<p>Analyzing chaotic dynamical systems is a challenging task, due to their strong sensitivity to initial conditions, implying that even the smallest deviation in the initial conditions will amplify exponentially over the time. As a result, the reconstruction of periodic or chaotic orbits of chaotic dynamical systems from time series represents a formidable task. In this paper, we consider this problem for the case of two-dimensional discrete chaotic systems (maps). In a previous paper presented at the ISMSI 2024 conference, this problem was addressed through a popular bio-inspired swarm intelligence technique, the cuckoo search algorithm with Lévy flights, and applied to the Hénon chaotic map. In this paper, the previous method is substantially extended with the addition of a local search procedure combined with the cuckoo search algorithm, a new model accounting for cubic polynomial functions for the chaotic map with up to two nonlinearities, and several modifications of the fitness function to specialize it to the cases of periodic and chaotic orbits. These new features improve the previous method significantly, allowing to address problems unsolvable with the previous approach. The method has been applied to several instances of periodic and chaotic orbits of three popular 2D chaotic maps: the Hénon map, the Duffing map and the Burger map, exhibiting challenging behaviors, such as several non-linearities, multi-branch chaotic attractors or quasiperiodicity. The graphical and numerical results show that the method performs very well and is able to recover the periodic and chaotic orbits of these maps with good accuracy. The method also shows great robustness and generalization ability, being able to adapt to different 2D chaotic maps without further modification. Some limitations of the current approach as well as some ideas for future work in the field are also outlined.</p>