I have found that many proofs regarding the stability of the $L_4$ and $L_5$ Lagrange points are based on linear approximations of the equations of motion near these points. However, from a dynamical systems perspective, Lagrange points are essentially fixed points of the phase flow. Linear approximations at fixed points yield a linear system of equations, such as $\dot{\phi}=A\phi$, where $\phi$ is a coordinate in phase space and $A$ is a constant matrix. If all eigenvalues of $A$ have negative real parts, then the fixed point is stable. If $A$ has an eigenvalue with a positive real part, then the fixed point is unstable, which corresponds to the case of $L_1$ to $L_3$ points. However, if all eigenvalues of $A$ are purely imaginary, we cannot determine the stability of the fixed point from the linear approximation of the phase flow. This is precisely the case for $L_4$ and $L_5$ points.
Many "proofs" I have come across only argue that the eigenvalues are imaginary, without considering higher-order approximations of the phase flow, cf. e.g. this Phys.SE post. I would like to inquire if such treatment is rigorous, and if there exists a rigorous proof regarding the stability of the $L_4$ and $L_5$ Lagrange points.
To illustrate the details, here is an example from "An Introduction to Dynamical Systems: Discrete and Continuous" by R. C. Robinson:
Consider a 2-D system: $$ \begin{aligned} \dot{x} &= -y+ax(x^2+y^2)\\ \dot{y} &= x+ay(x^2+y^2) \end{aligned} $$ The origin is a fixed point of this system, and through linear approximation, we can determine that $$ \begin{bmatrix} \delta\dot x\\ \delta\dot y \end{bmatrix}=\begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}\begin{bmatrix} \delta x\\ \delta y \end{bmatrix} $$ The eigenvalues of this linear approximation matrix are $\pm\mathrm{i}$, which are purely imaginary. However, if we consider this system in polar coordinates, using $r^2=x^2+y^2$, we have: $$ \begin{aligned} &r\dot r = x\dot x+y\dot y=x(-y+axr^2)+y(x+ayr^2)=ar^4\\ \Rightarrow\quad& \dot r= ar^3 \end{aligned} $$ It can be seen that when $a>0$, the origin is a repelling point and unstable; only when $a\le0$, the origin is a stable point.
