Wada structures in a binary black hole system

Abstract

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.

Chaotic exits from a weakly magnetized Schwarzschild black hole

“…In generalizing, we gain access to richer dynamics and are able to study how the critical escape energy depends on the strength of the interaction between the magnetic field and the charged particle. In addition, using newly proposed concepts and methods from nonlinear dynamics [7,8], we study some topological and structural aspects of the basins that were left uninvestigated in [6]. In particular, with the merging method of [8] we show that the basin boundaries are not just fractal, they also possess the peculiar Wada property, which means that they are boundaries shared by more than two basins.…”

“…In addition, using newly proposed concepts and methods from nonlinear dynamics [7,8], we study some topological and structural aspects of the basins that were left uninvestigated in [6]. In particular, with the merging method of [8] we show that the basin boundaries are not just fractal, they also possess the peculiar Wada property, which means that they are boundaries shared by more than two basins. We also quantify the "uncertainty" of our dynamical system, using the concept of basin entropy that was introduced in [7], and investigate how this uncertainty is affected by the strength of the interaction between the magnetic field and the charged particle.…”

How to detect Wada basins

“…Also, these new methods broaden the scope of application of the original idea of Yoneyama to unexpected fields [6], illustrating that it constitutes a very special property of chaotic dynamical systems usually indicating a lack of predictability and with a bright future ahead in spite of all the work that has been done so far.…”

“…The cumbersome structure of the Wada basins implies a particular kind of unpredictability [7], since a small perturbation in the initial conditions lying on a Wada boundary may lead the trajectory to any of the system's attractors. Since the pioneering works of Yorke and collaborators [16,20,21,23], the Wada property has been found in many different cases: chaotic scattering [24,14], Hamiltonian systems [2], fluid dynamics [28], interaction between waves [5], delayed systems [9], black hole shadows [6], etc.…”

“…In fact, many works have been devoted to the application of this method to just one dynamical system with fixed parameters [24,28,2,1]. The computation of the unstable manifold of the periodic orbit can be burdensome in some cases [9,6]. Also the method is restricted to ODEs, Hamiltonians and maps showing the connected Wada property.…”