Systems with Escapes

Contopoulos, G.; Harsoula, M.

doi:10.1196/annals.1350.012

Cited by 7 publications

(4 citation statements)

References 40 publications

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“…The properties of MP geometries were studied in detail by Hartle & Hawking [14] and Chandrasekhar [15]. Contopoulos revealed that the binary MP spacetime exhibits self-similarity and chaotic dynamics [16][17][18][19][20][21][22]. Influential perspectives on the role of chaos in binary systems in relativity have followed from Yurtsever [23], Dettmann [24,25], Cornish [26,27], Frankel [28], Levin [29,30] and many others [31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Binary black hole shadows, chaotic scattering and the Cantor set

Shipley¹,

Dolan²

2016

Class. Quantum Grav.

124

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We investigate the qualitative features of binary black hole shadows using the model of two extremally charged black holes in static equilibrium (a Majumdar-Papapetrou solution). Our perspective is that binary spacetimes are natural exemplars of chaotic scattering, because they admit more than one fundamental null orbit, and thus an uncountably infinite set of perpetual null orbits which generate scattering singularities in initial data. Inspired by the three-disc model, we develop an appropriate symbolic dynamics to describe planar null geodesics on the double black hole spacetime. We show that a one-dimensional (1D) black hole shadow may constructed through an iterative procedure akin to the construction of the Cantor set; thus the 1D shadow is self-similar.Next, we study non-planar rays, to understand how angular momentum affects the existence and properties of the fundamental null orbits. Taking slices through 2D shadows, we observe three types of 1D shadow: regular, Cantor-like, and highly chaotic. The switch from Cantor-like to regular occurs where outer fundamental orbits are forbidden by angular momentum. The highly chaotic part is associated with an unexpected feature: stable and bounded null orbits, which exist around two black holes of equal mass M separated by a 1 < a < √ 2a 1 , where a 1 = 4M/ √ 27. To show how this possibility arises, we define a certain potential function and classify its stationary points. We conjecture that the highly chaotic parts of the 2D shadow possess the Wada property.Finally, we consider the possibility of following null geodesics through event horizons, and chaos in the maximally extended spacetime. * joshipley1@sheffield.ac.uk

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Section: Introductionmentioning

confidence: 99%

Binary black hole shadows, chaotic scattering and the Cantor set

Shipley¹,

Dolan²

2016

Class. Quantum Grav.

124

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“…This escape can be interpreted as a reaction by crossing the potential energy barrier. This model system and its high dimensional analog has been studied in great detail in nonlinear dynamical systems, statistical mechanics, for developing molecular simulation algorithms [25][26][27][28][29] . In this study, we will define the three exits as follows:…”

Section: H éNon-heiles System and Reaction Dynamicsmentioning

confidence: 99%

Support vector machines for learning reactive islands

Naik¹,

Krajňák²,

Wiggins³

2021

Preprint

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We develop a machine learning framework that can be applied to data sets derived from the trajectories of Hamilton's equations. The goal is to learn the phase space structures that play the governing role for phase space transport relevant to particular applications. Our focus is on learning reactive islands in two degrees-of-freedom Hamiltonian systems. Reactive islands are constructed from the stable and unstable manifolds of unstable periodic orbits and play the role of quantifying transition dynamics. We show that support vector machines (SVM) is an appropriate machine learning framework for this purpose as it provides an approach for finding the boundaries between qualitatively distinct dynamical behaviors, which is in the spirit of the phase space transport framework. We show how our method allows us to find reactive islands directly in the sense that we do not have to first compute unstable periodic orbits and their stable and unstable manifolds. We apply our approach to the Hénon-Heiles Hamiltonian system, which is a benchmark system in the dynamical systems community. We discuss different sampling and learning approaches and their advantages and disadvantages.

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“…In other words, the equipotential curves or, equivalently, the zerovelocity curves (ZVCs) are open containing several channels through which a particle can escape from the dynamical system. The phenomenon of escapes from a dynamical system, especially the escape of stars from stellar systems has been an active field of research during the last decades [13,15,[17][18][19]23,30,40,41,46]. The reader can find more illuminating details on the subject of escapes in the review [42] and also in [14].…”

mentioning

confidence: 99%

Revealing the evolution, the stability, and the escapes of families of resonant periodic orbits in Hamiltonian systems

Zotos

2013

Nonlinear Dyn

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We investigate the evolution of families of periodic orbits in a bisymmetrical potential made up of a twodimensional harmonic oscillator with only one quartic perturbing term, in a number of resonant cases. Our main objective is to compute sufficiently and accurately the position and the period of the periodic orbits. For the derivation of the above quantities (position and period) we deploy in each resonance case semi-numerical methods. The comparison of our semi-numerical results with those obtained by the numerical integration of the equations of motion indicates that, in every case the relative error is always less than 1% and therefore, the agreement is more than sufficient. Thus, we claim that semi-numerical methods are very effective tools for computing periodic orbits. We also study in detail, the case when the energy of the orbits is larger than the escape energy. In this case, the periodic orbits in almost all resonance families become unstable and eventually escape from the system. Our target is to calculate the escape period and the escape position of the periodic orbits and also monitor their evolution with respect to the value of the energy.

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Systems with Escapes

Cited by 7 publications

References 40 publications

Binary black hole shadows, chaotic scattering and the Cantor set

Binary black hole shadows, chaotic scattering and the Cantor set

Support vector machines for learning reactive islands

Revealing the evolution, the stability, and the escapes of families of resonant periodic orbits in Hamiltonian systems

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