HW 2: Equilibria and Stability

Important

Due:

  • Friday, February 6, 9:30 AM

Introduction

This homework explores equilibrium points (also called fixed points) and stability in one-dimensional dynamical systems. (If you’d like to and have time, feel free to look at the two-dimensional challenge problems.) Traditionally, the field of dynamical systems uses Newton’s notation for a derivative: \(\dot x\) is the first derivative of \(x\), \(\ddot x\) is the second derivative, and so forth.

Exercises

Exercise 1: Equilibria and Stability in 1D Maps

Choose three of the dynamical systems below. Find all the equilibrium points, and figure out if they’re stable, unstable, or conditionally stable (stable from one direction and not the other).

  • \(\dot x = x^2 - 1\)

  • \(\dot x = \sin x\)

  • \(\dot x = x - \cos x\)

  • \(\dot N = rN\left(1 - \frac{N}{K}\right)\) for \(r, K > 0\)

  • \(\dot N = a N \ln(bN)\) for \(a,b > 0\)

  • \(\dot x = ax - x^3\), for \(a \in \mathbb{R}\) (explore different values of \(a\) here)

  • \(\dot x = 1 + \frac{1}{2} \cos x\)

Exercise 2: Linear Stability

You’ll keep working with the dynamical systems you looked at in Exercise 1. For notation, let \(f(x)\) (or the appropriate other variable) be the right side of the equation. In other words, \(\dot x = f(x)\).

For each equilibrium point \(x^*\) that you found in Exercise 1, find the derivative \(f'(x^*)\). What do you notice about this value and stability of \(x^*\)? Try to explain what’s going on here

Exercise 3: Reading Questions

📖 Answer these questions based on the reading: The Math of Catastrophe

  • What two ideas stood out to you most from the tipping points article?

  • Based on the article, how would you describe a bifurcation?

Challenge Exercise: 2D Flows

Dynamical systems can have multiple variables we’re interested in. Here are a couple systems to explore. What kinds of behaviors or patterns are there? What aspects are stable or unstable?

In this first one, let \(\omega\) be a constant real parameter: \[\begin{align*} \dot x &= v \\ \dot v &= - \omega^2 x \end{align*}\] (Note: this one is really \(\ddot x = - \omega^2 x\), but it’s more convenient to write things as just first derivatives!)

In this second one, let \(a\) be a constant real parameter (strongly recommend exploring different values of \(a\)): \[\begin{align*} \dot x &= ax \\ \dot y &= -y \end{align*} \]