# Non-stable vs non-linear orbits

In orbital dynamics there are often two 'classes'- stable and non-stable, or periodic and non-periodic. Is this just a specific class of linear and non-linear harmonics? Would Klemperer's rosette orbit be an example of a non-linear orbit because it is neither closed nor stable?

• The only linear (non-curved) orbital paths are directly towards or away from the more massive body, but the motion along these paths is non-linear due to acceleration from gravity. Thus, all orbital motion is nonlinear in $\mathbb{R}^3$. One might argue that a circular orbit is linear in a spherical coordinate system aligned perpendicular to the orbital plane, since then $r$, $\phi$, and $\dot{\rho}$ are constant, which I suppose is true, but not very useful. Commented Feb 24, 2022 at 18:40
• @ConnorGarcia I interpreted the OP as asking about "linear harmonic oscillation", not linear paths. Commented Feb 28, 2022 at 12:13
• @CarlWitthoft Under that interpretation, there is no simple (linear) harmonic motion since gravitational forces scale with $r^2$. The obvious only exception is when $r$ is fixed, for a circular orbit, in which the motion could be considered harmonic along each dimension in the orbital plain: en.wikipedia.org/wiki/…. Commented Feb 28, 2022 at 20:15