Most of these exercises are of an advanced nature. However, any of the exercises from Chapter 8 may be done using odesolve, so look there for more elementary exercises.
The Rossler System
xʹ = −y−z
yʹ = x+ay
zʹ = b − cz + xz
provides another example of a strange attractor similar to that of the Lorenz System.
a) Set the parameters to a = 0.2, b = 0.2,c = 2.5, and use the initial conditions x (0) = y(0) = z(0) = 1. Set the maximum step size of the solver to 0.5. Select 3D plot and select the variablesx, y, and z. solve and plot the solution on the time interval [0,250]. Export the data to the command window workspace as the structure odedata*, where 8 is a small integer. Then execute t = odedata*. t,x = odedata*. x, y = odedata*. y, and z = odedata*. z. Eliminate transient behavior with N=find (t>200), followed by plot(x(N) ,y(N) to plot in the xy-plane.
b) Repeat the instructions of part a) for c = 3, c = 4, and c = 4.2. Obtain xy-plots for each c-value. Be sure to eliminate transient behavior as was done in part a).
c) Set c = 5 and and the time interval to [0,1000], Repeat part a), but restrict the xy-plot to the time interval [500, 1000].
d). The progression in parts a) and b) is called period doubling on the route to chaos, which is illustrated in part c). Can you explain why?
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