Review article: Dynamical systems, algebraic topology and the climate sciences

The definition of climate itself cannot be given without a proper understanding of the key ideas of long-term behavior of a system, as provided by dynamical systems theory. Hence, it is not surprising that concepts and methods of this theory have percolated into the climate sciences as early as the 1960s. The major increase in public awareness of the socio-economic threats and opportunities of climate change has led more recently to two major developments in the climate sciences: (i) the Intergovernmental Panel on Climate Change's successive Assessment Reports and (ii) an increasing understanding of the interplay between natural climate variability and anthropogenically driven climate change. Both of these developments have benefited from remarkable technological advances in computing resources, relating throughput as well as storage, and in observational capabilities, regarding both platforms and instruments. Starting with the early contributions of nonlinear dynamics to the climate sciences, we review here the more recent contributions of (a) the theory of non-autonomous and random dynamical systems to an understanding of the interplay between natural variability and anthropogenic climate change and (b) the role of algebraic topology in shedding additional light on this interplay. The review is thus a trip leading from the applications of classical bifurcation theory to multiple possible climates to the tipping points associated with transitions from one type of climatic behavior to another in the presence of time-dependent forcing, deterministic as well as stochastic.</p

Vortex dipolar structures in a rigid model of the larynx at flow onset

Starting jet airflow is investigated in a channel with a pair of consecutive slitted constrictions approximating the true and false vocal folds in the human larynx. The flow is visualized using the Schlieren optical technique and simulated by solving the Navier-Stokes equations for an incompressible two-dimensional viscous flow. Laboratory and numerical experiments show the spontaneous formation of three different classes of vortex dipolar structures in several regions of the laryngeal profile under conditions that may be assimilated to those of voice onset.Fil: Chisari, Nora Elisa. University of Princeton; Estados Unidos. Universidad de Buenos Aires. Facultad de Ingeniería; ArgentinaFil: Artana, Guillermo Osvaldo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería; ArgentinaFil: Sciamarella, D.. Centre National de la Recherche Scientifique; Franci

Noise-driven Topological Changes in Chaotic Dynamics

Noise modifies the behavior of chaotic systems in both quantitative and qualitative ways. To study these modifications, the present work compares the topological structure of the deterministic Lorenz (1963) attractor with its stochastically perturbed version. The deterministic attractor is well known to be "strange" but it is frozen in time. When driven by multiplicative noise, the Lorenz model's random attractor (LORA) evolves in time. Algebraic topology sheds light on the most striking effects involved in such an evolution. In order to examine the topological structure of the snapshots that approximate LORA, we use Branched Manifold Analysis through Homologies (BraMAH) -- a technique originally introduced to characterize the topological structure of deterministically chaotic flows -- which is being extended herein to nonlinear noise-driven systems. The analysis is performed for a fixed realization of the driving noise at different time instants in time. The results suggest that LORA's evolution includes sharp transitions that appear as topological tipping points.Comment: 12 pages and 4 figure

Modeling voice production with time-delay systems: the larynx tube

International audienceTime-delay systems (shortly, TDS) are also called systems with aftereffect or dead-time, hereditary systems, equations with deviating argument or differential-difference equations. In voice production, TDS play a role in voice-production modeling when source and tract are coupled allowing for delayed feedback on the vocal fold dynamics [2,3]. This work undertakes the incorporation of the larynx tube to this modeling scheme, following an approach inspired in the assimilation of the larynx tube to a Helmholtz resonator, introduced four decades ago to study the singing formant

Mechanism of and Threshold Biomechanical Conditions for Falsetto Voice Onset

The sound source of a voice is produced by the self-excited oscillation of the vocal folds. In modal voice production, a drastic increase in transglottal pressure after vocal fold closure works as a driving force that develops self-excitation. Another type of vocal fold oscillation with less pronounced glottal closure observed in falsetto voice production has been accounted for by the mucosal wave theory. The classical theory assumes a quasi-steady flow, and the expected driving force onto the vocal folds under wavelike motion is derived from the Bernoulli effect. However, wavelike motion is not always observed during falsetto voice production. More importantly, the application of the quasi-steady assumption to a falsetto voice with a fundamental frequency of several hundred hertz is unsupported by experiments. These considerations suggested that the mechanism of falsetto voice onset may be essentially different from that explained by the mucosal wave theory. In this paper, an alternative mechanism is submitted that explains how self-excitation reminiscent of the falsetto voice could be produced independent of the glottal closure and wavelike motion. This new explanation is derived through analytical procedures by employing only general unsteady equations of motion for flow and solids. The analysis demonstrated that a convective acceleration of a flow induced by rapid wall movement functions as a negative damping force, leading to the self-excitation of the vocal folds. The critical subglottal pressure and volume flow are expressed as functions of vocal fold biomechanical properties, geometry, and voice fundamental frequency. The analytically derived conditions are qualitatively and quantitatively reasonable in view of reported measurement data of the thresholds required for falsetto voice onset. Understanding of the voice onset mechanism and the explicit mathematical descriptions of thresholds would be beneficial for the diagnosis and treatment of voice diseases and the development of artificial vocal folds

An unfinished tale of nonlinear PDEs: Do solutions of 3D incompressible Euler equations blow-up in finite time

Abstract The solutions of time-dependent PDEs may show a bewildering variety of behaviors. The example of the KuramotoSivashinsky (KS) equation illustrates how simple an equation can be with extremely complex solutions. The equations for incompressible fluids are also notorious for the extremely complex behavior of their solutions. But in the latter case, the situation is in a sense far less satisfactory than for the KS equations, since one still ignores if their solution remains smooth as time goes on, with smooth initial data in 3D. This leaves in a quite uncertain state all theories of 3D turbulence, which may be either a way of coping with our ignorance of the true dynamics of real fluids, or a way of analyzing the behavior of the smooth solutions of the Navier-Stokes equations. Below, in this homage to Professor Kuramoto, we present some remarks on this question, limited to the 3D Euler equations without viscosity. The existence of solutions of the 3D incompressible Euler equations for fluids blowing-up in finite time remains an open question if the initial energy is finite. The consensus seems to be that the singularity, if there is any, is local in space and time, although it is difficult to find a straight statement supporting such a claim in the literature. Based on the properties of self-similar equations, we claim that this is not possible at least if one assumes such a self-similar blow-up, including some refinements in the time dependence that extend the class of self-similar solutions. Actually, the possible blow-up solutions are very much constrained by the conservation of energy and of circulation. Based on scaling transformations, we argue that the singularity cannot be a simple self-similar blow-up, but requires some sort of oscillations on a logarithmic time-scale. The space dependence cannot be simple either. The various constraints lead to a self-similar collapse towards a line (at least for an axisymmetric flow with swirl), with a wavelength along the axis decreasing by steps. Our final result is a set of explicit equations such that, if they have a smooth solution satisfying certain conditions, the original Euler equations have a finite-time singularity

The templex approach in Lagrangian Analysis

International audienceLagrangian analysis is central to understanding how fluid particles are organized in their time evolution. Of particular interest in geophysics are rotating fluids in the presence of a radial temperature gradient, similar to those that control the dynamics of the atmosphere and oceans on a planetary scale. The phenomenon can be studied with the help of simplified kinematic models, such as Shadden’s driven double gyre, which simulates oceanic patterns, or the Bickley Jet, which reproduces the dynamics of zonal jets in the atmosphere. The use of Branched Manifold Analysis through Homologies (BraMAH) has proven useful for detecting Lagrangian Coherent Sets (LCSs) from single particle time series [Charó et al., 2019, 2020, 2021]. Here we examine how the templex approach, introduced by Charó, Letellier and Sciamarella in 2022, improves the description of the dynamics of the different particle sets

From the Rössler attractor to the templex

Methods to accomplish a topological description of the structure of a flow in high-dimensional state space (in more than three dimensions) have a long history, full of partially fruitful attempts, but the ideal mathematical object to achieve this task seems to be what we now call templex. Cell complexes can be traced back to Poincaré’s papers during 1900 and the study of chaotic attractors using cell complexes to the 1990s. Since then, algebraic topology has been regarded as the most promising mathematical formalism to describe chaos beyond three dimensions, overcoming the restrictions that templates, based on the knot content of attractors, cannot represent. In this talk, we present the road leading to the templex starting from the Rössler attractor and ending with a four-dimensional system designed on the basis of a set of equations proposed by Deng

Observability of laminar bidimensional fluid flows seen as autonomous chaotic systems

Lagrangian transport in the dynamical systems approach has so far been investigated disregarding the connection between the whole state space and the concept of observability. Key issues such as the definitions of Lagrangian and chaotic mixing are revisited under this light, establishing the importance of rewriting nonautonomous flow systems derived from a stream function in autonomous form, and of not restricting the characterization of their dynamics in subspaces. The observability of Lagrangian chaos from a reduced set of measurements is illustrated with two canonical examples: the Lorenz system derived as a low-dimensional truncation of the Rayleigh-Benard convection equations and the driven double-gyre system introduced as a kinematic model of configurations observed in the ocean. A symmetrized version of the driven double-gyre model is proposed. Published under license by AIP Publishing