at the linear approximation, $\theta_\alpha(t) \simeq \lambda_\alpha t$ and $\theta_{\alpha+3}(t) \simeq -\lambda_\alpha t$, it follows then $\theta_\alpha(t) \simeq -\theta_{\alpha+3}(t)$.

Dynamics of the fermionic string
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Chaotic behaviour
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We consider now the dynamics of the fermionic string governed by eq. \[equSchro\] with the Hamiltonian $H^{eff}=\sigma^i \otimes (X_i-x_i)$. The dynamics of the string is indirectly affected by the fluctuations included in the dynamics of $X$ and $x$. Following ref. [@Asano; @Gur; @Hanada], these dynamics are chaotic. We can verify this fact by studying the dominant asymptotic Lyapunov exponent $\underline \lambda$ of the dynamics of $X$, fig. \[lyap\].

![\[lyap\] $\frac{1}{t} \ln \frac{\|X(t)-X_0(t)\|}{\|X(0)-X_0(0)\|}$ (computed with the Sprott algorithm [@Gur]) with $X(t)$ and $X_0(t)$ the dynamics of the D2-brane respectively with and without fluctuations in the initial conditions, with $\|X\| = \sqrt{\|X_i\|_2 \|X^i\|_2}$ ($\|\cdot \|_2$ being the usual matrix norm). The system is a fuzzy sphere with the initial fluctuations choosen following a gaussian law with $\sigma=0.01$. The number of D0-branes is $N=201$. The curve tends to the dominant Lyapunov exponent $\underline \lambda=0.75$.](LyapSphere201.eps){width="6.4cm"}

$\underline \lambda>0$ confirming its chaotic behaviour. The numerical simulations show that the value of $\underline \lambda$ is highly dependent from the model and from the initial fluctuations. The spin of the 