I was trying to construct ODE system with solution stays in constrained area and end up with this: $$\frac{\mathrm dx}{\mathrm dt} = -\cos\left(\frac{y}{z+1}\right)$$ $$\frac{\mathrm dy}{\mathrm dt} = -\sin\left(\frac{x}{z+1}\right)$$ $$\frac{\mathrm dz}{\mathrm dt} = \cos(x) - \sin(z) $$ With initial conditions $$x_0=0.14\,,\; y_0=3.52\,,\; z_0=1.35$$ it looks like this with $t$ in range $[0,100000]$ and step $0.1$ by $8$ order RK solver (scipy library solve_ivp):
Visualization of phase space $x,y,z-t$ annotated by color, blue at start $[0]$ and red at end $[100000]$.
I wonder if there is something specific about this system? Does solution always stays at some radius about empty region with $\,t\to\infty\,?$