Examinando por Autor "Daza, Alvar"

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A test for fractal boundaries based on the basin entropy
(Elsevier, 2020-10-31) Puy, Andreu; Daza, Alvar; Wagemakers, Alexandre; Sanjuán, Miguel AF
In dynamical systems, basins of attraction connect a given set of initial conditions in phase space to their asymptotic states. The basin entropy and related tools quantify the unpredictability in the final state of a system when there is an initial perturbation or uncertainty in the initial state. Based on the basin entropy, the log2 criterion allows for efficient testing of fractal basin boundaries at a fixed resolution. Here, we extend this criterion into a new test with improved sensitivity that we call the fractality test. Using the same single scale information, the fractality test allows for the detection of fractal boundaries in many more cases than the log2 criterion. The new test is illustrated with the paradigmatic driven Duffing oscillator, and the results are compared with the classical approach given by the uncertainty exponent. We believe that this work can prove particularly useful to study both high-dimensional systems and experimental basins of attraction.
Ascertaining when a basin is Wada: the merging method
(Nature, 2018-07-02) Daza, Alvar; Wagemakers, Alexandre; Sanjuán, Miguel A. F.
Trying to imagine three regions separated by a unique boundary seems a difficult task. However, this is exactly what happens in many dynamical systems showing Wada basins. Here, we present a new perspective on the Wada property: A Wada boundary is the only one that remains unaltered under the action of merging the basins. This observation allows to develop a new method to test the Wada property, which is much faster than the previous ones. Furthermore, another major advantage of the merging method is that a detailed knowledge of the dynamical system is not required.
Basin entropy: a new tool to analyze uncertainty in dynamical systems
(Nature, 2016-08-12) Daza, Alvar; Wagemakers, Alexandre; Georgeot, Bertrand; Guéry-Odelin, David; Sanjuán, Miguel A. F.
In nonlinear dynamics, basins of attraction link a given set of initial conditions to its corresponding final states. This notion appears in a broad range of applications where several outcomes are possible, which is a common situation in neuroscience, economy, astronomy, ecology and many other disciplines. Depending on the nature of the basins, prediction can be difficult even in systems that evolve under deterministic rules. From this respect, a proper classification of this unpredictability is clearly required. To address this issue, we introduce the basin entropy, a measure to quantify this uncertainty. Its application is illustrated with several paradigmatic examples that allow us to identify the ingredients that hinder the prediction of the final state. The basin entropy provides an efficient method to probe the behavior of a system when different parameters are varied. Additionally, we provide a sufficient condition for the existence of fractal basin boundaries: when the basin entropy of the boundaries is larger than log2, the basin is fractal.
Chaotic dynamics and fractal structures in experiments with cold atoms
(APS, 2017-01-25) Daza, Alvar; Georgeot, Bertrand; Guéry-Odelin, David; Wagemakers, Alexandre; Sanjuán, Miguel A. F.
We exploit tools from nonlinear dynamics to the detailed analysis of cold atom experiments. A powerful example is provided by the recent concept of basin entropy which allows to quantify the final state unpredictability that results from the complexity of the phase space geometry. We show here that this enables one to reliably infer the presence of fractal structures in phase space from direct measurements. We exemplify the method with numerical simulations in an experimental configuration made of two crossing laser guides and originally used as a matter wave splitter.
Characterization of Fractal Basins Using Deep Convolutional Neural Networks
(International Journal of Bifurcation and Chaos, 2022-07-12) Valle, David; Wagemakers, Alexandre; Daza, Alvar; Sanjuán, Miguel A.F.
Neural network models have recently demonstrated impressive prediction performance in complex systems where chaos and unpredictability appear. In spite of the research efforts carried out on predicting future trajectories or improving their accuracy compared to numerical methods, not sufficient work has been done on using deep learning techniques in which the unpredictability can be characterized of chaotic systems or give a general view of the global unpredictability of a system. In this work, we propose a novel approach based on deep learning techniques to measure the fractal dimension of the basins of attraction of the Duffing oscillator for a variety of parameters. As a consequence, we provide an algorithm capable of predicting fractal dimension measures as accurately as the box-counting algorithm, but with a computation speed about ten times faster.
Classifying basins of attraction using the basin entropy
(Elsevier, 2022) Daza, Alvar; Wagemakers, Alexandre; Fernandez Sanjuán, Miguel A.
A basin of attraction represents the set of initial conditions leading to a specific asymptotic state of a given dynamical system. Here, we provide a classification of the most common basins found in nonlinear dynamics with the help of the basin entropy. We have also found interesting connections between the basin entropy and other measures used to characterize the unpredictability associated to the basins of attraction, such as the uncertainty exponent, the lacunarity or other different parameters related to the Wada property.
Propagation of waves in high Brillouin zones: Chaotic branched flow and stable superwires
(2021-09-27) Daza, Alvar; Heller, Eric J.; Graf, Anton M.; Räsänen, Esa
We report unexpected classical and quantum dynamics of a wave propagating in a periodic potential in high Brillouin zones. Branched flow appears at wavelengths shorter than the typical length scale of the ordered periodic structure and for energies above the potential barrier. The strongest branches remain stable indefinitely and may create linear dynamical channels, wherein waves are not confined directly by potential walls as electrons in ordinary wires but rather, indirectly and more subtly by dynamical stability. We term these superwires since they are associated with a superlattice.
Strong sensitivity of the vibrational resonance induced by fractal structures
(2013-07-01) Daza, Alvar; Wagemakers, Alexandre; Sanjuán, Miguel A. F.
We consider a nonlinear system perturbed by two harmonic forcings of different frequencies. The slow forcing drives the system into an oscillatory regime while the fast perturbation enhances the effect of the slow periodic drive. The vibrational resonance occurs when this enhancement is optimal, usually when the fast perturbation has an amplitude much higher than the slow periodic forcing. We show that this resonance can also happen when the amplitude of the fast perturbation is far below the amplitude of the slow periodic forcing due to a peculiar condition of the phase space. Moreover, this resonance presents an extreme sensitivity to small variations of the fast perturbation. We explore here this phenomenon that we call ultrasensitive vibrational resonance.
Testing for Basins of Wada
(Nature, 2015-11-10) Daza, Alvar; Wagemakers, Alexandre; Sanjuán, Miguel A. F.; Yorke, James A.
Nonlinear systems often give rise to fractal boundaries in phase space, hindering predictability. When a single boundary separates three or more different basins of attraction, we say that the set of basins has the Wada property and initial conditions near that boundary are even more unpredictable. Many physical systems of interest with this topological property appear in the literature. However, so far the only approach to study Wada basins has been restricted to two-dimensional phase spaces. Here we report a simple algorithm whose purpose is to look for the Wada property in a given dynamical system. Another benefit of this procedure is the possibility to classify and study intermediate situations known as partially Wada boundaries.
The saddle-straddle method to test for Wada basins
(Elsevier, 2020-01-03) Wagemakers, Alexandre; Daza, Alvar; Sanjuán, Miguel AF
First conceived as a topological construction, Wada basins abound in dynamical systems. Basins of attraction showing the Wada property possess the particular feature that any small perturbation of an initial condition lying on the boundary can lead the system to any of its possible outcomes. The saddle-straddle method, described here, is a new method to identify the Wada property in a dynamical system based on the computation of its chaotic saddle in the fractalized phase space. It consists of finding the chaotic saddle embedded in the boundary between the basin of one attractor and the remaining basins of attraction by using the saddle-straddle algorithm. The simple observation that the chaotic saddle is the same for all the combinations of basins is sufficient to prove that the boundary has the Wada property.
Unpredictability and basin entropy
(IOP, 2023-02-01) Daza, Alvar; Wagemakers, Alexandre; Sanjuán, Miguel A. F.
The basin entropy is a simple idea that aims to measure the the final state unpredictability of multistable systems. Since 2016, the basin entropy has been widely used in different contexts of physics, from cold atoms to galactic dynamics. Furthermore, it has provided a natural framework to study basins of attraction in nonlinear dynamics and new criteria for the detection of fractal boundaries. In this article, we describe the concept as well as fundamental applications. In addition, we provide our perspective on the future challenges of applying the basin entropy idea to understanding complex systems.
Using the basin entropy to explore bifurcations
(Elsevier, 2023) Wagemakers, Alexandre; Daza, Alvar; F. Sanjuán, Miguel A.
Bifurcation theory is the usual analytic approach to study the parameter space of a dynamical system. Despite the great power of prediction of these techniques, fundamental limitations appear during the study of a given problem. Nonlinear dynamical systems often hide their secrets and the ultimate resource is the numerical simulation of the equations. This paper presents a method to explore bifurcations by using the basin entropy. This measure of the unpredictability can detect transformations of phase space structures as a parameter evolves. We present several examples where the bifurcations in the parameter space have a quantitative effect on the basin entropy. Moreover, some transformations, such as the basin boundary metamorphoses, can be identified with the basin entropy but are not reflected in the bifurcation diagram. The correct interpretation of the basin entropy plotted as a parameter extends the numerical exploration of dynamical systems.
Vibrational resonance in a time-delayed genetic toggle switch
(Elsevier, 2013-02-01) Daza, Alvar; Wagemakers, Alexandre; Rajasekar, Shanmuganathan; Sanjuán, Miguel A. F.
Biological oscillators can respond in a surprising way when they are perturbed by two external periodic forcing signals of very different frequencies. The response of the system to a low-frequency signal can be enhanced or depressed when a high-frequency signal is acting. This is what is known as vibrational resonance (VR). Here we study this phenomenon in a simple time-delayed genetic toggle switch, which is a synthetic gene-regulatory network. We have found out how the low-frequency signal changes the range of the response, while the high-frequency signal influences the amplitude at which the resonance occurs. The delay of the toggle switch has also a strong effect on the resonance since it can also induce autonomous oscillations.
Wada property in systems with delay
(Elsevier, 2017-02-01) Daza, Alvar; Wagemakers, Alexandre; Sanjuán, Miguel A. F.
Delay differential equations take into account the transmission time of the information. These delayed signals may turn a predictable system into chaotic, with the usual fractalization of the phase space. In this work, we study the connection between delay and unpredictability, in particular we focus on the Wada property in systems with delay. This topological property gives rise todramatic changes in the final state for small changes in the history functions.
Wada structures in a binary black hole system
(2018-10-29) Daza, Alvar; O Shipley, Jake; Dolan, Sam R.; Sanjuán, Miguel AF
A key goal of the Event Horizon Telescope is to observe the shadow cast by a black hole. Recent simulations have shown that binary black holes, the progenitors of gravitational waves, present shadows with fractal structure. Here we study the binary shadow structure using techniques from nonlinear dynamics, recognizing shadows as exit basins of open Hamiltonian dynamical systems. We apply a recently developed numerical algorithm to demonstrate that parts of the Majumdar-Papapetrou binary black hole shadow exhibit the Wada property: any point of the boundary of one basin is also on the boundary of at least two additional basins. We show that the algorithm successfully distinguishes between the fractal and regular (i.e., nonfractal) parts of the binary shadow.