HW 3: Bifurcations

Important

Due:

  • Friday, February 13, 9:30 AM

Exercises

Exercise 1: Exploring Bifurcations

Part (a)

Abby started looking at \(\dot{x} = rx - x^2\). (You can see the beginning of her work in the drive linked on the Week 2 page.)

Finish this up: what equilibrium points are there for which values of \(x\)? What is their stability?

Then draw the bifurcation diagram (\(r\) on the horizontal axis, equilibrium values \(x^*\) on the vertical axis).

Types of Bifurcations

For these 1D systems, a bifurcation is a change in a parameter (like \(r\) above) that changes the behavior of the system, like number of equilibria or their stability.

A saddle-node bifurcation (also called a fold or blue sky bifurcation) is a bifurcation in which two equilibrium points collide and annihilate each other. From class, \(\dot{x} = r + x^2\) was one of these.

A transcritical bifurcation is a bifurcation in which the equilibrium points always exist but they exchange stability at some value of the parameter. Part (a) was one of these.

A pitchfork bifurcation is a transition from one equilibrium point to three equilibrium points. From class, \(\dot{x} = rx - x^3\) and \(\dot{x} = rx + x^3\) were this kind.

Part (b)

For each of these systems, find their equilibrium points across all values of the parameter and classify their stability. Then sketch the bifurcation diagram and classify the type of bifurcation.

  • \(\dot{x} = x + \frac{rx}{1+x^2}\)
  • \(\dot{x} = 1 + rx + x^2\)
  • \(\dot{x} = r + x - x^3\)

Exercise 2:

The last system from Exercise 1, \(\dot{x} = r + x - x^3\), exhibits interesting behavior called hysteresis. It will help throughout this to look at your bifurcation diagram from Exercise 1.

  • Start with \(r = -1\). What (approximately) is the equilibrium point? What is its stability?
  • Take that equilibrium point as your initial \(x\) value. Imagine slowly decreasing \(r\) to 0. When \(r = 0\), what equilibrium value do you end up at and why?
  • Now slowly increase \(r\) to 0.5. What equilibrium value do you end up at here, and why?
  • Finally, slowly decrease \(r\) back to 0. What equilibrium value are you at now?

The system involving an albedo of either about 0.3 or 0.7 that we looked at in class in Week 2 behaves in a similar way. How does this relate to recovery after a tipping point (like Earth freezing over in the albedo case)?

Exercise 3: Reading Questions

Answer these questions based on one of the two following readings:

📖 Reading Option 1: How Climate Scientists Saw the Future

🎧Listening/Reading Option 2: Modeling Paleoclimates

  • What two ideas stood out to you most from your reading?

  • What kinds of mathematical or statistical considerations are there in modeling the past, present, or future Earth system?

Challenge Exercise

Another system to look at: what’s happening with \(\dot{x} = rx - \sin(x)\)?