Editing Appendix/Ramblings/NonlinarOscillation (section)

====Quartic Solution====
Here, we will draw from the [https://en.wikipedia.org/wiki/Quartic_function Wikipedia discussion of the quartic function].  The generic form is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>ax^4 + bx^3 + cx^2 + dx + e \,.</math>
  </td>
</tr>
</table>
</div>
Relating this to our specific quartic function, we should ultimately make the following assignments:
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>a</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>9</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>b</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>20 - \delta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>c</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>12 -  3\delta  = 3(4-\delta )</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>d</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-3 \delta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>e</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\delta</math>
  </td>
</tr>
</table>
</div>

We need to evaluate the following expressions:
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>p</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{8ac-3b^2}{8a^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2^3 \cdot 3^3(4-\delta )-3(20-\delta )^2}{2^3\cdot 3^4}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>q</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{b^3 - 4abc + 8a^2d}{8a^3}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{(20 - \delta )^3 - 2^2\cdot 3^3(4-\delta ) (20 - \delta ) - 2^3\cdot 3^5\delta }{2^3 \cdot 3^6}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Delta_0</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>c^2 - 3bd + 12ae</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3^2(4-\delta )^2 + 3^2\delta (20 - \delta ) - 2^2\cdot 3^3\delta</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>144</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Delta_1</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>2c^3 - 9bcd + 27b^2e+27ad^2 - 72ace</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2\cdot 3^3(4-\delta)^3 + 3^4(20-\delta)(4-\delta)\delta - 3^3(20-\delta)^2 \delta+3^7\delta^2 + 2^3\cdot 3^5(4-\delta)\delta</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3^3(128 + 32\delta + \delta^2) \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
Note:  &nbsp; <math>\Delta_1^2 - 4\Delta_0^3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3^6(2^{13}\delta + 2^8\cdot 5 \delta^2 + 2^6\delta^3 + \delta^4) \, .</math>
  </td>
</tr>
</table>
</div>

For a given value of <math>\delta</math>, then, the pair of real roots is:
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>\epsilon_\pm</math>
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  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{b}{4a} + S \pm \frac{1}{2}\biggl[ -4S^2 - 2p - \frac{q}{S} \biggr]^{1/2} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>S</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{1}{2}\biggl[- \frac{2p}{3} + \frac{1}{3a}\biggl(Q + \frac{\Delta_0}{Q}\biggr)  \biggr]^{1/2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>Q</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\Delta_1 + \sqrt{\Delta_1^2 - 4\Delta_0^3}}{2} \biggr]^{1/3} \, .
</math>
  </td>
</tr>
</table>
</div>

We have used an Excel spreadsheet to evaluate these expressions.  The following table identifies <math>\epsilon_\pm</math> pairs (the middle two columns of numbers) for twenty different values of the external pressure; more specifically, for twenty values of <math>0 \le \delta \le 0.69938</math>, equally spaced between the two limits.  The corresponding pairs of <math>\tilde\xi_\pm</math> are also listed (rightmost pair of columns).

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<table border="0" align="center"><tr><th align="center">
<font size="+1">Table 1</font></th></tr>
<tr><td align="center">

<table border="1" align="center">
<tr><th align="center">
<font size="+1">Sets of Paired Models from Quartic Solution</font>
</th></tr>
<tr><td align="left">
<pre>
P_e/P_norm   delta	  eps_+   eps_-	         xi_+ 	 xi_-
160.867	    0.00000	0.00000  0.00000	3.00000	3.00000
158.664	    0.03681	0.05747	-0.05338	3.17241	2.83986
156.497	    0.07362	0.08253	-0.07435	3.24759	2.77695
154.363	    0.11043	0.10227	-0.09000	3.30681	2.72999
152.264	    0.14724	0.11927	-0.10291	3.35781	2.69127
150.198	    0.18405	0.13452	-0.11407	3.40355	2.65780
148.164	    0.22086	0.14852	-0.12398	3.44556	2.62805
146.163	    0.25767	0.16159	-0.13296	3.48476	2.60111
144.193	    0.29448	0.17391	-0.14120	3.52174	2.57640
142.254	    0.33129	0.18563	-0.14883	3.55690	2.55351
140.345	    0.36809	0.19685	-0.15596	3.59055	2.53212
138.466	    0.40490	0.20764	-0.16266	3.62291	2.51203
136.617	    0.44171	0.21805	-0.16898	3.65415	2.49305
134.796	    0.47852	0.22814	-0.17498	3.68442	2.47505
133.003	    0.51533	0.23794	-0.18070	3.71382	2.45791
131.239	    0.55214	0.24748	-0.18615	3.74245	2.44154
129.501	    0.58895	0.25680	-0.19138	3.77039	2.42586
127.790	    0.62576	0.26590	-0.19640	3.79770	2.41081
126.106	    0.66257	0.27481	-0.20122	3.82444	2.39634
124.447	    0.69938	0.28355	-0.20587	3.85065	2.38238
</pre>
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</table>

</td></tr>
</table>
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