Search

In vector field analysis, saddle points have two different types of invariant manifolds, namely stable ones and unstable ones. The invariant manifolds represent separatrices that partition the domain of trajectories into invariant regions of different dynamics. In this work, we analyze the basins of attraction of two different stable nodes by reconstructing the separatrices of a saddle point. To this purpose we present a computational algorithm that detects the points lying on the manifold, considering the plane generated by the two stable eigenvectors of the saddle point. Finally we reconstruct the surface by using the moving least-squares approximant method.

Cooperation is a common behavior between the members of predators species, because it can improve theirs skill in hunt, especially in endangered eco-systems. This behavior it is well known to induce the Strong Allee effect, that can induce the extinction when the initial populations’ is under a critical density called ”Allee threshold ”. Here we investigate the impact of the pack hunting in a predator-prey system in which the predator suffers of an infectious disease with frequency and vertical transmission. The result is a three dimensional system with the predators population divided into susceptible and infected individuals. Studying the system dynamics a scenario was identified in which the model presents a bistability. However for a strong hunting cooperation the Allee threshold becomes almost zero, ensuring the survival of the predators. Thus we present a study to analyze this critical density by considering the basins of attraction of the stable equilibrium points. This paper addresses the question of finding the point lying on the surface which partitions the phase plane. Therefore a Moving Least Square (MLS) method based on compactly supported radial functions has been adopted to reconstruct the separatrix manifold.

The theory of random matrices, or random matrix theory, RMT in what follows, has been developed at the beginning of the fties to describe the sta- tistical properties of energy levels of complex quantum systems, [1], [2], [3]. In the early eighties it has enjoyed renewed interest since it has been recognized as a very useful tool in the study of numerous physical systems. Specically, it is very useful in the analysis of chaotic quantum systems. In fact, in the last years many papers appeared about the problem of quantum chaos which implies the quantization of systems whose underlying classical dynamics is irregular (i.e. chaotic). The simplest models considered in this eld are billi- ards of various shapes. From the the classical point of view, a point particle in a 2-dimensional billiard displays regular or irregular motion depending on the shape of the billiard; for instance motion in a rectangular or circular billi- ard is regular thanks to the symmetries of the boundary. On the other hand, billiards of arbitrary shapes imply chaotic motion, i.e. exponential diver- gence of initially nearby trajectories. In order to study quantum billiards we have to consider the Schroedinger equation in various 2-dimensional domains. The eigenvalues of the Schroedinger equation represent the allowed energy levels of our quantum particle in the billiard under consideration, while the eigenfunction norms represent the probability density of nding the particle in a certain position. The question of quantum chaos is whether the charac- ter of the classical motion (regular or chaotic) can in uence some properties