Ernest Fontich

The parameterization method for invariant manifolds III: overview and applications

We describe a method to establish existence and regularity of invariant manifolds and, at the same time to find simple maps which are conjugated to the dynamics on them. The method establishes several invariant manifold theorems. For instance, it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in Banach spaces. We also present several other applications of the method.

The splitting of separatrices for analytic diffeomorphisms

We study families of diffeomorphisms close to the identity, which tend to it when the parameter goes to zero, and having homoclinic points. We consider the analytical case and we find that the maximum separation between the invariant manifolds, in a given region, is exponentially small with respect to the parameter. The exponent is related to the complex singularities of a flow which is taken as an unperturbed problem. Finally several examples are given.

Invariant manifolds for a class of parabolic points

The authors consider a class of maps having the origin as a parabolic fixed point with a nondiagonalizable linear part, degenerate in the sense that it has a line of fixed points through it, and they give conditions for the existence and regularity of invariant manifolds. This class is motivated from Poincare maps of flows appearing in celestial mechanics.

General scaling law in the saddle–node bifurcation: a complex phase space study

Saddle–node bifurcations have been described in a multitude of nonlinear dynamical systems modeling physical, chemical, as well as biological systems. Typically, this type of bifurcation involves the transition of a given set of fixed points from the real to the complex phase space. After the bifurcation, a saddle remnant can continue influencing the flows and generically, for non-degenerate saddle–node bifurcations, the time the flows spend in the bottleneck region of the ghost follows the inverse square root scaling law. Here we analytically derive this scaling law for a general one-dimensional, analytical, autonomous dynamical system undergoing a not necessarily non-degenerate saddle–node bifurcation, in terms of the degree of degeneracy by using complex variable techniques. We then compare the analytic calculations with a one-dimensional equation modeling the dynamics of an autocatalytic replicator. The numerical results are in agreement with the analytical solution.

Exponentially small splitting of invariant manifolds of parabolic points

Notation and main results Analytic properties of the homoclinic orbit of the unperturbed system Parameterization of local invariant manifolds Flow box coordinates The extension theorem Splitting of separatrices References.

Hamiltonian systems with orbits covering densely submanifolds of small codimension

The existence of a transition chain in a Hamiltonian system leads to the existence of orbits shadowing it, if some lambda lemma can be applied. This fact has been used to prove the existence of diffusion in perturbations of integrable a priori stable systems. We prove that, under suitable conditions, there are orbits which cover densely codimension two submanifolds of the (2n - 1)-dimensional energy level. We also construct examples, near general integrable systems, where the computations to verify the sufficient conditions for the appearance of such phenomenon can be performed.

Bifurcations analysis of oscillating hypercycles

We investigate the dynamics and transitions to extinction of hypercycles governed by periodic orbits. For a large enough number of hypercycle species (n>4) the existence of a stable periodic orbit has been previously described, showing an apparent coincidence of the vanishing of the periodic orbit with the value of the replication quality factor Q where two unstable (non-zero) equilibrium points collide (named QSS). It has also been reported that, for values below QSS, the system goes to extinction. In this paper, we use a suitable Poincaré map associated to the hypercycle system to analyze the dynamics in the bistability regime, where both oscillatory dynamics and extinction are possible. The stable periodic orbit is identified, together with an unstable periodic orbit. In particular, we are able to unveil the vanishing mechanism of the oscillatory dynamics: a saddle-node bifurcation of periodic orbits as the replication quality factor, Q, undergoes a critical fidelity threshold, QPO. The identified bifurcation involves the asymptotic extinction of all hypercycle members, since the attractor placed at the origin becomes globally stable for values Q<QPO. Near the bifurcation, these extinction dynamics display a periodic remnant that provides the system with an oscillating delayed transition. Surprisingly, we found that the value of QPO is slightly higher than QSS, thus identifying a gap in the parameter space where the oscillatory dynamics has vanished while the unstable equilibrium points are still present. We also identified a degenerate bifurcation of the unstable periodic orbits for Q=1.

Homoclinic orbits to parabolic points

Abstract. We give a proof of the Poincaré-Melnikov method in the case of non-Hamiltonian perturbations of one and a half degrees of freedom Hamiltonians, having orbits homoclinic to degenerate periodic orbits of parabolic type.

On analytical properties of normal forms

An analytic Hamiltonian with one degree of freedom can be transformed to Birkhoff normal form near a nondegenerate critical point. In this paper we describe a method for studying analytical properties of the normal form, by means of a differential equation which is satisfied by it. As an application we determine a domain of analyticity for the pendulum Hamiltonian.

ON THE METAPOPULATION DYNAMICS OF AUTOCATALYSIS: EXTINCTION TRANSIENTS RELATED TO GHOSTS

One of the theoretical approaches to study spatially-extended ecosystems is given by metapopulation models, which consider fragmented populations inhabiting discrete patches linked by migration. Most of the metapopulation models assume exponential growth of the local populations and few works have explored the role of cooperation in fragmented ecosystems. In this letter, we study the dynamics and the bifurcation scenarios of a minimal, two-patch metapopulation Turing-like model given by nonlinear differential equations with an autocatalytic reaction term together with diffusion. We also analyze the extinction transients of the metapopulations focusing on the effect of coupling two local populations undergoing delayed transition phenomena due to ghost saddle remnants. We find that increasing diffusion rates enhance the delaying capacity of the ghosts. We finally propose the saddle remnant as a new class of transient generator mechanism for ecological systems.