Existence/uniqueness homework (SOLVED)

Your answers must be uploaded on Canvas, as a single PDF, by
5pm Eastern Time.
This homework covers 1. Existence/uniqueness; 2. Non-dimensionalization/scaling; 3. Bifurcation diagrams;
and 4. Saddle-node bifurcations.
These topic are covered in §2.4-3.1 in Strogatz.
For your bifurcation diagrams:
i. Indicate stable fixed points with a solid line and unstable fixed points with a dashed line
ii. Show your calculations for how you determined the fixed points
iii. Explain how you determined stability and/or show your calculations
iv. Clearly indicate any bifurcation(s) (if they exist)
v. Clearly identify and label bifurcation(s) (saddle-node, transcritical, pitchfork, if they exist)
1. (Problem 3.1.3 in Strogatz) Consider the following non-dimensional equation
x˙ = r + x − ln(1 + x)
a) Sketch all qualitatively different phase portraits – remember to label fixed points, indicate their stability,
and indicate the flow direction on the horizontal axis. [Note: “phase portrait” is a generic term for what
I’ve been calling the phase line in class].
b) Show that a saddle-node bifurcation occurs at some critical value of r.
c) Sketch the bifurcation diagram.
2. The following non-dimensional equation models population growth
N˙ = RN − N(1 − N)
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(1)
a) Draw a bifurcation diagram for this equation as R varies.
b) At each bifurcation, the system’s qualitative dynamics change, so that the phase portrait differs from one
side of the bifurcation to the other. Identify regions with similar dynamics and sketch a phase portrait for
each region. [Note: “phase portrait” is a generic term for what I’ve been calling the phase line in class].
c) Pick one region and sketch several trajectories, N(t), for several different initial conditions. On your plot
of N(t), indicate the fixed points.
d) Use Matlab to check your answer to part c. (You don’t need to turn this in – but it’s a good idea to use
Matlab to check your work when you can).
3. Consider the equation
x˙ = x
1/3
(2)
a) Using Matlab, perform the following two simulations:
i. Simulate this equation, running time backwards from t = 1.52 to t = 0, starting from x(t = 1.52) = 1.
ii. Simulate this equation, running time forwards from t = 0 to t = 1.52, starting from x(t = 0) = 0.
b) If you’ve done the simulations correctly (or at least in the same way that I did), then the trajectories
cross. In class, I claimed that trajectories cannot cross. Explain this apparent contradiction.
c) Calculate the exact solution for the simulation in part i. Is this solution unique? How does it differ from
the computed solution? Explain the source of any differences.
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d) Calculate the exact solution for the simulation in part ii. Is this solution unique? How does it differ from
the computed solution? Explain the source of any differences.
4. In class (and on Homework 1), we’ve been discussing the non-dimensional form of an equation modeling
the production of a protein:
dpˆ
dtˆ
= −pˆ+ α
pˆ
2
pˆ
2 + 12
a) Suppose α = 4. Linearize about each fixed point to determine stability. Which one is the most stable
(i.e., where do small perturbations decay the fastest)?
b) For how big an initial perturbation is the linear approximation good? Use Matlab to explore this. To do
so, you’ll need to define what a “good” approximation is (e.g., error less than 1%, 0.1%), and then use Matlab
to calculate the difference between the “exact” solution, generated by Matlab, and the linear approximation.
5. Turn in a completed version of worksheet 2, which you worked on during class on September 8.
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