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#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 27 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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