$T^4$ Strikes Again!

This looks like the simplest oscillating system ever,

$m\cfrac { d^{ 2 }x }{ dt^{ 2 } } =-2{ mc^{ 2 } }ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } }  } x))+\psi _{ d }$

$\cfrac { d^{ 2 }x }{ dt^{ 2 } } =-2{ c^{ 2 } }ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } }  } x))+\cfrac { \psi _{ d } }{ m }$

Compared to,

$m\cfrac{d^2x}{dt^2}=-kx$

it is not linear.  So, consider,

$(2{ mc^{ 2 } }ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } }  } x)))^{'}=c{ \sqrt { { 2m } }  }G.tanh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } }  } x)$

$\lim\limits_{x\to\,large}{c{ \sqrt { { 2m } }  }G.tanh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } }  } x)}=c{ \sqrt { { 2m } }  }G$

Then we have,

$m\cfrac { d^{ 2 }x }{ dt^{ 2 } } =-c{ \sqrt { { 2m } }  }G.x+\psi _{ d }$

and so, the approximate natural oscillating frequency is,

$f_{osc}=\sqrt{\cfrac{c{ \sqrt { { 2m } }  }G}{m}}=\left(\cfrac{2}{m}\right)^{{1}/{4}}\sqrt{{cG }}$

where $m$ is the mass density of the particle and $G$ is the constant of integration from solving $F_{\rho}$.  $c$ is the speed of light.

$\left(f_{osc}\right)^4=\cfrac{2c^2G^2}{m}$

It is expected that since the lower gradient values are less than or equal to the approximate value, as we let $x\to\,large$, the actual oscillation frequencies given a large population of particles, spread to higher values of $f_{osc}$.  In other words, blue.