## Lyapunov's exponents as a function of the detuning for trajectories as those illustrated in figure 3 (shown in the same order) and in figure 4

**Figure 6.** Lyapunov's exponents as a function of the detuning for trajectories as those illustrated in figure 3 (shown in the same order) and in figure 4. The eigenvectors of the biggest Lyapunov exponents of the apparently quasi-periodic trajectories were found to involve always the axial motion. The other parameters of the beam and their coupling to the atom are *l* = 2, *k*_{⊥} = 0.66 μm^{−1}, *g*_{peak} = 47.5 Γ.

**Abstract**

We characterize the semiclassical dynamics of dilute thermal atom clouds located in three-dimensional optical lattices generated by stationary optical Bessel beams. The dynamics of the cold atoms is explored in the quasi-Hamiltonian regime that arises using laser beams with far-off resonance detuning. Although the transverse structure of Bessel beams exhibits a complex topological structure, it is found that the longitudinal motion along the main propagation axis of the beam is the detonator of a high sensitivity of the atoms' motion to the initial conditions. This effect would not be properly described by bidimensional models. We show that an experimental implementation can be highly simplified by an analysis of the behaviour of the dynamical system under scale transformations. Experimentally feasible signatures of the chaotic dynamics of the atom clouds are also identified.