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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:
 The Lorenz Differential Equations
Before we start, we import some preliminary libraries. We will also import (below) the accompanying lorenz.py file, which contains the actual solver and plotting routine.
Step1:  We explore the Lorenz system of differential equations
Step2:  For the default set of parameters, we see the trajectories swirling around two points, called attractors. 
The object returned by interactive is a Widget object and it has attributes that contain the current result and arguments
Step3:  After interacting with the system, we can take the result and perform further computations. In this case, we compute the average positions in \(x\), \(y\) and \(z\).
Step4:  Creating histograms of the average positions (across different trajectories) show that, on average, the trajectories swirl about the attractors.
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Python Code:
%matplotlib inline
from ipywidgets import interactive, fixed
Explanation: The Lorenz Differential Equations
Before we start, we import some preliminary libraries. We will also import (below) the accompanying lorenz.py file, which contains the actual solver and plotting routine.
End of explanation
from lorenz import solve_lorenz
w=interactive(solve_lorenz,sigma=(0.0,50.0),rho=(0.0,50.0))
w
Explanation: We explore the Lorenz system of differential equations:
$$
\begin{aligned}
\dot{x} & = \sigma(y-x) \
\dot{y} & = \rho x - y - xz \
\dot{z} & = -\beta z + xy
\end{aligned}
$$
Let's change (\(\sigma\), \(\beta\), \(\rho\)) with ipywidgets and examine the trajectories.
End of explanation
t, x_t = w.result
w.kwargs
Explanation: For the default set of parameters, we see the trajectories swirling around two points, called attractors. 
The object returned by interactive is a Widget object and it has attributes that contain the current result and arguments:
End of explanation
xyz_avg = x_t.mean(axis=1)
xyz_avg.shape
Explanation: After interacting with the system, we can take the result and perform further computations. In this case, we compute the average positions in \(x\), \(y\) and \(z\).
End of explanation
from matplotlib import pyplot as plt
plt.hist(xyz_avg[:,0])
plt.title('Average $x(t)$');
plt.hist(xyz_avg[:,1])
plt.title('Average $y(t)$');
Explanation: Creating histograms of the average positions (across different trajectories) show that, on average, the trajectories swirl about the attractors.
End of explanation
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