Exit Basins Research Articles

Relativistic chaotic scattering: Unveiling scaling laws for trapped trajectories.

In this paper we study different types of phase space structures which appear in the context of relativistic chaotic scattering. By using the relativistic version of the Hénon-Heiles Hamiltonian, we numerically study the topology of different kind of exit basins and compare it with the case of low velocities in which the Newtonian version of the system is valid. Specifically, we numerically study the escapes in the phase space, in the energy plane, and in the β plane, which richly characterize the dynamics of the system. In all cases, fractal structures are present, and the escaping dynamics is characterized. In every case a scaling law is numerically obtained in which the percentage of the trapped trajectories as a function of the relativistic parameter β and the energy is obtained. Our work could be useful in the context of charged particles which eventually can be trapped in the magnetosphere, where the analysis of these structures can be relevant.

Period-doubling bifurcations and islets of stability in two-degree-of-freedom Hamiltonian systems.

In this paper, we show that the destruction of the main Kolmogorov-Arnold-Moser (KAM) islands in two-degree-of-freedom Hamiltonian systems occurs through a cascade of period-doubling bifurcations. We calculate the corresponding Feigenbaum constant and the accumulation point of the period-doubling sequence. By means of a systematic grid search on exit basin diagrams, we find the existence of numerous very small KAM islands ("islets") for values below and above the aforementioned accumulation point. We study the bifurcations involving the formation of islets and we classify them in three different types. Finally, we show that the same types of islets appear in generic two-degree-of-freedom Hamiltonian systems and in area-preserving maps.

E&P Notes (November 2021)

E&P Notes (November 2021)

Final state sensitivity in noisy chaotic scattering

The unpredictability in chaotic scattering problems is a fundamental topic in physics that has been studied either in purely conservative systems or in the presence of weak perturbations. In many systems noise plays an important role in the dynamical behavior and it models their internal irregularities or their coupling with the environment. In these situations the unpredictability is affected by both the chaotic dynamics and the stochastic fluctuations. In the presence of noise two trajectories with the same initial condition can evolve in different ways and converge to a different asymptotic behavior. For this reason, even the exact knowledge of the initial conditions does not necessarily lead to the predictability of the final state of the system. Hence, the noise can be considered as an important source of unpredictability that cannot be fully understood using the conventional methods of nonlinear dynamics, such as the exit basins and the uncertainty exponent. By adopting a probabilistic point of view, we develop the concepts of probability basin, uncertainty basin and noise-sensitivity exponent, that allow us to carry out both a quantitative and qualitative analysis of the unpredictability on noisy chaotic scattering problems.

Exit basins for the Sitnikov problem with variable mass

In the present paper, we perform a numerical study of the Sitnikov problem aiming to characterize the orbits of a variable mass particle (e.g., comet, rocket, asteroid or spacecraft) and determine the uncertainty in the prediction of the final state of the test particle. The classification of final states was done through the well-known exit basins, while the determination of the uncertainty was calculated using a new tool named Basin entropy. It is found that for small values of the initial mass of the test particle, the number of initial conditions leading to bounded orbits gets increased, thus reducing the uncertainty in the final states. The same behavior in uncertainty is observed for increasing values of the exponent in Jeans law for the variation of the mass. Our results allow us to conclude that: i) an accelerated fuel consumption in the initial stages of stabilization of a satellite can keep the object in an oscillatory state around the primaries and ii) if the mass of the satellite is less than one hundredth of the mass of each primary, it is possible to predict with a very high certainty the final state of the satellite, regardless of the accuracy in the initial conditions of the system.

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.

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.

Wada structures in a binary black hole system

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, recognising 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., non-fractal) parts of the binary shadow.

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.

Fractal boundaries in chaotic hamiltonian systems

Fractal structures are typically present in the dynamics of chaotic orbits in non-integrable open Hamiltonian systems and result from the extremely complicated nature of the invariant manifolds of unstable periodic orbits. Exit basins, the set of initial conditions leading to orbits escaping through a given exit, have very frequently fractal boundaries. In this work we analyze exit basin boundaries in a dynamical system of physical interest, namely the motion of charged particles in a magnetized plasma subjected to electrostatic drift waves, and characterize in a quantitative way the fractality of these structures and their observable consequences, as the final-state uncertainty.

Chaotic magnetic field lines and fractal structures in a tokamak with magnetic limiter

Open hamiltonian systems have typically fractal structures underlying chaotic dynamics with a number of physical consequences on transport. We consider such fractal structures related to the formation of a chaotic magnetic field line region near the tokamak wall and the corresponding field line dynamics, described by a two-dimensional area-preserving map. We focus on the exit basins, which are sets of points which originate orbits escaping through some exit and are typically fractal, also exhibiting the so-called Wada property. We show qualitative as well as quantitative evidences of these properties.

Fractal structures in the chaotic motion of charged particles in a magnetized plasma under the influence of drift waves

Abstract Chaotic dynamics in open Hamiltonian dynamical systems typically presents a number of fractal structures in phase space derived from the interwoven structure of invariant manifolds and the corresponding chaotic saddle. These structures are thought to play an important role in the transport properties related to the chaotic motion. Such properties can explain some aspects of the non-uniform nature of the anomalous transport observed in magnetically confined plasmas. Accordingly we consider a theoretical model for the interaction of charged test particles with drift waves. We describe the exit basin structure of the corresponding chaotic orbit in phase space and interpret it in terms of the invariant manifold structure underlying chaotic dynamics. As a result, the exit basin boundary is shown to be a fractal curve, by direct calculation of its box-counting dimension. Moreover, when there are more than two basins, we verify the existence of the Wada property, an extreme form of fractality.

Escape dynamics and fractal basin boundaries in the planar Earth–Moon system

The escape of trajectories of a spacecraft, or comet or asteroid in the presence of the Earth–Moon system is investigated in detail in the context of the planar circular restricted three-body problem, in a scattering region around the Moon. The escape through the necks around the collinear points $$L_1$$ and $$L_2$$ as well as the leaking produced by considering collisions with the Moon surface, taking the lunar mean radius into account, were considered. Given that different transport channels are available as a function of the Jacobi constant, four distinct escape regimes are analyzed. Besides the calculation of exit basins and of the spatial distribution of escape time, the qualitative dynamical investigation through Poincare sections is performed in order to elucidate the escape process. Our analyses reveal the dependence of the properties of the considered escape basins with the energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. Finally, we observe the plentiful presence of stickiness motion near stability islands which plays a remarkable role in the longest escape time behavior. The application of this analysis is important both in space mission design and study of natural systems, given that fractal boundaries are related with high sensitivity to initial conditions, implying in uncertainty between safe and unsafe solutions, as well as between escaping solutions that evolve to different phase space regions.

Effects of periodic forcing in chaotic scattering.

The effects of a periodic forcing on chaotic scattering are relevant in certain situations of physical interest. We investigate the effects of the forcing amplitude and the external frequency in both the survival probability of the particles in the scattering region and the exit basins associated to phase space. We have found an exponential decay law for the survival probability of the particles in the scattering region. A resonant-like behavior is uncovered where the critical values of the frequencies ω≃1 and ω≃2 permit the particles to escape faster than for other different values. On the other hand, the computation of the exit basins in phase space reveals the existence of Wada basins depending of the frequency values. We provide some heuristic arguments that are in good agreement with the numerical results. Our results are expected to be relevant for physical phenomena such as the effect of companion galaxies, among others.

Weakly noisy chaotic scattering

The effect of a weak source of noise on the chaotic scattering is relevant to situations of physical interest. We investigate how a weak source of additive uncorrelated Gaussian noise affects both the dynamics and the topology of a paradigmatic chaotic scattering problem as the one taking place in the open nonhyperbolic regime of the Hénon-Heiles Hamiltonian system. We have found long transients for the time escape distributions for critical values of the noise intensity for which the particles escape slower as compared with the noiseless case. An analysis of the survival probability of the scattering function versus the Gaussian noise intensity shows a smooth curve with one local maximum and with one local minimum which are related to those long transients and with the basin structure in phase space. On the other hand, the computation of the exit basins in phase space shows a quadratic curve for which the basin boundaries lose their fractal-like structure as noise turned on.

Controlling unpredictability in the randomly driven Hénon–Heiles system

Noisy scattering dynamics in the randomly driven Hénon–Heiles system is investigated in the range of initial energies where the motion is unbounded. In this paper we study, with the help of the exit basins and the escape time distributions, how an external perturbation, be it dissipation or periodic forcing with a random phase, can enhance or mitigate the unpredictability of a system that exhibit chaotic scattering. In fact, if basin boundaries have the Wada property, predictability becomes very complicated, since the basin boundaries start to intermingle, what means that there are points of different basins close to each other. The main responsible of this unpredictability is the external forcing with random phase, while the dissipation can recompose the basin boundaries and turn the system more predictable. Therefore, we do the necessary simulations to find out the values of dissipation and external forcing for which the exit basins present the Wada property. Through these numerical simulations, we show that the presence of the Wada basins have a specific relation with the damping, the forcing amplitude and the energy value. Our approach consists on investigating the dynamics of the system in order to gain knowledge able to control the unpredictability due to the Wada basins.

Noisy scattering dynamics in the randomly driven Hénon-Heiles oscillator

Noisy scattering dynamics in the randomly driven Hénon-Heiles oscillator is investigated when the energy is above the threshold to permit particles to escape from the scattering region. First, some basic simulation procedures are briefly introduced and the fractal exit basins appear to be robust when the bounded noisy excitation is imposed on the oscillator. Second, several key fractal characteristics of the sample basin boundaries, such as the delay-time function and the uncertainty dimension, are estimated from which this oscillator is found to be structurally unstable against the bounded noisy excitation. Moreover, the stable and unstable manifolds of some sample chaotic invariant sets are estimated and illustrated in a special two-dimensional Poincaré section. Lastly, several previous methods are developed to identify three arbitrarily chosen noisy scattering time series of the randomly driven Hénon-Heiles oscillator, from which the quasiperiodic-dominant and the chaotic-dominant dynamical behaviors are distinguished.

Dynamical properties of the weak stability boundary and associated sets

In this contribution the weak stability boundary algorithmic definition was numerically accomplished with the inclusion of lunar and earth collisional sets and a subclassification of the unstable set. Then, the associated sets to WSB definition were analyzed and characterized according to relevant dynamical properties in order to clarify their applicability in earth-moon transfer orbit design. The obtained stable, unstable, and collisional sets are defined as a function of the osculating ellipse eccentricity for prograde and retrogade initial conditions. The stable sets, candidates to ballistic capture transfers, are subclassified according to chosen specific criteria, namely, the Jacobi constant intervals defined by distinct classes of Hill regions, the location of the final state after a complete cycle with respect to the Hill sphere, the permanence in the lunar sphere of influence in a full cycle around the moon, and exit basins for retrograde evolution. By the first time, with this investigation, elucidative criteria based on three-body problem elements are employed to identify initial condition subsets with required properties to design ballistic capture transfers.

BIFURCATIONS AND CHAOS IN HAMILTONIAN SYSTEMS

This paper deals with the use of recent computational techniques in the numerical study of qualitative properties of two degrees of freedom of Hamiltonian systems. These numerical methods are based on the computation of the OFLI2 Chaos Indicator, the Crash Test and exit basins and the skeleton of symmetric periodic orbits. As paradigmatic examples, three classical problems are studied: the Copenhagen and the (n + 1)-body ring problems and the Hénón–Heiles Hamiltonian. All the numerical integrations have been done by using the state-of-the-art numerical library TIDES based on the extended Taylor series method.