## 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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