The current research focuses on the prediction of the maximum axial compression load a cylindrical shell is able to bear. This maximum axial compression load is the value at which a cylindrical shell loses stability and abruptly buckles. After the buckling event the shells are permanently damaged and become unfit for any structural function. Hence, buckling is a critical not admissible failure mode in the design of shell structures. Standard methodologies available in the literature such as linear stability analysis fail to deliver an accurate prediction and overestimate the value of the load at which collapse occurs. The reason for this deviation between real shells and theory is the presence of geometric imperfections that are unique for each shell. In fact, nominally identical shells exhibit large variations in their load bearing capability. This variability means that the only reliable approach able to obtain the real loading bearing capability previous to this research is a destructive compression test. In this thesis, a new conceptual approach to describe the behavior of cylindrical shells is introduced. The new description is based on the dynamical systems approach used to study turbulence in the field of fluid dynamics. Using the dynamical systems approach applied to a non-linear formulation of the shell equations, fix points of the dynamical system are calculated and their stability under finite amplitude perturbations characterised. The boundary delimiting the transitions to a buckled state or returning to the unbuckled one is also characterized. The basin enclosed by this boundary, the basin of attraction, becomes smaller as the axial load is increased, vanishing at the compression load at which a cylindrical shell buckles. The shrinking of the basin of attraction can be characterized by evaluating its extension at different axial loads. This can be done by probing cylindrical shells. This probing at different axial compression loads defines a landscape that can be used to extrapolate the load at which the landscape, together with the basin of attractions vanishes. This axial compression load is the buckling load of the shell. The framework described theoretically is implemented in a series of test campaigns to show the predictive capability of stability landscapes in real shells. Stability landscapes are the experimental technique employed to explore and characterize the basin of attraction associated with each fix point. The exploration of the basin of attraction is performed using a single poker that represents only one of the directions to perturb a fix point of the dynamical system towards the boundary of the basin of attraction. The success of extrapolating the buckling load with stability landscapes constructed with a single poker at a single location does not provide a perfect predictive capability in real shells. The reason for this is the complex interaction between the imperfections present in real shells. The complex interaction between imperfections was also studied during this research, revealing that the buckling load of cylindrical shells is a function of all the imperfections present in the shell. The buckling load of a cylindrical shell is not dictated by the strongest imperfection alone, but by the combination of all defects. The key output of the current research is a non-destructive test procedure based on constructing stability landscapes at different locations of a cylindrical shell