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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Final
Chinmai Raman
5/21/2016
Given three real-valued functions of time x(t), y(t), z(t), consider the following coupled first-order ODEs
Step1:  Problem 2 (c = 2, 3, 4, 4.15, 4.2, 5.7)
c = 2 is plotted above
c = 3
Step2:  Already we can see a bifurcation occuring in the y vs x and z vs y vs x graphs that were not there in the case of c =2. The nonlinearity in the z variable has begun to become active in that the trajectory is leaving the x-y plane. 
The x vs t graph shows us that the x-values are now alternating between four values, as opposed to two previously. This is identical to the behavior we saw from the logistic update map on the midterm.
c = 4
Step3:  Another bifurcation has now occured and is apparent in the y vs x graph. The limits of the x-values are now eight-fold; the number of values that x converges to has doubled again. The influence of the non-linearity in z is now very obvious.
c = 4.15
Step4:  The period doubling is occuring at an increasing rate. This is demonstrated by the thicker lines in the xy and xyz graphs. This period doubling phase will soon end as the system approaches complete chaos and the our predictive power decreases greatly.
c = 4.2
Step5:  The lines are getting thicker as the bifurcations increase at an increasing rate. It is now not immediately apparent how many asymptotic values x approaches from the x vs t graph.
c = 5.7
Step6:  The period doubling cascade has given rise to a chaotic attractor with a single lobe. This is an example of spiral-type chaos and exhibits the characteristic sensitivity to initial conditions. The oscillations in the x vs t graph are now completely chaotic and irregular in amplitude. The logistic update map also displays the same behavior of a period doubling cascade giving rise to a chaotic system. As we increase c, this system, like the logistic update map from the midterm as ve vary initial conditions, also demonstrates the stretching and folding quality that we discussed in class. The xyz graph is also reminiscent of a mobius strip in that the underside becomes the upperside via the portion of the graph not in the x-y plane.
Problem 3
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Python Code:
ros = p1.Rossler(2)
ros.run()
ros.plotx()
ros.ploty()
ros.plotz()
ros.plotxy()
ros.plotyz()
ros.plotxz()
ros.plotxyz()
Explanation: Final
Chinmai Raman
5/21/2016
Given three real-valued functions of time x(t), y(t), z(t), consider the following coupled first-order ODEs:
$x˙ = −y − z, y˙ = x + ay, z˙ = b + z(x − c)$
where a = b = 0.2 and c is a parameter that we will tune. Note that this system has a single nonlinear term
xz. 
I will be exploring the consequences of this nonlinearity.
Problem 1 (c = 2)
End of explanation
ros3 = p1.Rossler(3)
ros3.run()
ros3.plotx()
ros3.plotxy()
ros3.plotxyz()
Explanation: Problem 2 (c = 2, 3, 4, 4.15, 4.2, 5.7)
c = 2 is plotted above
c = 3
End of explanation
ros4 = p1.Rossler(4)
ros4.run()
ros4.plotx()
ros4.plotxy()
ros4.plotxyz()
Explanation: Already we can see a bifurcation occuring in the y vs x and z vs y vs x graphs that were not there in the case of c =2. The nonlinearity in the z variable has begun to become active in that the trajectory is leaving the x-y plane. 
The x vs t graph shows us that the x-values are now alternating between four values, as opposed to two previously. This is identical to the behavior we saw from the logistic update map on the midterm.
c = 4
End of explanation
ros415 = p1.Rossler(4.15)
ros415.run()
ros415.plotx()
ros415.plotxy()
ros415.plotxyz()
Explanation: Another bifurcation has now occured and is apparent in the y vs x graph. The limits of the x-values are now eight-fold; the number of values that x converges to has doubled again. The influence of the non-linearity in z is now very obvious.
c = 4.15
End of explanation
ros42 = p1.Rossler(4.2)
ros42.run()
ros42.plotx()
ros42.plotxy()
ros42.plotxyz()
Explanation: The period doubling is occuring at an increasing rate. This is demonstrated by the thicker lines in the xy and xyz graphs. This period doubling phase will soon end as the system approaches complete chaos and the our predictive power decreases greatly.
c = 4.2
End of explanation
ros57 = p1.Rossler(5.7)
ros57.run()
ros57.plotx()
ros57.plotxy()
ros57.plotxyz()
Explanation: The lines are getting thicker as the bifurcations increase at an increasing rate. It is now not immediately apparent how many asymptotic values x approaches from the x vs t graph.
c = 5.7
End of explanation
p1.plotmaxima('x')
p1.plotmaxima('y')
p1.plotmaxima('z')
Explanation: The period doubling cascade has given rise to a chaotic attractor with a single lobe. This is an example of spiral-type chaos and exhibits the characteristic sensitivity to initial conditions. The oscillations in the x vs t graph are now completely chaotic and irregular in amplitude. The logistic update map also displays the same behavior of a period doubling cascade giving rise to a chaotic system. As we increase c, this system, like the logistic update map from the midterm as ve vary initial conditions, also demonstrates the stretching and folding quality that we discussed in class. The xyz graph is also reminiscent of a mobius strip in that the underside becomes the upperside via the portion of the graph not in the x-y plane.
Problem 3
End of explanation
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