HW 10 ungraded
Math 527,
These are practice problems meant to give you some practice solving 3-d linear systems and in linearizing
nonlinear systems.
Problems 1-3. Find the general solution of the linear system. Boldface indicates a vector. E.g. x is a
vector with components x, y, z (or x1, x2, x3 if you prefer).
1. x
0 =
1 4 2
4 −1 −2
0 0 6
x
2. x
0 =
−1 1 0
1 2 1
0 3 −1
x
3. x
0 =
2 5 1
−5 −6 −4
0 0 2
x
Problem 4. The Van der Pol oscillator is the second-order nonlinear ordinary equation
x
00 − µ(1 − x
2
)x
0 + x = 0
where µ is a nonnegative parameter and primes indicate differentiation in time, e.g. x
0 = dx/dt.
(a) Convert the 2nd-order nonlinear ODE into a nonlinear system x
0 = f(x) where x =
x
y
using the
substitution y = x
0
.
(b) Show that the origin (x, y) = (0, 0) is the only equilibrium of x
0 = f(x) for all µ.
(c) Linearize the dynamics x
0 = f(x) for small perturbations about the origin. Your answer should be
in the form
x
0 = Df x
where Df is the matrix of partial derivatives of f evaluated at the origin and x is assumed small.
(d) Find the general solution of the linear system from (c) for µ = 1. Is the equilibrium at the origin
stable or unstable?
(e) The eigenvalues of Df and hence the nature of the solutions of (c) change form at a couple specific
values of µ ≥ 0. Determine what these values are, and then specify what kind of solution of (c) occurs
over what region of µ.
For example, your asnwer might be “for 0 ≤ µ < 11, the solutions to the linearized dynamics are stable
oscillations, due to complex eigenvalues with negative real part, and for 11 ≤ µ, the solutions are stable
real-valued exponentials, because both eigenvalues are real and negative.”
This problem is easier than it looks!
