Unstable Orbits Research Articles

Irreversible mixing by unstable periodic orbits in buoyancy dominated stratified turbulence

We consider turbulence driven by a large-scale horizontal shear in Kolmogorov flow (i.e. with sinusoidal body forcing) and a background linear stable stratification with buoyancy frequency $N_{B}^{2}$ imposed in the third, vertical direction in a fluid with kinematic viscosity $\unicode[STIX]{x1D708}$. This flow is known to be organised into layers by nonlinear unstable steady states, which incline the background shear in the vertical and can be demonstrated to be the finite-amplitude saturation of a sequence of instabilities, originally from the laminar state. Here, we investigate the next order of motions in this system, i.e. the time-dependent mechanisms by which the density field is irreversibly mixed. This investigation is achieved using ‘recurrent flow analysis’. We identify (unstable) periodic orbits, which are embedded in the turbulent attractor, and use these orbits as proxies for the chaotic flow. We find that the time average of an appropriate measure of the ‘mixing efficiency’ of the flow $\mathscr{E}=\unicode[STIX]{x1D712}/(\unicode[STIX]{x1D712}+{\mathcal{D}})$ (where ${\mathcal{D}}$ is the volume-averaged kinetic energy dissipation rate and $\unicode[STIX]{x1D712}$ is the volume-averaged density variance dissipation rate) varies non-monotonically with the time-averaged buoyancy Reynolds numbers $\overline{Re}_{B}=\overline{{\mathcal{D}}}/(\unicode[STIX]{x1D708}N_{B}^{2})$, and is bounded above by $1/6$, consistently with the classical model of Osborn (J. Phys. Oceanogr., vol. 10 (1), 1980, pp. 83–89). There are qualitatively different physical properties between the unstable orbits that have lower irreversible mixing efficiency at low $\overline{Re}_{B}\sim O(1)$ and those with nearly optimal $\mathscr{E}\lesssim 1/6$ at intermediate $\overline{Re}_{B}\sim 10$. The weaker orbits, inevitably embedded in more strongly stratified flow, are characterised by straining or ‘scouring’ motions, while the more efficient orbits have clear overturning dynamics in more weakly stratified, and apparently shear-unstable flow.

On the Periodic Logistic Map

In this paper, the famous logistic map is studied in a new point of view. We study the boundedness and the periodicity of non-autonomous logistic map $${x_{n + 1}} = {r_n}{x_n}\left( {1 - {x_n}} \right),n = 0,1, \ldots ,$$ where {r n } is a positive p-periodic sequence. The sufficient conditions are given to support the existence of asymptotically stable and unstable p-periodic orbits. This appears to be the first study of the map with variable parameter r.

Laser-generated Richtmyer–Meshkov and Rayleigh–Taylor instabilities. III. Near-peripheral region of Gaussian spot

AbstractDynamics and organization of laser-generated three-dimensional (3D) Richtmyer–Meshkov (RMI) and Rayleigh–Taylor instabilities (RMI and RTI) on metal target in the semiconfined configuration are different in the central region (CR) (Lugomer, 2016), near central region (NCR) (Lugomer, 2017) and the near periphery region (NPR) of the Gaussian-like spot. The RMI/RTI in the NPR evolve from the shock and series of reshocks associated with lateral expansion and increase of the vapor density, decrease of the Atwood number and momentum transfer. Scanning electron micrographs show irregular (chaotic) web of the base-plane walls, and mushroom spikes on its nodal points with disturbed two-dimensional (2D) lattice organization. Lattice disturbance is caused by the incoherent wavy motion of background fluid due to fast reshocks, which after series of reflections change their strength and direction. Reconstruction of the disturbed lattice reveals rectangular lattice of mushroom spikes with p2mm symmetry. The splitting (bifurcation) of mushroom diameter distribution on the large and small mushroom spikes increases with radial distance from the center of Gaussian-like spot. Dynamics of their evolution is represented by the orbits or stable periods in 2D phase space. The constant mushroom diameter – stable circulation or the stable periodic orbits – are the limit cycles between the unstable spiral orbits. Those with increasing periods represent supercritical Hopf bifurcation, while those leading to decrease and disappearance represent subcritical Hopf bifurcation. The empirical models of RMI, although predict dependence of the growth rate on radial distance (distance the reshocks travel to reach the interface), show many limitations. More appropriate interpretation of the simultaneous growth and lattice organization of small and large spikes give the fundamental model based on the interference of the perturbation modes depending on their amplitude, relative phase, and the symmetry. The late-time instability in the base-plane evolves into line solitons, vortex filaments and wave–vortex structures with chaotic rather than stochastic features.

High-Speed "4D" Computational Microscopy of Bacterial Surface Motility.

Bacteria exhibit surface motility modes that play pivotal roles in early-stage biofilm community development, such as type IV pili-driven "twitching" motility and flagellum-driven "spinning" and "swarming" motility. Appendage-driven motility is controlled by molecular motors, and analysis of surface motility behavior is complicated by its inherently 3D nature, the speed of which is too fast for confocal microscopy to capture. Here, we combine electromagnetic field computation and statistical image analysis to generate 3D movies close to a surface at 5 ms time resolution using conventional inverted microscopes. We treat each bacterial cell as a spherocylindrical lens and use finite element modeling to solve Maxwell's equations and compute the diffracted light intensities associated with different angular orientations of the bacterium relative to the surface. By performing cross-correlation calculations between measured 2D microscopy images and a library of computed light intensities, we demonstrate that near-surface 3D movies of Pseudomonas aeruginosa translational and rotational motion are possible at high temporal resolution. Comparison between computational reconstructions and detailed hydrodynamic calculations reveals that P. aeruginosa act like low Reynolds number spinning tops with unstable orbits, driven by a flagellum motor with a torque output of ∼2 pN μm. Interestingly, our analysis reveals that P. aeruginosa can undergo complex flagellum-driven dynamical behavior, including precession, nutation, and an unexpected taxonomy of surface motility mechanisms, including upright-spinning bacteria that diffuse laterally across the surface, and horizontal bacteria that follow helicoidal trajectories and exhibit superdiffusive movements parallel to the surface.

Proxima may be an interloper from far away

Proxima Centauri's orbit is currently bound to Alpha Centauri A and B, but it is unclear if it will remain stable, say Fabo Feng and Hugh Jones at the University of Hertfordshire in the UK. They modeled 100 Proxima clones in orbits around their binary partners. Over a simulated 10 billion years, Proxima Centauri became unbound from the pair due to an unstable orbit 25 per cent of the time. If it had been born in the system, it is unlikely the orbit would be unstable aeons later. So, Proxima may have been captured by the stellar pair.

Quasinormal modes and absorption cross-sections of Born–Infeld–de Sitter black holes

In this paper, we have studied QNM modes and absorption cross-sections of Born–Infeld–de Sitter black holes. WKB approximation is employed to compute the QNM modes of massless scalar fields. We have also used null geodesics to compute quasinormal modes in the eikonal approximation. In the eikonal limit, QNMs of black holes are determined by the parameters of the circular null geodesics. Unstable circular null orbits are derived from the effective metric which is obeyed by light rays under the influence of a nonlinear electromagnetic field. Comparison is shown with the QNM of the linear electromagnetic counterpart, the Reissner–Nordström black hole. Furthermore, the null geodesics are employed to compute the absorption cross-sections in the high frequency limit via the sinc approximation.

Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibro-impact systems

In this paper, unstable dynamics is considered for the models of vibro-impact systems with linear differential equations coupled to an impact map. To provide a skeleton for the organization of chaotic attractors, we propose a method for detecting unstable periodic orbits embedded in chaotic attractors through a combination of unconstrained optimization technique and Poincaré map. Three numerical examples from different vibro-impact models demonstrate that the strategy can efficiently detect unstable periodic orbits in chaotic attractors. In order to explore the mechanism responsible for the creation of multi-dimensional tori attractors, we also present another method to detect unstable quasiperiodic orbits embedded multi-dimensional tori attractors by examining a specially transient time series. The upper bound and lower bound of the transient time series (in the Poincaré map) can be obtained by analyzing transient Lyapunov exponent and transient Lyapunov dimension. Some examples verify the effectiveness of the numerical algorithm.

Detecting unstable periodic orbits in chaotic time series using synchronization.

An alternative approach of detecting unstable periodic orbits in chaotic time series is proposed using synchronization techniques. A master-slave synchronization scheme is developed, in which the chaotic system drives a system of harmonic oscillators through a proper coupling condition. The proposed scheme is designed so that the power of the coupling signal exhibits notches that drop to zero once the system approaches an unstable orbit yielding an explicit indication of the presence of a periodic motion. The results shows that the proposed approach is particularly suitable in practical situations, where the time series is short and noisy, or it is obtained from high-dimensional chaotic systems.

Dynamics of tethered satellites in the vicinity of the Lagrangian point L 2 $L_{2}$ of the Earth–Moon system

This paper analyzes the dynamical evolution of satellites formed by two masses connected by a cable—tethered satellites. We derive the Lagrangian equations of motion in the neighborhood of the collinear equilibrium points, especially for the $L_{2} $ , of the restricted problem of three bodies. The rigid body configuration is expanded in Legendre polynomials up to fourth degree. We present some numerical simulations of the influence of the parameters such as cable length, mass ratio and initial conditions in the behavior of the tethered satellites. The equation for the collinear equilibrium point is derived and numerically solved. The evolution of the equilibria with the variation of the cable length as a parameter is studied. We also present a discussion of the linear stability around these equilibria. Based on this analysis calculate some unstable Lyapunov orbits associated to these equilibrium points. We found periodic orbits in which the tether travels parallel to itself without involving the angular motion. The numerical applications are focused on the Earth–Moon system. However, the general character of the equations allows applications to the $L_{1}$ equilibrium and obviously to systems other than the Earth–Moon.

On uniqueness of static spacetimes with non-trivial conformal scalar field

We discuss the uniqueness of static spacetimes with non-trivial conformal scalar field. We can show that the spacetime is unique to be the Bocharova–Bronnikov–Melnikov–Bekenstein solution outside the surface composed of the unstable circular orbit of a photon (photon surface). In addition, we see that multiple disconnected photon surfaces having the same values of the scalar field do not exist.

Periodic Perturbation Method for Controlling Chaos for a Positive Output DC-DC Luo Converter

<p>A simple, non-feedback method of controlling chaos is implemented for a DC-DC converter. The weak periodic perturbation (WPP) is the control technique applied to stabilize an unstable orbit in a current-mode controlled Positive Output Luo (POL) DC-DC converter operating in a chaotic regime. With WPP, the operation of the converter is limited to stable period-1 orbit that exists in the original chaotic attractor. The proposed control strategy is implemented using simulations and the results are verified with hardware setup. The experimental results of the converter with WPP control are presented which shows the effectiveness of the control strategy.</p>

Whirling orbits around twirling black holes from conformal symmetry

Dynamics in the throat of rapidly rotating Kerr black holes is governed by an emergent near-horizon conformal symmetry. The throat contains unstable circular orbits at radii extending from the ISCO down to the light ring. We show that they are related by conformal transformations to physical plunges and osculating trajectories. These orbits have angular momentum arbitrarily higher than that of ISCO. Using the conformal symmetry we compute analytically the radiation produced by the physical orbits. We also present a simple formula for the full self-force on such trajectories in terms of the self-force on circular orbits.

Scaling in global tidal dissipation of the Earth-Moon system

The Moon migrated to ▪ cm over a characteristic time r/v=10 Gyr by tidal interaction with Earth’s oceans at a present velocity of v=3.8 cm yr−1. We model global dissipation to cover the entire history over the past 4.52 Gyr. We use scaling and numerical integration to model the off-resonance tidal interactions at relatively short tidal periods in the past to a near-resonance state at present. The global properties of the complex spatio-temporal dynamics and dissipation in broad spectrum ocean waves is modeled by damping ϵ=hF/(2Q0), where h is the tidal wave amplitude, F is the tidal frequency, and Q0 is the Q-factor at the present time. It satisfies Q0 ≃ 14 for consistency of migration time and age of the Moon consistent with observations for a near-resonance state today. Numerical results reveal the need for scaling with amplitude. It shows a startlingly fast eviction of the Moon from an unstable near-synchronous orbit close to the Roche limit, probably in a protolunar disk. Rapid spin down of Earth from an initial ∼ 30% of break-up by the Moon favored early formation of a clement global climate. Our theory suggests moons may be similarly advantageous to potentially habitable exoplanets.

Noise induced transitions and topological study of a periodically driven system

Noise induced transitions of an overdamped periodically driven oscillator are investigated theoretically and numerically in the limit of weak noise due to the Freidlin-Wentzell large deviation theory. Heteroclinic trajectories are found to approach the unstable orbit with fluctuational force tending to zeros. The global minimizer of the action functional corresponds to the most probable escape path and it shows a good agreement with statistical results. We then study the origins of singularities from a topological point of view by considering structures of the Lagrangian manifold and action surface. The switching line and cusp point turn out to have physical significance since they may impact the prehistory distributions, making the optimal path invalid.

The Coupled Orbit-Attitude Dynamics and Control of Electric Sail in Displaced Solar Orbits

Displaced solar orbits for spacecraft propelled by electric sails are investigated. Since the propulsive thrust is induced by the sail attitude, the orbital and attitude dynamics of electric-sail-based spacecraft are coupled and required to be investigated together. However, the coupled dynamics and control of electric sails have not been discussed in most published literatures. In this paper, the equilibrium point of the coupled dynamical system in displaced orbit is obtained, and its stability is analyzed through a linearization. The results of stability analysis show that only some of the orbits are marginally stable. For unstable displaced orbits, linear quadratic regulator is employed to control the coupled attitude-orbit system. Numerical simulations show that the proposed strategy can control the coupled system and a small torque can stabilize both the attitude and orbit. In order to generate the control force and torque, the voltage distribution problem is studied in an optimal framework. The numerical results show that the control force and torque of electric sail can be realized by adjusting the voltage distribution of charged tethers.

Quasinormal modes of nonlinear electromagnetic black holes from unstable null geodesics

The expressions for the quasinormal modes (QNMs) of black holes with nonlinear electrodynamics, calculated in the eikonal approximation, are presented. In the eikonal limit QNMs of black holes are determined by the parameters of the circular null geodesics. The unstable circular null orbits are derived from the effective metric that is the one obeyed by light rays under the influence of a nonlinear electromagnetic field. As an illustration we calculate the QNMs of four nonlinear electromagnetic black holes, two singular and two regular, namely from Euler-Heisenberg and Born-Infeld theories, for singular, and the magnetic Bardeen black hole and the one derived by Bronnikov for regular ones. Comparison is shown with the QNMs of the linear electromagnetic counterpart, their Reissner-Nordstr\"{o}m black hole.

Escape and trapping of low-frequency gravitationally lensed rays by compact objects within plasma

We consider the gravitational lensing of rays emitted by a compact object (CO) within a distribution of plasma with power-law density $\propto r^{-h}$. For the simplest case of a cloud of spherically symmetric cold non-magnetized plasma, the diverging effect of the plasma and the converging effect of gravitational lensing compete with one another. When $h<2$, the plasma effect dominates over the vacuum Schwarzschild curvature, potentially shifting the radius of the unstable circular photon orbit outside the surface of the CO. When this occurs, we define two relatively narrow radio-frequency bands in which plasma effects are particularly significant. Rays in the escape window have $\omega_{0} < \omega \leq \omega_{+}$ and are free to propagate to infinity from the CO surface. To a distant observer, the visible portion of the CO surface appears to shrink as the observed frequency is reduced, and vanishes entirely at $\omega_{0}$, in excess of the plasma frequency at the CO surface. We define the anomalous propagation window for frequencies $\omega_{-}< \omega \leq \omega_{0}$. Rays emitted from the CO surface within this frequency range are dominated by optical effects from the plasma and curve back to the surface of the CO, effectively cloaking the star from distant observers. We conclude with a study of neutron star (NS) compactness ratios for a variety of nuclear matter equations of state (EoS). For $h=1$, NSs generated from stiff EoS should display significant frequency dependence in the EW, and lower values of $h$ with softer EoS can also show these effects.

Global embeddings and hydrodynamic properties of Kerr black hole

In the presence of a rotating Kerr black hole, we investigate hydrodynamics of the massive particles and massless photons to construct relations among number density, pressure and internal energy density of the massive particles and photons around the rotating Kerr black hole and to study an accretion onto the black hole. On equatorial plane of the Kerr black hole, we investigate the bound orbits of the massive particles and photons around the black hole to produce their radial, azimuthal and precession frequencies. With these frequencies, we study the black holes GRO J1655-40 and 4U 1543-47 to explicitly obtain the radial, azimuthal and precession frequencies of the massive particles in the flow of perfect fluid. We next consider the massive particles in the stable circular orbit of radius of 1.0 ly around the supernovas SN 1979C, SN 1987A and SN 2213-1745 in the Kerr curved spacetime, and around the potential supermassive Schwarzschild black holes M87, NGC 3115, NGC 4594, NGC 3377, NGC 4258, M31, M32 and Galatic center, to estimate their radial and azimuthal frequencies, which are shown to be the same results as those in no precession motion. The photon unstable orbit is also discussed in terms of the impact parameter of the photon trajectory. Finally, on the equatorial plane of the Kerr black hole, we construct the global flat embedding structures possessing (9 + 3) dimensionalities outside and inside the event horizon of the rotating Kerr black hole. Moreover, on the plane, we investigate the warp products of the Kerr spacetime.

Lux in obscuro: photon orbits of extremal black holes revisited

It has been shown in the literature that the event horizon of an asymptotically flat extremal Reissner–Nordström black hole is also a stable photon sphere. We further clarify this statement and give a general proof that this holds for a large class of static spherically symmetric black hole spacetimes with an extremal horizon. In contrast, in the Doran frame, an asymptotically flat extremal Kerr black hole has an unstable photon orbit on the equatorial plane of its horizon. In addition, we show that an asymptotically flat extremal Kerr–Newman black hole exhibits two equatorial photon orbits if , one of which is on the extremal horizon in the Doran frame and is stable, whereas the second one outside the horizon is unstable. For , there is only one equatorial photon orbit, located on the extremal horizon, and it is unstable. There can be no photon orbit on the horizon of a non-extremal Kerr–Newman black hole.

Spherical FSO receivers for UAV communication: geometric coverage models

We present several analytical models to calculate the intersection surface of a spherical cap with a solid cone where either one of them is experiencing perturbations in its attitude. These models are developed to quantitatively analyze the mutual alignment, and thus performance, of mobile free-space optical (FSO) communication nodes used in air-to-ground and ground-to-air FSO links. Same models can be also applied to various geometrical alignment problems in the presence of instability. Examples include coverage area of imaging, radar, and optical communication satellites while orbiting unstable orbits. These models provide a closed-form direct mapping between platform dynamics and FSO link performance facilitating a priori performance prediction and easing feedback requirements in real-time tracking systems.