#1 Consider the system of planar nonlinear differential equations,
x' = xy-x² y + y³
y'= x² + y ³ -xy²
a) Convert the system to polar coordinates. b) Analyze the system and sketch a phase portrait.
2 Consider the predator prey model
x' = ax - bxy
y' = cxy-dy
This system has an equilibrium at (x,y) = (2,a). Transform the equilibrium point to the origin and
convert to polar coordinates. Compute G = so g(8)d0 and show that it is zero in this case. This proves
that the fixed point is a nonlinear center in this example (note that the system is not Hamiltonian)
#3 Consider the system
y' = x.
Find a reversor S, identify Fix(S) and show that the origin is a nonlinear center.
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