Not Lorentz Attractor

Particle in oscillation and circular motion when $v_{cir}$ is real, in a $\psi$ cloud,

Particle in oscillation and circular motion when $v_{cir}$ is complex, in a $\psi$ cloud,

$O$ is the center of the containing particle.  The oscillating particle turns 90^o when the circular motion radius is zero, $r=0$.  The bottom part of the diagrams should be a smaller loop.  There can be cases where the top and bottom loops are almost equal.

This illustrate the complex case where the negative root of the factor,

$\left\{ -2{ c^{ 2 } }cos(\theta )ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } }  } x))+\cfrac { \psi _{ d } }{ m }  \right\}$

in the expression for $v_{cir}$,

$v^{ 2 }_{ cir }=cos(\theta )(x+x_{ z })\left\{ -2{ c^{ 2 } }cos(\theta )ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } }  } x))+\cfrac { \psi _{ d } }{ m }  \right\}$

occurs before the root at $x=-x_z$ due the the factor $(x+x_z)$.

These are not Lorenz attractors because of the 90^o turn.

Note:  In order that the velocity along $OP$ reverses, there has to be a sign change in $v_{\small{SHM}}$ that is accommodated here because we have an expression for $v^2_{\small{SHM}}$.  This change in direction occurs when $\cfrac{dv}{dx}=0$.