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\title{Bifurcation events}
\date{April 26, 2021}

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Sometimes a small change in a parameter in a differential equation can have a big effect on the behavior of the system.
This is called a \textbf{bifurcation event} (and is a key feature of chaotic systems).
We will study the bifurcations that occur in the system \[\boxed{ \vx[y]' = \begin{bmatrix} \alpha & 1 \\ -4 & 0 \end{bmatrix}\vx[y] }\]

% \prob{
% Download and run \href{./pplane.jar}{pplane.jar}.
% Set the Display Window so that both $x$ and $y$ range between $-2$ and $2$.
% Graph the phase plane for $\alpha = -5$,  $\alpha = 0$,  $\alpha = 2$,  $\alpha = 4$, and $\alpha = 8$. 
% (Hint: for $\alpha = 8$ you'll need to enter the system of equations $x' = 8x+y$ and $y' = -4x$).
% Compare and contrast the phase plane graphs. 
% }

\prob{
Find the eigenvalues of  $\displaystyle \begin{bmatrix} \alpha & 1 \\ -4 & 0 \end{bmatrix}$ as a function of $\alpha$ (use the quadratic formula).
\begin{enumerate}[a)]
\item For what values of $\alpha$ does the matrix have distinct real eigenvalues?
\item For what values of $\alpha$ does the matrix have repeated real eigenvalues?
\item For what values of $\alpha$ does the matrix have imaginary eigenvalues?
\end{enumerate}
}

\prob{
Choose any value of $\alpha$ you want and solve the IVP (note that solutions are hard to find for some choices of $\alpha$): \[ \vx[y]' = \begin{bmatrix} \alpha & 1 \\ -4 & 0 \end{bmatrix}\vx[y],\ \vx[y](0) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}. \]
Plot the solution in \href{https://www.desmos.com/calculator}{Desmos (link)} and sketch it.
You can plot the space curve $\displaystyle \langle x(t), y(t) \rangle = \langle f(t), g(t) \rangle$ by entering \texttt{(f(t), g(t))} and making sure the parameter $t$ ranges over a large enough interval to show the distinctive features of the curve.
Show any real-valued eigenvectors with a dashed line.
}


\prob{
Use the \href{https://www.geogebra.org/m/utcMvuUy}{GeoGebra phase plane plotter (link)} to plot the phase plane for the following values of $\alpha$.
Identify each system as a source, a sink, a saddle, a spiral source, a spiral sink, an ellipse, or none of the above.
\begin{enumerate}[a)]
\item Your value from the previous problem.
\item $\alpha =  -5$.
\item $\alpha =  0$.
\item $\alpha =  2$.
\item $\alpha =  4$.
\item $\alpha =  8$.
\end{enumerate}
}

\end{document}