Editing Appendix/Ramblings/NonlinarOscillation (section)

===Two Example Eigenfunctions===

The following figure is fundamentally a reproduction of [[SSC/FreeEnergy/Powerpoint#Figure3|Figure 3 from an accompanying discussion]].  It presents the "Case M" equilibrium sequence from both an order-of-magnitude analysis (marked by light-blue squares) and a detailed force-balance analysis (light-green triangles).  The dark green circular dot identifies the configuration at the pressure maximum of the sequence &#8212; <math>P_\mathrm{max}/P_\mathrm{norm} = 160.867</math> &#8212; and the red circular dot identifies the location along the sequence where the transition from stable to dynamically unstable configurations occurs  &#8212; <math>P_e/P_\mathrm{norm} = 124.447</math>.  (All pressures have been normalized to <math>P_\mathrm{max}</math> in the figure.)

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<tr><th align="center"><font size="+1">Figure 1</font></th></tr>
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[[File:CaseMpairings0.png|400px|center|Case M equilibrium sequences]]
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At any <math>P_e</math> between these two limiting values, a pair of ''stable'' equilibrium configurations exist; approximately twenty example pairings are listed in Table 1.  The horizontal, black dashed line in the figure has been drawn at <math>P_e/P_\mathrm{norm} = 158.664</math>.  The pair of equilibrium configurations associated with this pressure is identified graphically by the two points at which this dashed line intersects the detailed force-balance equilibrium sequence; as is detailed in the second row of Table 1, the configurations correspond to models having <math>\tilde\xi = 2.83986</math> (right intersection) and <math>\tilde\xi = 3.17241</math> (left intersection).  The left-hand panel of Figure 2 shows how the Lagrangian radial coordinate varies with mass, <math>r_\xi(m_\xi)</math>, throughout the interior of these two equilibrium configurations (the locus of green and orange dots, respectively); for reference, the profile of the configuration at <math>P_\mathrm{max}</math> is presented as well (locus of black dots). The right-hand panel of Figure 2 shows the same paired configuration profiles, but ''relative'' to the profile of the configuration at <math>P_\mathrm{max}</math>.

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<table border="0" align="center">
<tr><th align="center"><font size="+1">Figure 2</font></th></tr>
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[[File:CaseMeigen2.png|600px|center|Case M eigenfunction#1]]
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The horizontal, black dot-dash line in Figure 1 has been drawn at <math>P_e/P_\mathrm{norm} = 124.447</math>.  The pair of equilibrium configurations associated with this pressure is identified graphically by the two points at which this dot-dash line intersects the detailed force-balance equilibrium sequence; as is detailed in the last row of Table 1, the configurations correspond to models having <math>\tilde\xi = 2.83986</math> (right intersection) and  <math>\tilde\xi = 3.17241</math> (left intersection), which is the configuration that marks the onset of a dynamical instability.  The left-hand panel of Figure 3 shows how the Lagrangian radial coordinate varies with mass, <math>r_\xi(m_\xi)</math>, throughout the interior of these two ''paired'' equilibrium configurations (the locus of green and orange dots, respectively); again, for reference, the profile of the configuration at <math>P_\mathrm{max}</math> is presented as well (locus of black dots). The right-hand panel of Figure 3 shows the same paired configuration profiles, but ''relative'' to the profile of the configuration at <math>P_\mathrm{max}</math>.


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<tr><th align="center"><font size="+1">Figure 3</font></th></tr>
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[[File:CaseMeigen1.png|600px|center|Case M eigenfunction#2]]
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