Manuel Falconi

Global Dynamics of Mechanical Systems with Cubic Potentials

We study the behavior of solutions of mechanical systems with polynomial potentials of degree 3 by using a blow up of McGehee type. We first state some general properties for positive degree homogeneous potentials. In particular, we prove a very general property of transversality of the invariant manifolds of the flow along the homothetic orbit. The paper focuses in the study of global flow in the case of homogeneous polynomial potentials of degree 3 for negative energy. The flow is fairly simple because of its gradient-like structure, although for some values of the polynomial coefficients we have diverse behaviour of the separatrices on the infinity manifold, which are essential to describe the global flow.

n-1 Independient firts integrals for linear differential systems in Rn and Cn

We prove that every linear system with constant coefficients on Rn or Cn is Darboux integrable by providing a complete explicit list ofn−1 independent Darboux first integrals.

Stability and global dynamic of a stage-structured predator-prey model with group defense mechanism of the prey

Dynamics of predator-prey model where the predator feeds on two age classes of prey.Existence of defense mechanism and the interference of the prey and the age.The coexistence of the predator and the prey populations is possible.There exists a bounded range of the carrying capacity for co-existence. In this paper we describe the flow of a predator-prey model where the predator feeds on two age classes of prey. Some basic features of the flow are proved under very mild hypothesis on the functional responses. To study the relationship between the ecological parameters and the behavioral ones, we also analyze a special case of the model where modified versions of the Holling type II functional responses are considered to take into account both a certain interference of the age classes of the prey on the predator activity and a defense mechanism of the juvenile class of the prey. Indeed, we find that independently of the profit that each one of the prey age class provides to the predator, the coexistence of the predator and the prey populations is only possible for a bounded range of the carrying capacity due to the joint action of the defense mechanism and the interference of the prey and the age.

Chaotic dynamics and coexistence in a three species interaction model

This work deals with a three-dimensional system, which describes a food web model consisting of a prey, a specialist predator and a top predator which is generalist as it consumes the other two species. Using tools of dynamical systems we prove that the trajectories of system are bounded and that open subsets of parameters exist, such that the system in the first octant has at most two singularities. For an open subset of the parameters space, the system is shown to have an invariant compact set and this is a topologically transitive attractor set. Finally, we find another open set in the parameters space, such that the system has two limit cycles each contained in different invariant planes. The work is completed with a numeric simulation showing the attractor is a strange attractor.

Asymptotic measures of random logistic maps

In this paper, we consider a system whose state x changes to if a perturbation occurs at the time t, for and the state x changes to the new state at the time t, for . Here, and are logistic maps. We assume that the number of perturbations in the interval is a discrete random variable . We show that under certain conditions on the parameters and , the system has, even for the non-contractive case, an unique stationary probability measure, the support of which can be either a Cantor set or an interval.

A Generalist Predator and the Planar Zero-Hopf Bifurcation

A typical approach for searching periodic orbits of planar dynamical systems is through the Hopf bifurcation. In this work we present a family of predator–prey models with a generalist predator which does not exhibit a Hopf bifurcation, but a planar zero-Hopf bifurcation; that means, in the whole bifurcation process the eigenvalues of the linear approximation around the equilibrium points remain as pure imaginary. Similar models with a nongeneralist predator always possess a Hopf bifurcation.

On the coexisting dynamics in the alternate iteration of two logistic maps

We consider a two parameter family of maps which are obtained as composition of two logistic maps. We say that a map presents a coexistence of dynamics if it has two attractors. In this article, we give conditions on the parameters to have coexistence of dynamics and we give a topological description of the parameter space.

On the parameter dependence of the real cubic surfaces in Arnold's form

AbstractWe consider the topological classification of cubic surfaces which are obtained as intersection of the sphere

HOPF BIFURCATION IN PREDATOR-PREY MODELS WITH AN AGE-STRUCTURED PREY PROCEEDINGS

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