Multiple Periodicity expressed in Terms of a Torus

Such periodicity divides the infinite $u$-domain of the dynamical system into countably many equivalent subdomains of size $\Delta u=2\pi/\omega_1$. Taking into account that the ultimate goal is to uncover the statistical properties of the motion and to make a perturbational analysis of the system, one desires to compare the motion in successive subdomains. This is achieved by viewing them as a single equivalence class, a circle whose perimeter is $\Delta u=2\pi/\omega_1$. The traversal of successive subdomains corresponds to rotations by $2\pi$ on this circle.

Analogous statements hold for the other three coordinates $v$, $x$, and $y$. They are cyclic, which expresses the fact that the dynamical system is invariant under translation into these three directions. This implies invariance under discrete but arbitrary translations, say, $\Delta v=2\pi/\omega_2$, $\Delta x=L_x$, and $\Delta y=L_y$. These magnitudes are to be justified later in the presence of perturbations.

For the same reason and in the same manner as was done with the $u$-domain, one divides each of the $v$, $x$, and $y$ domains into three equivalence classes which are three circles with respective perimeters $\Delta v=2\pi/\omega_2$, $\Delta x=L_x$, and $\Delta y=L_y$. The compactification of the four rectilinear spacetime coordinates into four circles means that Minkowski spacetime $\mathbb{R}^4$ has been compactified (for the purpose of dynamical sytems analysis) into a four-dimensional torus,

$$\mathbb{T}^4=S^1\times S^1\times S^1\times S^1~.$$ (33)

It is a topological expression of the four-fold periodicity of the dynamical system. The spacetime trajectory of a particle gets represented as a winding trajectory on this torus.