MA 342N
Assignment 2
Due 16 March 2016
Id:
342N-s2016-2.m4,v 1.3 2016/05/13 14:01:35 john Exp john
1. For the following system
u′ = v + uw,
v ′ = u + vw,
w ′ = w + uv
(a) find the linearisation at the equilibrium (0, 0, 0),
(b) find the eigenvectors and eigenvalues of the coefficient matrix,
(c) find the dimensions of the stable manifold, unstable manifold and
centre manifold,
(d) find the lowest order for which there is a resonance,
(e) find a linear change of variable which diagonalises the coefficient
matrix,
(f) remove the quadratic part of the non-linearity via a quadratic
change of variable.
2. (a) Find the equilibria of the equation
y ′ = y 3 − zy 2 + 2y 2 − zy + z 2 − 2z.
Hint: The cubic factors into a quadratic times a linear factor.
1
Id:
342N-s2016-2.m4,v 1.3 2016/05/13 14:01:35 john Exp john 2
(b) What are the dimensions of the stable, unstable and centre manifolds at these equilibria?
(c) Draw a diagram of the stable and unstable equilibria in the z-η
plane.
(d) There are three bifurcation points, z = 0, z = 1 and z = 4. Are
these transcritical, saddle-node, Hopf or pitchfork bifurcations?