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 90o 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 90o 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 \(\cfrac{dv}{dx}=0\).
