Basin Entropy Research Articles

Basins of convergence of equilibrium points in the restricted three-body problem with modified gravitational potential

Abstract This article aims to investigate the points of equilibrium and the associated convergence basins in the restricted problem with two primaries, with a modified gravitational potential. In particular, for one of the primary bodies, we add an external gravitational term of the form 1/r3, which is very common in general relativity and represents a gravitational field much stronger than the classical Newtonian one. Using the well-known Newton–Raphson iterator we numerically locate the position of the points of equilibrium, while we also obtain their linear stability. Furthermore, for the location of the points of equilibrium, we obtain semi-analytical functions of both the mass parameter and the transition parameter. Finally, we demonstrate how these two variable parameters affect the convergence dynamics of the system as well as the fractal degree of the basin diagrams. The fractal degree is derived by computing the (boundary) basin entropy.

Equilibrium Points and Basins of Convergence in the Triangular Restricted Four-Body Problem with a Radiating Body

In this paper, we extend the basic equilateral four-body problem by introducing the effect of radiation pressure, Poynting–Robertson drag, and solar wind drag. In our setup, three primaries lie at the vertices of an equilateral triangle and move in circular orbits around their common center of mass. Here, one of the primaries is a radiating body and the fourth body (whose mass is negligible) does not affect the motion of the primaries. We show that the existence and the number of equilibrium points of the problem depend on the mass parameters and radiation factor. Consequently, the allowed regions of motion, the regions of the basins of convergence for the equilibrium points, and the basin entropy will also depend on these parameters. The present dynamical model is analyzed for three combinations of mass for the primaries: equal masses, two equal masses, different masses. As the main results, we find that in all cases the libration points are unstable if the radiation factor is larger than 0.01 and hence able to destroy the stability of the libration points in the restricted four-body problem composed by the Sun, Jupiter, Trojan asteroid and a test (dust) particle. Also, we conclude that the number of fixed points decreases with the increase of the radiation factor.

Measuring the transition between nonhyperbolic and hyperbolic regimes in open Hamiltonian systems

We show that the presence of KAM islands in nonhyperbolic chaotic scattering has deep implications on the unpredictability of open Hamiltonian systems. When the energy of the system increases, the particles escape faster. For this reason, the boundary of the exit basins becomes thinner and less fractal. Hence, we could expect a monotonous decrease in the unpredictability as well as in the fractal dimension. However, within the nonhyperbolic regime, fluctuations in the basin entropy have been uncovered. The reason is that when increasing the energy, both the size and geometry of the KAM islands undergo abrupt changes. These fluctuations do not appear within the hyperbolic regime. Hence, the fluctuations in the basin entropy allow us to ascertain the hyperbolic or nonhyperbolic nature of a system. In this manuscript, we have used continuous and discrete open Hamiltonian systems in order to show the relevant role of the KAM islands on the unpredictability of the exit basins, and the utility of the basin entropy to analyze this kind of systems.

Determining the Properties of the Basins of Convergence in the Generalized Hénon–Heiles System

We examine the convergence properties of the generalized Hénon–Heiles system, by using the multivariate version of the Newton–Raphson iterative scheme. In particular, we numerically investigate how the perturbation parameter [Formula: see text] influences several aspects of the method, such as its speed and efficiency. Color-coded diagrams are used for revealing the basins of convergence on the configuration plane. Additionally, we compute the degree of fractality of the convergence basins on the configuration space, as a function of the perturbation parameter, by using different tools, such the uncertainty dimension and the (boundary) basin entropy. Our analysis suggests that the perturbation parameter strongly influences the number of the equilibrium points, as well as the geometry and the structure of the associated basins of convergence. Furthermore, the highest degree of fractality, along with the appearance of nonconverging points, occur near the critical values of the perturbation parameter.

Nonlinear dynamics and chaos in micro/nanoelectromechanical beam resonators actuated by two-sided electrodes

Abstract We investigate theoretically the nonlinear dynamics and the emergence of chaos in suspended beam micro/nanoelectromechanical (MEMS/NEMS) resonators actuated by two-sided electrodes. Through the analysis of phase diagrams we have found that the system presents a rich and complex nonlinear behavior. Multistability is observed in a significant region of the relevant parameter space, involving periodic and chaotic attractors. Complex and varied routes to chaos were also found. Basins of attraction with strongly intermingled attractors provide further evidence of multistability. The basins are analyzed in greater detail. Their fractal dimensions and uncertainty exponent are calculated using the well known box counting and uncertainty methods. The results for the uncertainty exponent are compared with those obtained with yet another approach, based on the recently proposed basin entropy method. The comparison provides a test for the new approach, which we conclude that is a reliable alternative method of calculation. Very low uncertainty exponents have been obtained, indicating that some basins have extremely intermingled attractors, what may have significant influence in the experimental investigation and practical applications of the resonators. We also conclude that the observation of chaos in this system is favored by lower frequencies of excitation and comparatively small quality factors (larger dissipation).

On the Convergence Dynamics of the Sitnikov Problem with Non-spherical Primaries

We investigate, using numerical methods, the convergence dynamics of the Sitnikov problem with non-spherical primaries, by applying the Newton-Raphson (NR) iterative scheme. In particular, we examine how the oblateness parameter $A$ influences several aspects of the method, such as its speed and efficiency. Color-coded diagrams are used for revealing the convergence basins on the plane of complex numbers. Moreover, we compute the degree of fractality of the convergence basins on the complex space, as a relation of the oblateness, by using different computational tools, such the fractal dimension as well as the (boundary) basin entropy.

Basin entropy behavior in a cyclic model of the rock-paper-scissors type

We deal with stochastic network simulations in a model with three distinct species that compete under cyclic rules which are similar to the rules of the popular rock-paper-scissors game. We investigate the Hamming distance density and then the basin entropy behavior, running the simulations for some typical values of the parameters mobility, predation and reproduction and for very long time evolutions. The results show that the basin entropy is another interesting tool of current interest to investigate chaotic features of the network simulations that are usually considered to describe aspects of biodiversity in the cyclic three-species model.

A special type of explosion of basin boundary

We study a special type of explosion of a basin boundary set in an archetypal oscillator. A typical feature is that the basin boundaries change the number of basins separating at the same time. Before the explosion, a basin boundary contains some Wada points of ten basins. After the explosion, the basin boundary contains some Wada points of eighteen basins. The underlying mechanism for the explosion is investigated by the heteroclinic tangency and Lambda lemma. Basin entropy and boundary basin entropy are also used to describe the nature of basins of attraction and the basin boundary explosion.

Resonant behavior and unpredictability in forced chaotic scattering

Chaotic scattering in open Hamiltonian systems is a topic of fundamental interest in physics, which has been mainly studied in the purely conservative case. However, the effect of weak perturbations in this kind of systems has been an important focus of interest in the last decade. In a previous work, the authors studied the effects of a periodic forcing in the decay law of the survival probability, and they characterized the global properties of escape dynamics. In the present paper, we add two important issues in the effects of periodic forcing: the fractal dimension of the set of singularities in the scattering function, and the unpredictability of the exit basins, which is estimated by using the concept of basin entropy. Both the fractal dimension and the basin entropy exhibit a resonant-like decrease as the forcing frequency increases. We provide a theoretical reasoning, which could justify this decreasing in the fractality near the main resonant frequency, that appears for $\omega \approx 1$. We attribute the decrease in the basin entropy to the reduction of the area occupied by the KAM islands and the basin boundaries when the frequency is close to the resonance. Finally, the decay rate of the exponential decay law shows a minimum value of the amplitude, $A_c$, which reflects the complete destruction of the KAM islands in the resonance. We expect that this work could be potentially useful in research fields related to chaotic Hamiltonian pumps, oscillations in chemical reactions and companion galaxies, among others.

Revealing the Newton–Raphson basins of convergence in the circular pseudo-Newtonian Sitnikov problem

In this paper we numerically explore the convergence properties of the pseudo-Newtonian circular restricted problem of three and four primary bodies. The classical Newton–Raphson iterative scheme is used for revealing the basins of convergence and their respective fractal basin boundaries on the complex plane. A thorough and systematic analysis is conducted in an attempt to determine the influence of the transition parameter on the convergence properties of the system. Additionally, the roots (numerical attractors) of the system and the basin entropy of the convergence diagrams are monitored as a function of the transition parameter, thus allowing us to extract useful conclusions. The probability distributions, as well as the distributions of the required number of iterations are also correlated with the corresponding basins of convergence.

On the Newton–Raphson basins of convergence of the out-of-plane equilibrium points in the Copenhagen problem with oblate primaries

The Copenhagen case of the circular restricted three-body problem with oblate primary bodies is numerically investigated by exploring the Newton–Raphson basins of convergence, related to the out-of-plane equilibrium points. The evolution of the position of the libration points is determined, as a function of the value of the oblateness coefficient. The attracting regions, on several types of two-dimensional planes, are revealed by using the multivariate Newton–Raphson iterative method. We perform a systematic and thorough investigation in an attempt to understand how the oblateness coefficient affects the geometry of the basins of convergence. The convergence regions are also related with the required number of iterations and also with the corresponding probability distributions. The degree of the fractality is also determined by calculating the fractal dimension and the basin entropy of the convergence planes.

Basins of Convergence in the Circular Sitnikov Four-Body Problem with Nonspherical Primaries

The Newton–Raphson basins of convergence, related to the equilibrium points, in the Sitnikov four-body problem with nonspherical primaries are numerically investigated. We monitor the parametric evolution of the positions of the roots, as a function of the oblateness coefficient. The classical Newton–Raphson optimal method is used for revealing the basins of convergence, by classifying dense grids of initial conditions in several types of two-dimensional planes. We perform a systematic and thorough analysis in an attempt to understand how the oblateness coefficient affects the geometry as well as the basin entropy of the convergence regions. The convergence areas are related with the required number of iterations and also with the corresponding probability distributions.

Uncertainty dimension and basin entropy in relativistic chaotic scattering.

Chaotic scattering is an important topic in nonlinear dynamics and chaos with applications in several fields in physics and engineering. The study of this phenomenon in relativistic systems has received little attention as compared to the Newtonian case. Here we focus our work on the study of some relevant characteristics of the exit basin topology in the relativistic Hénon-Heiles system: the uncertainty dimension, the Wada property, and the basin entropy. Our main findings for the uncertainty dimension show two different behaviors insofar as we change the relativistic parameter β, in which a crossover behavior is uncovered. This crossover point is related with the disappearance of KAM islands in phase space, which happens for velocity values above the ultrarelativistic limit, v>0.1c. This result is supported by numerical simulations and by qualitative analysis, which are in good agreement. On the other hand, the computation of the exit basins in the phase space suggests the existence of Wada basins for a range of β<0.625. We also studied the evolution of the exit basins in a quantitative manner by computing the basin entropy, which shows a maximum value for β≈0.2. This last quantity is related to the uncertainty in the prediction of the final fate of the system. Finally, our work is relevant in galactic dynamics, and it also has important implications in other topics in physics such as as in the Störmer problem, among others.

Dynamical analysis of bounded and unbounded orbits in a generalized Hénon–Heiles system

The Hénon–Heiles potential was first proposed as a simplified version of the gravitational potential experimented by a star in the presence of a galactic center. Currently, this system is considered a paradigm in dynamical systems because despite its simplicity exhibits a very complex dynamical behavior. In the present paper, we perform a series expansion up to the fifth-order of a potential with axial and reflection symmetries, which after some transformations, leads to a generalized Hénon–Heiles potential. Such new system is analyzed qualitatively in both regimes of bounded and unbounded motion via the Poincaré sections method and plotting the exit basins. On the other hand, the quantitative analysis is performed through the Lyapunov exponents and the basin entropy, respectively. We find that in both regimes the chaoticity of the system decreases as long as the test particle energy gets far from the critical energy. Additionally, we may conclude that despite the inclusion of higher order terms in the series expansion, the new system shows wider zones of regularity (islands) than the ones present in the Hénon–Heiles system.

Basins of convergence of equilibrium points in the generalized Hénon–Heiles system

We numerically explore the Newton–Raphson basins of convergence, related to the libration points (which act as attractors of the convergence process), in the generalized Hénon-Heiles system (GHH). The evolution of the position as well as of the linear stability of the equilibrium points is determined, as a function of the value of the perturbation parameter. The attracting regions, on the configuration (x,y) plane, are revealed by using the multivariate version of the classical Newton–Raphson iterative algorithm. We perform a systematic investigation in an attempt to understand how the perturbation parameter affects the geometry as well as of the basin entropy of the attracting domains. The convergence regions are also related with the required number of iterations.

Basins of Convergence of Equilibrium Points in the Generalized Hill Problem

The Newton–Raphson basins of attraction, associated with the libration points (attractors), are revealed in the generalized Hill problem. The parametric variation of the position and the linear stability of the equilibrium points is determined, when the value of the perturbation parameter [Formula: see text] varies. The multivariate Newton–Raphson iterative scheme is used to determine the attracting domains on several types of two-dimensional planes. A systematic and thorough numerical investigation is performed in order to demonstrate the influence of the perturbation parameter on the geometry as well as of the basin entropy of the basins of convergence. The correlations between the basins of attraction and the corresponding required number of iterations are also illustrated and discussed. Our numerical analysis strongly indicates that the evolution of the attracting regions in this dynamical system is an extremely complicated yet very interesting issue.

Chaotic dynamics and fractal structures in experiments with cold atoms

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.

Basin entropy: a new tool to analyze uncertainty in dynamical systems.

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.

Form and function in gene regulatory networks: the structure of network motifs determines fundamental properties of their dynamical state space.

Network motifs have been studied extensively over the past decade, and certain motifs, such as the feed-forward loop, play an important role in regulatory networks. Recent studies have used Boolean network motifs to explore the link between form and function in gene regulatory networks and have found that the structure of a motif does not strongly determine its function, if this is defined in terms of the gene expression patterns the motif can produce. Here, we offer a different, higher-level definition of the ‘function’ of a motif, in terms of two fundamental properties of its dynamical state space as a Boolean network. One is the basin entropy, which is a complexity measure of the dynamics of Boolean networks. The other is the diversity of cyclic attractor lengths that a given motif can produce. Using these two measures, we examine all 104 topologically distinct three-node motifs and show that the structural properties of a motif, such as the presence of feedback loops and feed-forward loops, predict fundamental characteristics of its dynamical state space, which in turn determine aspects of its functional versatility. We also show that these higher-level properties have a direct bearing on real regulatory networks, as both basin entropy and cycle length diversity show a close correspondence with the prevalence, in neural and genetic regulatory networks, of the 13 connected motifs without self-interactions that have been studied extensively in the literature.

Evaluation of attractors and basins of asynchronous random Boolean networks

We present an algebraic approach for determining the attractors and their basins of random Boolean networks under an asynchronous stochastic update based on the recently developed matrix semitensor product theory, which allows for converting the logical dynamics of a Boolean network into a standard iterative dynamics. In this setting, all attractors and basins are specified by the network transition matrices. We then devise procedures that can find all attractors and their basins exactly. We also discuss the issue of overlapping basins in asynchronous random Boolean networks, and we propose methods to compute the weight of each attractor and the basin entropy of the systems.