Math/CCM 404/504 - Assignment 2
Due:
Tuesday, January 24. Nothing accepted after Thursday, January 26. 2 points off for being late.
Please work by yourself. See me if you need help.
You are studying the same population of turtles that you studied in Assignment 1. Again x(t) is the number
of turtles at time t measured in 100's. Suppose that x(t) is governed by the following differential equation.
dx
= - 0.1 x + 0.025 x2 - 0.001 x3 - hx
dt
(1)
Here h is a parameter that measures the fishing intensity. Note that this equation has the form
dx
= f(x, h) where f(x, h) = - 0.1 x + 0.025 x2 - 0.001 x3 - hx.
dt
1. (2 points) If we consider the algebraic equation f(x, h) = 0, then the solutions lie on certain curves in the
h-x plane. Make a plot of the solution curves of the equation f(x, h) = 0 in the h-x plane. This plot is the
start of the bifurcation diagram of the differential equation. The curves show how the equilibrium points
of (1) vary with h. Make the plot with the h-axis horizontal and the x-axis vertical. Draw the curves in
some other color than the normal color you write with.
2. (2 points) The curves that you plotted in part 1 are where f(x, h) = 0. These curves separate the h-x
plane into regions. In each region f(x, h) is either positive or negative. Determine the sign of f(x, h) in
each region and indicate this in the bifurcation diagram by vertical arrows where a vertical arrow
pointing up indicates positive and a vertical arrow pointing down indicates negative.
3. (2 points) Using the information in part 2, determine which parts of the solution curves of f(x, h) = 0 are
sinks, which parts are sources, which parts are neither. Label this on the bifurcation diagram.
4. (2 points) Determine the bifurcation point(s) and classify them as to their type, e.g. saddle-node,
exchange, or pitchfork.
5. (4 points) Consider the electric circuit at the right. Let i be the current.
Assume the following.
i.
The voltage source supplies a constant voltage of VB = 6 volts.
ii. The voltage drop VR across the resistor is related to the current by
the formula i = eVR – 1.
iii. The inductor has an inductance of L = 3 henrys. Recall that the
voltage drop VL across an inductor is related to the current by the
di
formula VL = L dt .
di
Find a differential equation of the form dt = f(i) that i satisfies. The
function f(i) should involve no other variables except i. Recall that
Kirchoff’s voltage law says that the voltage change between two points
on the circuit is the same no matter what path you take between the two
points.