This project aims at the experimental and theoretical study of many-particle systems interacting through long-range forces. The experimental tool is a cloud of ions confined by a radio-frequency linear trap and cooled by laser techniques. The long-range interaction is thus provided by the Coulomb repulsion. Such sets of trapped ions can be theoretically described at the fully microscopic level, possibly including quantum effects, or using macroscopic descriptions as continuous media. These different approaches allow the various parameters of the problem, including the number of trapped atoms, the confining potential or the cooling limit temperature, to be connected to each other. Trapped ions also offer a convenient ground for investigating the detailed thermodynamics and dynamics of a many-body system, especially its nonlinear aspects, bringing together experts in the fields of nonlinear dynamics and statistical physics. Such system is sometimes referred as one-component plasma (OCP). They have been studied theoretically and, for the last 10 years, also experimentally with laser-cooled ions in traps. In the low temperature regime, and in the case of an infinite (bulk) plasma,the ions crystallise into the body-centered cubic lattice. BCC Wigner crystals have been observed in the center of very large ion cloud in Penning traps. In radio-frequency traps, where the number of ions that can be trapped is not sufficient for such regular crystals to be observed, the importance of surface effects gives rise to an interesting competition between the trapping potential (boundary conditions) and the size of the cloud. The observed structures range from 1D strings of ions to 3D spheroidal crystals, and include 2D planar shapes where the ions are usually set as concentric shells and effective 2D shaped with ions arranging themselves in a cylinder. In the fluid regime, trapped ions exhibit many features related to nonlinear dynamics, and one interest for the present project precisely lies in the possibility to detect and even monitor chaotic motion in the ion cloud. Of special interest is the freezing transition itself, as the cloud is laser-cooled from the fluid to crystalline regimes. Because the system is far below the macroscopic limit, finite-size effects may remain significant, not only on the structure itself, but on the dynamical and thermodynamical mechanisms. Some of these features, including the characterization of the finite-size phase transitions, have already been documented in a number of cases, but only for clouds confined by a linear quadrupolar trap. In this case, the ion density is essentially uniform in the cloud. Working with higher-order traps, the ions should move away from the center, leading to a whole new type of structures. We plan to investigate further the influence of the confining potential by performing numerical simulations with Monte Carlo and molecular dynamics particle-based methods of the stable structures of large sets of ions trapped in such traps. Once the structures are characterized and compared to simple continuous media theories, the thermodynamical and nonlinear dynamical behaviours will be investigated. The experimental tool chosen for the proposed study is a double linear radio-frequency trap designed specifically for this project. In linear traps, the radio-frequency is used to confine in the plane transverse to the symmetry axis, whereas confinement along this axis is reached by static voltage. The usual configuration is based on four electrodes, defining a quadrupole geometry for the electric field, resulting in an averaged pseudopotential with a parabolic shape. We propose to couple this trap to a second part consisting of a higher order geometry (2k-pole with k=4 or 6) where the confining potential can be described by r^(2k-2). These two different potential shapes result in different spatial atomic distributions, leading to different dependences of the dynamics. These differences are expected to modify the dynamics of the ion cloud, give a detailed insight useful for comparing with theoretical predictions. This original set-up offers many possibilities of modifying the internal state of the ion cloud, opening the way to trigger reversible phase transitions and giving access to 1D, 2D or 3D dynamics. As the cloud is laser cooled, the emergence of stable structures, temperature-induced shape transitions and transitions from chaotic to regular dynamics could be monitored. Higher-order traps (2k>4) lead, by definition, to nonlinear coupled equations of motions. Thus, the stability of a trajectory depends on the initial conditions as well as the trapping parameter. To increase the number of trapped atoms or to study the influence of these intial conditions, it is thus very useful to be able to load the high-order trap with ions already laser cooled in the quadrupolar part of the trap.