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\title{Math 365: Honors Analysis I\\
Non-linear Systems Exercises}
\date{December 6, 2000}

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Find the critical points of the following systems and discuss the stability
of each. Sketch an appropriate phase portrait based on this information.
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\begin{tabular}{llcl}
1. & $dx/dt$ &=& $(2 + x)(y - x)$\\
   & $dy/dt$ &=& $(4-x)(y+x)$\\
\\
2. & $dx/dt$ &=& $x - x^2 -xy$\\
   & $dy/dt$ &=& $\frac{1}{2}y - \frac{1}{4}y^2 -\frac{3}{4}xy$\\
\\
3. & $dx/dt$ &=& $-(x-y)(1 - x - y)$\\
   & $dy/dt$ &=& $x(2+y)$\\
\\
4. & $dx/dt$ &=& $(1+x)\sin(y)$\\
   & $dy/dt$ &=& $1 - x - \cos(y)$\\
\\
5. & $dx/dt$ &=& $1-xy$\\
   & $dy/dt$ &=& $x - y^3$\\
\end{tabular}

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