Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

MATH3063 Quiz 2 practice, 2021

1.  Consider the system of ODEs

=   y(z2 + az y + y2 ),

dt

=   z(8z2 + 3y2 )

(a) Find all values of the parameters for which the system is a gradient system and

construct a suitable potential function o(z, y).

(b) Find all values of the parameters for which the system is conservative and construct

a corresponding Hamiltonian function H(z, y).

(c) Prove that if 3 = 81 and a  0 then there are no limit cycles.

2.  Consider the following system of nonlinear ordinary differential equations: z˙ = z2 + y2 8 10,        y˙ = 3z2 8 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: z˙ = 8z3 8 3y2 ,        y˙ = y(7z 8 y2 ).

Prove that the origin is an asymptotically stable fixed point.   Carefully state all the conditions required to apply any results you use.