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MATH3063 Quiz 2 practice
1. Consider the system of ODEs
dx
dt
= y(x2 + αx y + y2),
dy
dt
= x(βx2 + γy2)
where α, β and γ are real parameters.
(a) Find all values of the parameters for which the system is a gradient system and
construct a suitable potential function φ(x, y).
(b) Find all values of the parameters for which the system is conservative and construct
a corresponding Hamiltonian function H(x, y).
(c) Prove that if γ = −1 and α 6= 0 then there are no limit cycles.
2. Consider the following system of nonlinear ordinary differential equations:
x˙ = x2 + y2 − 10, y˙ = 3x2 − y
(a) Find all the steady states and, where possible, classify them using linear stability
analysis.
(b) Sketch the phase plane showing all nullclines, steady states, direction arrows on each
nullcline and some representative trajectories. Draw all of the stable and unstable
manifolds of any saddle points on your sketch.
3. Consider the following system of nonlinear ordinary differential equations:
x˙ = −x3 − 3y2, y˙ = y(7x− y2).
Prove that the origin is an asymptotically stable fixed point. Carefully state all the
conditions required to apply any results you use.