A survey on the inverse integrating factor

The relation between limit cycles of planar differential systems and the inverse integrating factor was first shown in an article of Giacomini, Llibre and Viano appeared in 1996. From that moment on, many research articles are devoted to the study of the properties of the inverse integrating factor and its relationwith limit cycles and their bifurcations. This paper is a summary of all the results about this topic. We include a list of references together with the corresponding related results aiming at being as much exhaustive as possible. The paper is, nonetheless, self-contained in such a way that all the main results on the inverse integrating factor are stated and a complete overview of the subject is given. Each section contains a different issue to which the inverse integrating factor plays a role: the integrability problem, relation with Lie symmetries, the center problem, vanishing set of an inverse integrating factor, bifurcation of limit cycles from either a period annulus or from a monodromic ω-limit set and some generalizations.

Spontaneous Emergence of Chirality in the Limited Enantioselectivity Model: Autocatalytic Cycle Driven by an External Reagent

The emergence of chirality in enantioselective autocatalysis for compounds unable to transform according to the Frank-like reaction network is discussed with respect to the controversial limited enantioselectivity (LES) model composed of coupled enantioselective and non-enantioselective autocatalyses. The LES model cannot lead to spontaneous mirror symmetry breaking (SMSB) either in closed systems with a homogeneous temperature distribution or in closed systems with a stationary non-uniform temperature distribution. However, simulations of chemical kinetics in a two-compartment model demonstrate that SMSB may occur if both autocatalytic reactions are spatially separated at different temperatures in different compartments but coupled under the action of a continuous internal flow. In such conditions, the system can evolve, for certain reaction and system parameters, toward a chiral stationary state; that is, the system is able to reach a bifurcation point leading to SMSB. Numerical simulations in which reasonable chemical parameters have been used suggest that an adequate scenario for such a SMSB would be that of abyssal hydrothermal vents, by virtue of the typical temperature gradients found there and the role of inorganic solids mediating chemical reactions in an enzyme-like role.

Fragile X-syndrome: literature review and report of two cases

Fragile X-syndrome is caused by a mutation in chromosome X. It is one of the most frequent causes of learning disability. The most frequent manifestations of fragile X-syndrome are learning disability, different orofacial morphological alterations and an increase in testicle size. The disease is associated with cardiac malformations, joint hyperextension and behavioural alterations. We present two male patients aged 17 and 10 years, treated in our Service due to severe gingivitis. Both showed the typical facial and dental characteristics of the syndrome. In addition, we detected the presence of root anomalies such as taurodontism and root bifurcation, which had not been associated with fragile X-syndrome in the literature. In some cases these root malformations have been associated with other sex-linked congenital syndromes, though in none of the studies published in the literature have they been related with fragile X-syndrome. This syndrome is relevant due to its high prevalence, the presentation of certain oral and facial characteristics that can facilitate the diagnosis, and the few cases published to date

Reliable computation of robust response tori on the verge of breakdown

We prove the existence and local uniqueness of invariant tori on the verge of breakdown for two systems: the quasi-periodically driven logistic map and the quasi-periodically forced standard map. These systems exemplify two scenarios: the Heagy-Hammel route for the creation of strange non- chaotic attractors and the nonsmooth bifurcation of saddle invariant tori. Our proofs are computer- assisted and are based on a tailored version of the Newton-Kantorovich theorem. The proofs cannot be performed using classical perturbation theory because the two scenarios are very far from the perturbative regime, and fundamental hypotheses such as reducibility or hyperbolicity either do not hold or are very close to failing. Our proofs are based on a reliable computation of the invariant tori and a careful study of their dynamical properties, leading to the rigorous validation of the numerical results with our novel computational techniques.

Multifractal dimension of chaotic attractors in a driven semiconductor superlattice

The multifractal dimension of chaotic attractors has been studied in a weakly coupled superlattice driven by an incommensurate sinusoidal voltage as a function of the driving voltage amplitude. The derived multifractal dimension for the observed bifurcation sequence shows different characteristics for chaotic, quasiperiodic, and frequency-locked attractors. In the chaotic regime, strange attractors are observed. Even in the quasiperiodic regime, attractors with a certain degree of strangeness may exist. From the observed multifractal dimensions, the deterministic nature of the chaotic oscillations is clearly identified.