## 6 Highlights  
We discuss dynamical stability for a classical system of two coupled oscillators in a 3D RF (Paul) trap using a well known model from literature [16; 17; 18], based on two control parameters: the axial angular moment and the ratio between the radial and axial secular frequencies of the trap. We enlarge the model by performing a qualitative analysis, based on the eigenvalues associated to the Hessian matrix of the potential, in order to explicitly determine the critical points, the minima and saddle points. The bifurcation set consists of the degenerate critical points. Its image in the control parameter space establishes the catastrophe set of equations which establishes the separatrix. We also supply the bifurcation diagram particularized to the system under investigation.  
By illustrating the phase portraits we demonstrate that ion dynamics mainly consists of periodic and quasiperiodic trajectories, in the situation when the eigenfrequencies ratio is a rational number. In the scenario in which the eigenfrequencies ratio is an irrational number, the system is ergodic and it exhibits repetitive (iterative) rotations in the vicinity of a certain point. Our results also stand for ions with different masses or ions that exhibit different electrical charges, by generalizing the system investigated. By illustrating the phase portraits and the associated power spectra we show that ion dynamics is periodic or quasiperiodic for the parameter values employed in the numerical modeling.  
The model we introduce is then used to investigate quantum stability for \(N\) identical ions levitated in a 3D QIT, and we infer the elements of the Hessian matrix of the potential function \(U\) in a critical point. We then apply our model to explore dynamical stability for SCCS consisting of \(N\) identical ions confined in different types of 3D QIT (Paul, Penning, or combined traps) that exhibit cylindrical (axial) symmetry, and show they can be studied locally in the neighbourhood of the minimum configurations that describe ordered structures (Coulomb or ion crystals [46]). In order to perform a global description, we introduce collective models with a small number of degrees of freedom or the Coulomb potential can be approximated with specific potentials for which the \(N\)-particle potentials are integrable. Small enough perturbations maintain the quantum stability although the classical system may also exhibit a chaotic behavior.  
We obtain the Hamilton function associated to a combined 3D QIT, which we show to be the sum of the Hamilton functions of the CM and of the relative motion of the ions. The ion distribution in the trap can be modeled by means of numerical analysis through the Hamilton function provided.  
The results obtained bring new contributions towards a better understanding of the dynamical stability of charged particles in 3D QIT, and in particular in combined ion traps, with applications such as high precision mass spectrometry for elementary particles, search for spatio-temporal variations of the fundamental constants in physics at the cosmological scale, etc. Our approach is also very relevant in generalizing the parameters of different types of traps in a unified manner.