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Numerical Methods Projects Problem 3. (80 points) An interesting class of equations are of the form dx (2) = P(x(t − τ )) − αx(t) dt 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 ) = n − x(t). 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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%% Problem 3
%% Part A
for tau=1:3



    for k=N+1:2000
       if X > 0
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