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\noindent{\bf Math 3226
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Laboratory  A}\\
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{\Large\bf Bifurcations in Linear Systems\footnote{This laboratory is based on   Laboratory 3.1 in {\em ``Differential Equations"} by Blanchard, Devaney and Hall.\flushright\tiny\copyright\  H. Knaust. \today .}}\bigskip\end{center}
{\em Consider the following system of  first-order linear differential equations:
\[\left(\begin{array}{c}x'(t)\\y'(t)\end{array}\right)=\left(\begin{array}{rr}a&b\\-1&-1\end{array}\right)\cdot\left(\begin{array}{c}x(t)\\y(t)\end{array}\right),\]
where $a$ and $b$ are real parameters.}

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\noindent{\bf 1} For each value of $a$ and $b$, classify the system's equilibrium points (as sinks, spirals, etc.). Draw a picture in the ``$ab$"-plane, and indicate 
the regions corresponding to the various types (for instance: shade all $(a,b)$ values for which the origin is a sink red, the values for which the origin is a spiral sink orange, and so forth). Be sure to include all the computations necessary to draw the picture.

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{\em As the values of $a$ and $b$ are changed and the point $(a,b)$ moves from one region to another, the  ``equilibrium type" changes. Such a change is called a  {\bfseries bifurcation}. A typical bifurcation occurs when a harmonic oscillator changes from being underdamped to being overdamped.}

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\noindent{\bf 2} Which of the  bifurcations in your picture
affect the long term behavior of the solutions? 

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\noindent{\bf 3} What is happening at the boundary between  regions?
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