Phase portraits of \(X=x/h\), \(M=m/h\), and \(R=\sqrt{v}/h\) for uncorrected autonomous Adam.
Quadratic loss and rotation convention.
\[f(x)=\tfrac12x^\top A x,\qquad A=Q\Lambda Q^\top,\qquad Q=R_z(\theta_z)R_y(\theta_y)R_x(\theta_x).\]
Active, right-handed rotations act on column vectors; angles are entered in degrees.
Normalized autonomous map. Define
\[X=\frac{x}{h},\qquad M=\frac{m}{h},\qquad W=\frac{v}{h^2},\qquad R=\sqrt W=\frac{\sqrt v}{h},\qquad\delta=\frac{\varepsilon}{h}.\]
The implemented map is
\[g_n=AX_n,\qquad M_{n+1}=\beta_1M_n+(1-\beta_1)g_n,\]
\[W_{n+1}=\beta_2W_n+(1-\beta_2)g_n^{\odot2},\qquad X_{n+1}=X_n-\frac{M_{n+1}}{\sqrt{W_{n+1}}+\delta}.\]
There is no bias correction. Inputs for the third state use \(W_0=v_0/h^2\), while the third portrait displays \(R=\sqrt W\).
Ready.
All parameter changes automatically rerun the simulation after a short debounce. Visual styling changes redraw without recomputing the orbit.