Applied Dynamical Systems - ME215A
Fall 2008
Homework #1 - Due Wednesday Oct 8th, in class
1. (20 pts total) Consider the linear set of ordinary differential equations for x= (x, y):
˙
x=Ax,(1)
where
A= −1 100
0−10 !.(2)
(a) (2 pts) Show that Ais a non-normal matrix, i.e. AAT6=ATA.
(b) (4 pts) Find the eigenvalues and eigenvectors of A. Are the eigenvectors orthogonal?
(c) (8 pts) Find the exact solution for x(t) in terms of the initial conditions x(0) = (x0, y0).
You can do this by (i) going to new coordinates which diagonalize A, solving the equations
in these coordinates, then transforming back to the original coordinates, or (ii) by solving
the ˙yequation for y, plugging this into the ˙xequation and solving for xby finding a
solution to the homogeneous equation and a particular solution.
(d) (4 pts) Let E=x2+y2. Taylor expand E(t) about t= 0 up to linear order in time.
That is, find aand bin the equation E(t)≈a+bt valid for small t.
(e) (2 pts) Find one initial condition for which dE
dt t=0 >0, and one initial condition for
which dE
dt t=0 <0.
2. (20 pts total) Consider the set of ordinary differential equations
˙r=rcos 2θ+ sin θ≡f1(r, θ) (3)
˙
θ= cos θ1
r−2 sin θ≡f2(r, θ).(4)
(a) (2 pts) Verify that (r, θ) = (1, π/2) ≡(¯r, ¯
θ) is a fixed point for these equations.
(b) (5 pts) Find the eigenvalues of the Jacobian matrix
J|(¯r,¯
θ)=
∂f1
∂r (¯r,¯
θ)
∂f1
∂θ (¯r,¯
θ)
∂f2
∂r (¯r,¯
θ)
∂f2
∂θ (¯r,¯
θ)
.(5)
Is the fixed point (¯r, ¯
θ) asymptotically stable?
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