respect to the evolution of $X(t)$ and $x(t)$, we can draw the Lyapunov frequencies of the dynamical system with respect to $t$, see fig. \[loclyap\].

![\[loclyap\] Evolution of the local Lyapunov frequencies of the equation \[eqsd2\] (${\mathrm{Sp}}(\sqrt{I})$) for fuzzy sphere, ellipsoid, torus and cylinder. The initial fluctuations are choosen following a gaussian law with $\sigma=0.01$. The number of D0-branes is $N=201$.](locLyapSphere201.eps "fig:"){width="6.4cm"} ![\[loclyap\] Evolution of the local Lyapunov frequencies of the equation \[eqsd2\] (${\mathrm{Sp}}(\sqrt{I})$) for fuzzy sphere, ellipsoid, torus and cylinder. The initial fluctuations are choosen following a gaussian law with $\sigma=0.01$. The number of D0-branes is $N=201$.](locLyapEllipsoid201.eps "fig:"){width="6.4cm"}\
![\[loclyap\] Evolution of the local Lyapunov frequencies of the equation \[eqsd2\] (${\mathrm{Sp}}(\sqrt{I})$) for fuzzy sphere, ellipsoid, torus and cylinder. The initial fluctuations are choosen following a gaussian law with $\sigma=0.01$. The number of D0-branes is $N=201$.](locLyapTorus201.eps "fig:"){width="6.4cm"} ![\[loclyap\] Evolution of the local Lyapunov frequencies of the equation \[eqsd2\] (${\mathrm{Sp}}(\sqrt{I})$) for fuzzy sphere, ellipsoid, torus and cylinder. The initial fluctuations are choosen following a gaussian law with $\sigma=0.01$. The number of D0-branes is $N=201$.](locLyapCylinder201.eps "fig:"){width="6.4cm"}

The effect of the thermalization can be also viewed with the evolution of the Lyapunov frequencies.

Dynamics onto the invariant torus
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Now we want describe the 