Papers by Miguel A.F. Sanjuan
DergiPark (Istanbul University), Nov 30, 2021
In Physics, we have laws that determine the time evolution of a given physical system, depending ... more In Physics, we have laws that determine the time evolution of a given physical system, depending on its parameters and its initial conditions. When we have multi-stable systems, many attractors coexist so that their basins of attraction might possess fractal or even Wada boundaries in such a way that the prediction becomes more complicated depending on the initial conditions. Chaotic systems typically present fractal basins in phase space. A small uncertainty in the initial conditions gives rise to a certain unpredictability of the final state behavior. The new notion of basin entropy provides a new quantitative way to measure the unpredictability of the final states in basins of attraction. Simple methods from chaos theory can contribute to a better understanding of fundamental questions in physics as well as other scientific disciplines.
Communications in Nonlinear Science and Numerical Simulation, Feb 1, 2022
In this manuscript we show that a noise-activated escape phenomenon occurs in closed Hamiltonian ... more In this manuscript we show that a noise-activated escape phenomenon occurs in closed Hamiltonian systems. Due to the energy fluctuations generated by the noise, the isopotential curves open up and the particles can eventually escape in finite times. This drastic change in the dynamical behavior turns the bounded motion into a chaotic scattering problem. We analyze the escape dynamics by means of the average escape time, the probability basins and the average escape time distribution. We obtain that the main characteristics of the scattering are different from the case of noisy open Hamiltonian systems. In particular, the noiseenhanced trapping, which is ubiquitous in Hamiltonian systems, does not play the main role in the escapes. On the other hand, one of our main findings reveals a transition in the evolution of the average escape time insofar the noise is increased. This transition separates two different regimes characterized by different algebraic scaling laws. We provide strong numerical evidence to show that the complete destruction of the stickiness of the KAM islands is the key reason under the change in the scaling law. This research unlocks the possibility of modeling chaotic scattering problems by means of noisy closed Hamiltonian systems. For this reason, we expect potential application to several fields of physics such us celestial mechanics and astrophysics, among others.
International Journal of Bifurcation and Chaos, Sep 2, 2021
Machine learning and deep learning techniques are contributing much to the advancement of science... more Machine learning and deep learning techniques are contributing much to the advancement of science. Their powerful predictive capabilities appear in numerous disciplines, including chaotic dynamics, but they miss understanding. The main thesis here is that prediction and understanding are two very different and important ideas that should guide us about the progress of science. Furthermore, it is emphasized the important role played by nonlinear dynamical systems for the process of understanding. The path of the future of science will be marked by a constructive dialogue between big data and big theory, without which we cannot understand.
arXiv (Cornell University), Mar 9, 2018
arXiv (Cornell University), Dec 29, 2022
The basin entropy is a simple idea that aims to measure the the final state unpredictability of m... more 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.
PLOS ONE, Apr 18, 2018
In nonlinear systems long term dynamics is governed by the attractors present in phase space. The... more In nonlinear systems long term dynamics is governed by the attractors present in phase space. The presence of a chaotic saddle gives rise to basins of attraction with fractal boundaries and sometimes even to Wada boundaries. These two phenomena involve extreme difficulties in the prediction of the future state of the system. However, we show here that it is possible to make statistical predictions even if we do not have any previous knowledge of the initial conditions or the time series of the system until it reaches its final state. In this work, we develop a general method to make statistical predictions in systems with fractal basins. In particular, we have applied this new method to the Duffing oscillator for a choice of parameters where the system possesses the Wada property. We have computed the statistical properties of the Duffing oscillator for different phase space resolutions, to obtain information about the global dynamics of the system. The key idea is that the fraction of initial conditions that evolve towards each attractor is scale free-which we illustrate numerically. We have also shown numerically how having partial information about the initial conditions of the system does not improve in general the predictions in the Wada regions.
Physical Review E, Jan 20, 2004
We consider the tent map as the prototype of a chaotic system with escapes. We show analytically ... more We consider the tent map as the prototype of a chaotic system with escapes. We show analytically that a small, bounded, but carefully chosen perturbation added to the system can trap forever an orbit close to the chaotic saddle, even in presence of noise of larger, although bounded, amplitude. This problem is focused as a two-person, mathematical game between two players called ''the protagonist'' and ''the adversary.'' The protagonist's goal is to survive. He can lose but cannot win; the best he can do is survive to play another round, struggling ad infinitum. In the absence of actions by either player, the dynamics diverge, leaving a relatively safe region, and we say the protagonist loses. What makes survival difficult is that the adversary is allowed stronger ''actions'' than the protagonist. What makes survival possible is ͑i͒ the background dynamics ͑the tent map here͒ are chaotic and ͑ii͒ the protagonist knows the action of the adversary in choosing his response and is permitted to choose the initial point x 0 of the game. We use the ''slope 3'' tent map in an example of this problem. We show that it is possible for the protagonist to survive.
Communications in Nonlinear Science and Numerical Simulation, Feb 1, 2017
Delay differential equations take into account the transmission time of the information. These de... more 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 to dramatical changes in the final state for small changes in the history functions.
Proceedings of Entropy 2021: The Scientific Tool of the 21st Century, May 5, 2021
In nonlinear dynamics, basins of attraction are defined as the set of points that, taken as initi... more In nonlinear dynamics, basins of attraction are defined as the set of points that, taken as initial conditions, lead the system to a specific attractor. This notion appears in a broad range of applications where multistability is present, which is a common situation in neuroscience, economy, astronomy, ecology, and other disciplines. 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 call them Wada basins. Usually, Wada basins have been considered even more unpredictable than fractal basins. However, this particular unpredictability has not been fully unveiled until the introduction of the concept of basin entropy. The basin entropy provides a quantitative measure of how unpredictable a basin is. With the help of several paradigmatic dynamical systems, we illustrate how to identify the ingredients that hinder the prediction of the final state. The basin entropy together with two new tests of the Wada property have been applied to some physical systems such as experiments of chaotic scattering of cold atoms, models of shadows of binary black holes, and classical and relativistic chaotic scattering associated to the Hénon-Heiles Hamiltonian system in astrophysics.
arXiv (Cornell University), May 12, 2020
We present a review of the different techniques available to study a special kind of fractal basi... more We present a review of the different techniques available to study a special kind of fractal basins of attraction known as Wada basins, which have the intriguing property of having a single boundary separating three or more basins. We expose several approaches to identify this topological property that rely on different, but not exclusive, definitions of the Wada property.
Europhysics Letters
The basin entropy is a simple idea that aims to measure the the final state unpredictability of m... more 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.
Nonlinear Dynamics
In the context of nonhyperbolic chaotic scattering, it has been shown that the evolution of the K... more In the context of nonhyperbolic chaotic scattering, it has been shown that the evolution of the KAM islands exhibits four abrupt metamorphoses that strongly affect the predictability of Hamiltonian systems. It has been suggested that these metamorphoses are related to significant changes in the structure of the KAM islands. However, previous research has not provided an explanation of the mechanisms underlying the metamorphoses. Here, we show that they occur due to the formation of a homoclinic or heteroclinic tangle that breaks the internal structure of the main KAM island. We obtain similar qualitative results in a two-dimensional Hamiltonian system and a two-dimensional area-preserving map. The equivalence of the results obtained in both systems suggests that the same four metamorphoses play an important role in conservative systems.
arXiv (Cornell University), Feb 13, 2018
For low-dimensional chaotic attractors there is usually a single number of unstable dimensions fo... more For low-dimensional chaotic attractors there is usually a single number of unstable dimensions for all of its periodic orbits and we can say such attractors exhibit "mono-chaos". In high-dimensional chaotic attractors, trajectories are prone to travel through quite different regions of phase space, some far more unstable than others. This heterogeneity makes predictability even more difficult than in low-dimensional homogeneous chaotic attractors. A chaotic attractor is "multi-chaotic" if every point of the attractor is arbitrarily close to periodic points with different numbers of unstable dimensions. We believe that most physical systems possessing a high-dimensional attractor are of this type. We make three conjectures about multi-chaos which we explore using three twodimensional paradigmatic examples of multi-chaotic attractors. They can be thought of as smallscale examples that give insight for real high-dimensional phenomena. We find a single route from mono-chaos to multi-chaos if an attractor changes continuously as a parameter is varied. This multichaos bifurcation (MCB) is a periodic orbit bifurcation; one branch of periodic orbits is created with a number of unstable dimensions that is different from the mono-chaos.
arXiv: Chaotic Dynamics, 2020
In Physics, we have laws that determine the time evolution of a given physical system, depending ... more In Physics, we have laws that determine the time evolution of a given physical system, depending on its parameters and its initial conditions. When we have multi-stable systems, many attractors coexist so that their basins of attraction might possess fractal or even Wada boundaries in such a way that the prediction becomes more complicated depending on the initial conditions. Chaotic systems typically present fractal basins in phase space. A small uncertainty in the initial conditions gives rise to a certain unpredictability of the final state behavior. The new notion of basin entropy provides a new quantitative way to measure the unpredictability of the final states in basins of attraction. Simple methods from chaos theory can contribute to a better understanding of fundamental questions in physics as well as other scientific disciplines.
Communications in Nonlinear Science and Numerical Simulation, 2021
The noise-enhanced trapping is a surprising phenomenon that has already been studied in chaotic s... more The noise-enhanced trapping is a surprising phenomenon that has already been studied in chaotic scattering problems where the noise affects the physical variables but not the parameters of the system. Following this research, in this work we provide strong numerical evidence to show that an additional mechanism that enhances the trapping arises when the noise influences the energy of the system. For this purpose, we have included a source of Gaussian white noise in the Hénon-Heiles system, which is a paradigmatic example of open Hamiltonian system. For a particular value of the noise intensity, some trajectories decrease their energy due to the stochastic fluctuations. This drop in energy allows the particles to spend very long transients in the scattering region, increasing their average escape times. This result, together with the previously studied mechanisms, points out the generality of the noise-enhanced trapping in chaotic scattering problems.
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2018
The dynamics on a chaotic attractor can be quite heterogeneous, being much more unstable in some ... more The dynamics on a chaotic attractor can be quite heterogeneous, being much more unstable in some regions than others. Some regions of a chaotic attractor can be expanding in more dimensions than other regions. Imagine a situation where two such regions and each contains trajectories that stay in the region for all time—while typical trajectories wander throughout the attractor. Furthermore, if arbitrarily close to each point of the attractor there are points on periodic orbits that have different unstable dimensions, then we say such an attractor is “hetero-chaotic” (i.e., it has heterogeneous chaos). This is hard to picture but we believe that most physical systems possessing a high-dimensional attractor are of this type. We have created simplified models with that behavior to give insight into real high-dimensional phenomena.
Discrete & Continuous Dynamical Systems - B, 2018
Monthly Notices of the Royal Astronomical Society, 2015
The predictability of a system indicates how much time a computed orbit is close to an actual orb... more The predictability of a system indicates how much time a computed orbit is close to an actual orbit of the system, independent of its stability or chaotic nature. We derive a predictability index from the distributions of finite-time Lyapunov exponents of several prototypical orbits, both regular and irregular, in a variety of galactic potentials. In addition, by analysing the evolution of the shapes of the distributions with the finite-time intervals sizes, we get an insight into the time-scales of the model when the flow dynamics evolve from the local to the global regime.
Physical Review A, 2017
We use tools from nonlinear dynamics to the detailed analysis of cold atom experiments. A powerfu... more We use 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 illustrate the method with numerical simulations in an experimental configuration made of two crossing laser guides that can be used as a matter wave splitter.
Nonlinear Dynamics, 2016
Discrete dynamical systems where one or several of their parameters vary randomly every iteration... more Discrete dynamical systems where one or several of their parameters vary randomly every iteration are usually referred to as random maps in the literature. However, very few methodologies have been proposed to control these kinds of systems when chaos is present. Here, we propose an extension of the partial control method, that we call parametric partial control, that can be naturally applied to random maps. We show that using this control method it is possible to avoid escapes from a region of the phase space with a transient chaotic behavior. The main advantage of this method is that it allows to control the system even if the corrections applied to the parameter are smaller than the disturbances affecting it. To illustrate how the method works, we have applied it to three paradigmatic models in nonlinear dynamics, the logistic map, the Hénon map and the Duffing oscillator.
Uploads
Papers by Miguel A.F. Sanjuan