $A$, which part of $A$ indicates whether the equilibrium is stable / unstable? Which part indicates whether it's a sink/source vs spiral sink/source?
	\end{itemize}
	\end{goals}
\end{annotation}
\question
	For each of the following general solutions, sketch the phase portrait.
\begin{parts}
	\item $	\vec{r}(t) = c_1 \begin{bmatrix} 2 \\ 1 \end{bmatrix} e^{2t} + c_2 \begin{bmatrix} -1 \\ 1 \end{bmatrix} e^{5t}.$
	\item $	\vec{r}(t) = c_1 \begin{bmatrix} 2 \\ 1 \end{bmatrix} e^{-2t} + c_2 \begin{bmatrix} -1 \\ 1 \end{bmatrix} e^{-5t}.$	
\end{parts}





\standardonlynewpage

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%
%  MODULE - Analysis of Systems
%
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\begin{module}{Analysis of Models with Systems}
	\label{sys:analysis}

	\input{modules/module19-sys-analysis.tex}
	\input{modules/module19-sys-analysis-exercises.tex}
\end{module}



\begin{lesson}
	\Title{Analysis of Models with Systems}

\Heading{Textbook}
	\begin{itemize}
		\item Module 18
	\end{itemize}

\Heading{Objectives}
	\begin{itemize}
		\item Deduce properties of solutions of systems of ODEs using different approaches
	\end{itemize}
	
\Heading{Motivation} 

Analysis of a system of ODEs is important.

It's not always possible to find the solution of a model, but it's usually possible to sketch a rough phase portrait, to approximate the solution, or to just deduce some properties of the solutions.


\Heading{Preparation for Class}
\begin{itemize}
	\item Read textbook
	\item Solve the core exercise \ref{sys:clothes}.1
\end{itemize}


\Heading{Tutorials and Projects}
\begin{itemize}
	\item Project \ref{proj:pursuit}: \pursuittitle
	\item Project \ref{proj:epidemic}: \epidemictitle
	\item Project \ref{proj:lotkavolterra}: \lotkavolterratitle
	\item Project \ref{proj:arms}: \armstitle
\end{itemize}



\end{lesson}




\question \label{sys:clothes}
	Consider the following model for the sales from a designer clothing brand:
	\begin{itemize}
	\item $x_1(t) = $ purchases by ``common mortals'' (CM) at time $t$ in years since the beginning of 2015.
	\item $x_2(t) = $ purchases by ``famous people'' (FP) at time $t$.
	\end{itemize}
	
	Our model is based on the following two principles:
	\begin{enumerate}[label={($P_{\arabic*}$)}]
		\item CM will buy more items from the brand when CM or FP buy more.
		\item FP will buy less when CM buy them, but will buy more when FP buy it.
	\end{enumerate}

	The model we considered is:
	$$
	\vec{x}'(t) = 
	\begin{bmatrix}
 		a & b \\
 		-c & d
	\end{bmatrix}
	\vec{x}(t)
	$$
	
	\begin{parts}
		\item Suppose that at the beginning only CM buy this brand. Describe how $x_1(t)$ and $x_2(t)$ evolve as $t>0$.


		\item Suppose that at the beginning only FP buy this brand. Describe how $x_1(t)$ and $x_2(t)$ evolve as $t>0$.


		\item What conditions on the constants $a,b,c,d$ will guarantee that the solutions will spiral? In that case, is it a spiral source or spiral sink? Is it clockwise or counterclockwise?
		\item Are there constants $a,b,c,d>0$, such that the solution $\vec{x}$ is periodic?
\begin{annotation}
	\begin{goals}
		\Goal{If there isn't enough time to finish:}
		\begin{itemize}
			\item Take an extra lecture if there is enough to do
			\item Can leave .6 as a practice problem
		\end{itemize}
	\end{goals}
\end{annotation}
		\item Consider the constants $a=b=c=d=1$. Assume that initially CM were buying $c_0>0$ items and FP were buying $f_0>0$ items.
			What will happen to $x_1(t)$ and $x_2(t)$ as $t \to \infty$? Explain the results in terms of the evolution of purchases from CM and FP.
		\item Consider the constants $a=b=c=d=1$.  If $c_0=10$, $f_0=100$, then at what time will FP stop buying items? And at what time will FP be buying the maximum number of items?
	
	\end{parts}



\begin{notslides}

\bookonlynewpage


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\end{notslides}