Numerical Methods Projects
Problem 3. (80 points) An interesting class of equations are of the form
dx (2) = P(x(t − τ )) − αx(t)
for some fixed function P and constants α, τ . They are called delay-differential equations. Such an equation arises as one proposed theoretical model for the production of
red blood cells (where α is a decay constant and the ‘delay’ τ appears due to a specific
duration, roughly four days, for newly produced blood cells to mature).
Apply Euler’s tangent method to specific cases below A, B and C and produce various
plots to support your claims and observations for each. Note that in order to carry out
computations, you will need to keep track of past values of x, which can be simplified
if you select h to divide τ evenly (e.g. h = τ/100). Moreover, in specifying initial
conditions, you will need to giv not just e x(0), but also past values of x going to x(−τ ).
−1, X > 0; A. Let α = 0 and P(X) = Try out several cases for τ and experi- 1, else.
ment with numerical approximations. What kind of oscillations do you observe?
How do these oscillations depend on τ?
B. Let α = 0 and P(X) = −c · X. Verify that this will produce oscillations that
either increase or decrease in amplitude depending on the value of c relative to
τ . Experiment with values of c. Find c such that oscillations produced neither
increase nor decrease in amplitude. In this latter case, show that the period of
the oscillations observed is 4τ .
C. Let α = 1 and P(X) = 2X (such an ODE models production of blood cells 1+Xn
and is known to display chaotic dynamics). Do not confuse the notation here;
the ODE in question is:
(3) dx 2x(t − τ )
dt 1 + x(t − τ )
For this section, assume τ = 2 and study the effect of varying integer parameter
n. Experiment with different initial conditions and study asymptotic behavior
of approximate solutions (t → ∞).
Problem 4. (80 points) Write a program to numerically integrate the Lorenz system of
equations as described in class using the 3D generalization of Euler’s method. Look up
the dynamics of this system of equations, and in particular the notion of attractor.
Reproduce some approximate trajectories starting with various initial conditions. Experiment with constants appearing in the system of equations. Experiment with step size
h. Identify initial conditions near attractors (e.g. by starting off at random loca-tions in
phase space, running the trajectory for a long time until it nears an attractor, then use
the last point as initial condition near attractor). Now change initial condition (near
attractor) by small amount and see how long it takes for the sensitive dependence on
initial conditions to create a very large change in (x(t), y(t), z(t)). Produce plots and
quantify your observations and answers.
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if X > 0