Description
Problem 1 Simulate the response of the following system:
v˙ + |v|v = u
Assume that we apply a unit step input in thrust u, followed 5 seconds later by a negative unit
step input. Repeat with increasing the input u 10 times. Compare the results with linear
system:
v˙ + v = u
Problem 2 Simulate the response of:
x˙ = x − x
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from initial points x0 = −1.5, −1, −0.5, 0, 0.5, 1, 1.5.
Plot the result on one graph and discuss the behavior for each initial condition.
Problem 3 Simulate the response of Lotka-Volterra (predator-prey) equations:
(
x˙ = αx − βxy
y˙ = δxy − γy
with α = 2/3, = 4/3, γ = δ = 1.
Assume x, y quantify thousands each and predator/prey initial conditions from x0 = y0 = [0.9, 1.8],
in steps of 0.1.
Plot the all trajectories on (x, y) plane. Does the resulting trajectory represent the limit cycle?
Problem 4 You may face chaos even in familiar mechanical systems as double pendulum.
To check this, simulate the responses of the double pendulum nearby initial conditions around
θ1 = π/2, θ2 = π/2 and compare the resulting cartesian trajectories.
The great animation may be found here.
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