

In [2]:

    
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from IPython.html.widgets import interact 
from ipywidgets import StaticInteract, RangeWidget

plt.rcParams['figure.max_open_warning'] = 100









    



:0: FutureWarning: IPython widgets are experimental and may change in the future.

This notebook looks at properties of two functions which can be described by

\begin{equation} f_E(x) = A\sin(x) - B\cos(2 x) \end{equation}\begin{equation} f_O(x) = A\cos(x) + B\sin(2 x) \end{equation}

where $x$ is a linearly increasing function of time while $A$ and $B$ are arbitary constants. We will define $x$ as:

\begin{equation} x(t) = 2\pi f t + x_{0} \end{equation}

However, we don't in fact need to concern outselves with $f$, but we will think about $x_0$.

Motivation

I came across this problem while looking at modulations in the spin-down rate of pulsars. In the work of Jones and Andersson (2001) two types of modulation are found: geometric and electromagnetic. Both can potentially be observed by careful timing, and I found a curious relation which wasn't immediately obvious why it occured. In this work I am abstracting the behaviour into the function defined above. I will refer to the $-$ function as the even, and the $+$ as the odd. Hopefully it will make sense why I decided on these labels later on..

Even

Since we only really care about the relative size of $A$ and $B$ so we will define the even function as follows

\begin{equation} f_E(x) = \sin(x) - R \cos(2x) \end{equation}

Let us define the function in python:



In [3]:

    
def fEven(x, omega, x0, R):
    return np.sin(omega*x+x0) - R*np.cos(2*(omega*x+x0))

And now we will plot it, using the powerful interactive tools of the ipython notebook. We will plot a single cycle and so for all intents and purposes we will be ignoring the frequency.



In [5]:

    
def plot_fEven(phi0=0, R=1):
    omega=2*np.pi
    fig, ax = plt.subplots()
    x = np.linspace(0, 4*np.pi/omega, 300)
    ax.plot(x, fEven(x, omega, phi0, R))
    return fig

StaticInteract(plot_fEven, 
               phi0=RangeWidget(0, 2*np.pi, 1.0), 
               R=RangeWidget(0, 2.1, 2.1))

I have labelled this function as even due to the fact the function is even about any of the maxima (but not the minimia).

Odd

Again we redefine it as:

\begin{equation} f_O(x) = \cos(x) + R \sin(2x) \end{equation}



In [52]:

    
def fOdd(x, omega, x0, R):
    return np.cos(omega*x+x0) + R*np.sin(2*(omega*x+x0))

def plot_fOdd(phi0=0, R=1):
    omega=2*np.pi
    ax = plt.subplot(111)
    x = np.linspace(0, 4*np.pi/omega, 1000)
    ax.plot(x, fOdd(x, omega, phi0, R))
    plt.show()

interact(plot_fOdd, phi0=(0, 2*np.pi), R=(0, 2.1))









    












    Out[52]:





<function __main__.plot_fOdd>

Now we have the odd function, this is distinctly different from the even function. We now see the function is not even about any either the minima or maxima.

Transformations between the two functions

We can directly see why the two functions are different by attempting to transform one into the other by linear transformations of $x$:

\begin{align} f_E(x) &= \sin(x) - R \cos(2x) \\ f_E(x+\pi/2) &= \sin(x+\pi/2) - R\cos(2(x+\pi/2) \\ f_E(x+\pi/2) &= \cos(x) + R\cos(2x) \end{align}

Here we have failed to transform the even into the odd function suggesting they are from two distinct sets of solutions. While I am sure that there is a far better way to show this, this method is sufficient for the time being.



In [ ]:

Two harmonically related sinusoids

Motivation

Even

Odd

Transformations between the two functions