Lagrangian Descriptors for Stochastic Differential Equations: A Tool for Revealing the Phase Portrait of Stochastic Dynamical Systems

In this paper we introduce a new technique for depicting the phase portrait of stochastic differential equations. Following previous work for deterministic systems, we represent the phase space by means of a generalization of the method of Lagrangian descriptors to stochastic differential equations. Analogously to the deterministic differential equations setting, the Lagrangian descriptors graphically provide the distinguished trajectories and hyperbolic structures arising within the stochastic dynamics, such as random fixed points and their stable and unstable manifolds. We analyze the sense in which structures form barriers to transport in stochastic systems. We apply the method to several benchmark examples where the deterministic phase space structures are well-understood. In particular, we apply our method to the noisy saddle, the stochastically forced Duffing equation, and the stochastic double gyre model that is a benchmark for analyzing fluid transport

A Theoretical Framework for Lagrangian Descriptors

This paper provides a theoretical background for Lagrangian Descriptors (LDs). The goal of achieving rigourous proofs that justify the ability of LDs to detect invariant manifolds is simplified by introducing an alternative definition for LDs. The definition is stated for $n$-dimensional systems with general time dependence, however we rigorously prove that this method reveals the stable and unstable manifolds of hyperbolic points in four particular 2D cases: a hyperbolic saddle point for linear autonomous systems, a hyperbolic saddle point for nonlinear autonomous systems, a hyperbolic saddle point for linear nonautonomous systems and a hyperbolic saddle point for nonlinear nonautonomous systems. We also discuss further rigorous results which show the ability of LDs to highlight additional invariants sets, such as $n$-tori. These results are just a simple extension of the ergodic partition theory which we illustrate by applying this methodology to well-known examples, such as the planar field of the harmonic oscillator and the 3D ABC flow. Finally, we provide a thorough discussion on the requirement of the objectivity (frame-invariance) property for tools designed to reveal phase space structures and their implications for Lagrangian descriptors

Chaotic Dynamics in Nonautonomous Maps:Application to the Nonautonomous Hénon Map

In this paper, we analyze chaotic dynamics for two-dimensional nonautonomous maps through the use of a nonautonomous version of the Conley–Moser conditions given previously. With this approach we are able to give a precise definition of what is meant by a chaotic invariant set for nonautonomous maps. We extend the nonautonomous Conley–Moser conditions by deriving a new sufficient condition for the nonautonomous chaotic invariant set to be hyperbolic. We consider the specific example of a nonautonomous Hénon map and give sufficient conditions, in terms of the parameters defining the map, for the nonautonomous Hénon map to have a hyperbolic chaotic invariant set. </jats:p

Data on the generation of two Nr2e3 mouse models by CRISPR / Cas9D10A nickase

NR2E3 encodes an orphan nuclear receptor that plays a dual function as both transcriptional activator and repressor in photoreceptors, being necessary for cone fate inhibition as well as rod differentiation and homeostasis. Mutations in this gene cause retinitis pigmentosa (RP), enhanced S cone syndrome (ESCS) and Goldmann-Favre syndrome (GFS). There is one reported Nr2e3 isoform that contains all 8 exons and a second -previously unreported- shorter isoform, which only spans the first 7 exons and whose function is still unknown. In this data article, we designed and generated two new mouse models by targeting exon 8 of Nr2e3 using the CRISPR/Cas9-D10A nickase in order to dissect the role of the two isoforms in Nr2e3 function and elucidate the different disease mechanisms caused by NR2E3 mutations. This strategy generated several modified alleles that altered the coding sequence of the last exon thereby affecting functional domains of the transcription factor. Allele Δ27 is an in-frame deletion of 27 bp that ablates the dimerization domain, whereas allele ΔE8 (full deletion of exon 8), produces only the short isoform that lacks the dimerization and repressor domains. Morphological and functional alterations of both Δ27 and ΔE8 mutants are reported in the associated research article "Nr2e3 functional domain ablation by CRISPR-Cas9D10A identifies a new isoform and generated Retinitis Pigmentosa and Enhanced S-cone Syndrome models" (Aísa-Marín et al., 2020)

El proyecto de Oskar Hansen para Auschwitz y la monumentalización del debate sobre la guerra

La propuesta de Oskar Hansen para el Monumento a las Víctimas del Fascismo en Auschwitz renunciaba a tener una forma inmutable, tanto por la interacción que reclamaba del visitante como por el efecto del paso del tiempo. De esta manera manifestaba su incapacidad para representar el horror provocado por la guerra, y menos aún para explicarlo. Adoptando la forma de un pedestal vacío, lo que verdaderamente pretendía monumentalizar era el debate que debía suscitarse en su superficie, de posiciones cambiantes e incluso contradictorias, pero necesario para evitar el olvido.Oskar Hansen’s proposal for the Memorial to the Victims of Fascism in Auschwitz renounced to have an immutable shape, both because of the interaction it claimed from the visitor and the effect of the passage of time. Devoid of a focus for commemoration and surrounded by remnants of the camp that were meant to acquire a romantic outlook in the future, it proclaimed its inability to depict the horror provoked by the war, let alone to explain it. Its transgressor character aroused the suspicion of the victims, who did not feel themselves represented on Hansen’s empty pedestal. Its novelty was based on shifting the burden of memory from the object ( the traditional monument) to the subject ( the viewer), as Postmodernism would do later, and stimulating critical thinking about the past. But above all, and because of its reluctance to ‘ talk’, it aspired to remain a valid space for remembrance, regardless of the transformations that postwar society’s relationship with the architectural heritage of Nazism went through. This relationship has been swinging like a pendulum over the years, first ignoring or underplaying the symbolic power of these buildings. Later on, when German society felt ready to ‘ come to terms with the past’, almost every trace of Nazism was deemed worthy of preservation, and the birth of the countermonument ( many of which literally replicated the mechanisms that Hansen put in place in Auschwitz) helped to shape a critical review of this period. And finally, it seems that the excess of memory during the last decades of the twentieth century is giving way to the normalization of this legacy as just one more element in the urban landscape. Given the changing nature of these attitudes, rather than a specific response, what Hansen really intended to monumentalize was the debate that should be fostered on the surface of his monument, as an antidote to oblivion