Homework 3

Due at 11:59pm on Feb 15th, 2026. Submit to https://canvas.uw.edu/courses/1862386/assignments/11153452.

Problem 1

Given a nonlinear system, \(\dot{x}=f(x)\), one way to study local stability is to linearize the system. In particular, let \(x_e\) be an equilibrium (\(f(x_e)=0\)), and let \(A\) be the Jacobian of \(f\) evaluated at \(x_e\). Since for a small enough perturbation, \(\frac{d}{dt} (x_e+\Delta x)= \dot{\Delta x} \approx f(x_e)+A\Delta x\), we can say that \(x_e\) is a stable equilibrium if the eigenvalues of \(A\) all have negative real parts.

Given the following nonlinear system

\[\dot{x} = f(x)= \begin{bmatrix} x_2 \\ x_1-x_1^3-0.5x_2 \\ -x_3 \end{bmatrix}.\]

a) Find all equilibrium points.

b) Find the stable equilibrium points.

Problem 2

Consider a swing equation with losses:

\[\begin{align} \dot{\theta} &= \omega \\ \dot{\omega} & = -\omega+P-(g-g\cos(\theta)+b\sin(\theta)), \end{align}\]

and we take \(g=0.5\) and \(b=1\). Find all values of \(P\) where at least one stable equilibrium exists.

Problem 3

Linearization is fundamentally a local technique since it tells us what happens very close to an equilibrium point. A natural conjecture is that if we understand what happens locally at every point, we can piece them together to understand some global structure. Unfortunately, this is not true when the dimension of the problem is larger than 1.

Consider this system:

\[\dot{x}=\begin{bmatrix} -x_1+x_3 (x_1+x_2 x_3)^2 \\ -x_2-(x_1+x_2x_3)^2 \\ -x_3 \end{bmatrix}\]

a) Compute the eigenvalues of the Jacobian matrix linearized at some \(x\). Note, the eigenvalues are constants, e.g., they don’t depend on the value of \(x\).

b) Directly verify that \(x(t)=(18e^t, -12e^{2t}, e^{-t})\) is a solution.

As a side note, this problem is actually related to a famous problem in algebraic geometry about polynomials, simply called the Jacobian Conjecture.

Problem 4

One way to construct discrete time systems is as approximations to continuous time systems. Given \(\dot{x} = Ax\), we can approximate it as \(\frac{x_{t+1}-x_{t}}{\delta} =A x_t\), and rearranging gives

\[x_{t+1}=(I+\delta A) x_t=\hat{A} x_t.\]

Naturally, we expect that discretization shouldn’t change the stability of the system. Show that there exist a sufficiently small \(\delta>0\) such that \(\hat{A}\) is asympotically stable (in the discrete sense) if and only if \(A\) is asymptotically stable (in the continuous sense).