Exact Orbits Research Articles

Traveling front in a diffusive predator–prey model with Beddington–DeAngelis functional response

In this paper, we consider a singular diffusive predator–prey model with Beddington–DeAngelis functional response, employing geometric singular perturbation theory and Bendixson's criteria. Our investigation revolves around transforming the reaction–diffusion equation into a multi‐scale four‐dimensional slow–fast system with two different orders of small parameters. Through once singular perturbation analysis, our focus shifts towards exploring the existence of heteroclinic orbits in a three‐dimensional system. We analyze these dynamics through the perspective of the Fisher–KPP equation in two limit cases. In the first case, only the normal to the two‐dimensional slow manifold is unstable. This allows for the deduction of existence of heteroclinic orbits in the three‐dimensional system through investigating the dynamics on the two‐dimensional slow manifold. Consequently, we obtain both monotonic traveling fronts and non‐monotonic fronts with oscillatory tails. In the second case, the normal to the one‐dimensional slow manifold exhibits both stable and unstable directions, then it is impossible to restrict the dynamics of the three‐dimensional system entirely to the slow manifold. Instead, we integrate the slow orbits of the reduced system with the fast orbits of the layer system to construct a singular heteroclinic orbit. According to Fenichel's theorem, we discover the existence of exact heteroclinic orbits of three‐dimensional system and derive the monotonic traveling fronts under weaker parameter conditions. Additionally, we also discuss the nonexistence of traveling fronts. Finally, we demonstrate our theoretical results with numerical simulations.

SIMULACIÓN DE LA INYECCIÓN DE UN HAZ NEUTRO EN TOKAMAKS CONTRIANGULARIDAD POSITIVA Y NEGATIVA

In this paper the dynamics of high energy charged particles in negative and positive triangularity tokamaks is through the simulation of a neutral beam injection on analytical MHD equilibria using the exact orbit code FOCUS. It is observed that the negative triangularity configuration presents fewer losses of non-thermal particles, and an energy deposition more localized than its counterpart.

Effect of normal magnetic field on the exact coherent state of channel flow at large Reynolds number

Laminar-turbulent subcritical transition has been always a hot issue in fluid mechanics. Exact coherent states are important for predicting the path of transition and understanding the cycle of turbulent self-sustaining process. One of the most common methods of investigating subcritical transition is dynamical system method and many different forms of invariant solutions have been obtained in many shear flows. In order to study the effect of magnetic field on the exact coherent state of channel flow at large Reynolds number, the direct numerical simulation combined with bisection method is used to calculate the exact coherent state-periodic orbit solution with different magnetic field strength at Re ⩾ 9000, and the structure and morphology in the flow field are compared. The results show that the magnetic field strength does not change the structure and shape of the solution and the scaling law between the transition threshold and Reynolds number does not change significantly, which shows obviously self-similarity. As the magnetic field strength increases, the period of the exact coherent state decreases and the perturbation energy in each direction exhibits periodical oscillation changes. In addition the shape of the amplitude curve also changes due to the magnetic field effect. The above results show that the magnetic field has a certain inhibition effect on the disturbance at large Reynolds number, and the flow field remains relatively stable.

The radiometric environment for Mars limb observations by the Mars Sample Return Earth Return Orbiter

After launching from the martian surface via the Mars Ascent Vehicle (MAV), the MAV and the Orbiting Sample (OS) capsule containing the samples collected on Mars by the Perseverance rover are to be identified by the Narrow Angle Camera (NAC) on the Earth Return Orbiter (ERO) spacecraft in order to determine the exact orbit of the capsule before rendezvous. To ensure detection of the OS, noise and straylight contributions to the NAC must be well characterized. Here, we assess the radiometric environment at Mars likely to be encountered by the NAC—from the surface through the middle atmosphere—using the High Resolution Stereo Camera (HRSC) onboard Mars Express (MEx) and the Mars Climate Sounder (MCS) onboard the Mars Reconnaissance Orbiter (MRO). The results show that the radiance values in general tend to increase as phase angle increases, as the season progresses from Ls = 60° to Ls = 230°, and as altitude decreases. We compare HRSC and MCS profiles where observing conditions were similar and find good agreement. At specific latitudes, high-altitude aerosols are present in 1–5% of observations and significantly increase the worst-case radiance contribution above 50 km. We construct envelope profiles from the maximum radiances at 5 km intervals from 0 to 90 km that provide important input for straylight calculations of the NAC and for the validation of models that may be used as input for straylight calculations.

Is every pseudo-orbit of some homeomorphism near an exact orbit of a nearby homeomorphism?

We propose a relaxed notion of shadowing. In particular, for a homeomorphism f on a metric space, we ask whether every approximate orbit is near some actual orbit of some nearby system. We distinguish the cases where the nearby homeomorphism is close or arbitrarily close to f. We prove the relations between these notions and ordinary shadowing and present various examples. We finally discuss an application, a \(C^0\)-closing lemma for chain recurrent points of a homeomorphism on a topological manifold, not necessarily compact. This result leads to a characterization of explosion phenomena for various recurrent sets on such manifolds.

The anatomy of Boris type solvers and the Lie operator formalism for deriving large time-step magnetic field integrators

This work gives a Lie operator derivation of various Boris solvers via a detailed study of trajectory errors in a constant magnetic field. These errors in the gyrocenter location and the gyroradius are the foundational basis for why Boris solvers existed, independent of any finite-difference schemes. This work shows that there are two distinct ways of eliminating these errors so that the trajectory of a charged particle in a constant magnetic field is exactly on the cyclotron orbit. One way reproduces the known second-order symmetric Boris solver. The other yields a previously unknown, but also on-orbit solver, not derivable from finite-difference schemes. By revisiting some historical calculations, it is found that many publications do not distinguish the poorly behaved leap-frog Boris solver from the symmetric second-order Boris algorithm. This symmetric second-order Boris solver's trajectory is much more accurate and remains close to the exact orbit in a combined nonuniform electric and magnetic field at time-steps greater than the cyclotron period. Finally, this operator formalism showed that Buneman's cycloid fitting scheme is mathematically identical to Boris' on-orbit solver and that Boris' E-B splitting is unnecessary.

Weakly shadowable vector fields on non-oriented surfaces

We say that a vector field has the weakly shadowing property if for any there exists such that for every d-pseudo orbit there exists an exact orbit whose -neighbourhood containing the pseudo orbit. It is proved in Li Ming and Zhongjie Liu [Weak shadowing property for flows on oriented surfaces, Proc. Amer. Math. Soc. 145(6) (2017), pp. 2591–2605.] that vector fields in the C 1-interior of the set of vector fields on an oriented smooth closed surface having the weakly shadowing property are structurally stable. In this paper, we show that the above conclusion does not hold for non-oriented surfaces. Precisely, we construct a non--stable vector field on the Klein bottle which has the weakly shadowing property robustly.

First-principles study of magnetic disordering and alloying effects on phase stability and elastic constants of Co<sub>2</sub>Cr<i>Z</i> (<i>Z</i> = Ga, Si, Ge) alloys

Using the exact Muffin-Tin orbital method combined with the coherent potential approximation, the effects of magnetic disordering and alloying effects on the phase stability of L2<sub>1</sub>- and D0<sub>22</sub>-Co<sub>2</sub>Cr<i>Z</i> (<i>Z</i> = Ga, Si, Ge) alloys are systematically investigated at 0 K in the present work. It is shown that at 0 K, the lattice parameter, bulk modulus, magnetic moments, and elastic constants of the studied L2<sub>1</sub> alloys are in line with the available theoretical and experimental data. In the ferromagnetic state, these alloys possess L2<sub>1</sub> structure; with the magnetic disordering degree (<i>y</i>) increasing, the energy of the phase increases relatively and finally turns from lower than D0<sub>22</sub> phase to higher than D0<sub>22</sub> phase. As a result, when <i>y</i> ≥ 0.1 (0.2), then <i>Z</i> = Si and Ge (<i>Z</i> = Ga) alloys are stabilized by the D0<sub>22</sub> phase. With <i>y</i> increasing, the tetragonal shear elastic modulus (<i>C</i><i>'</i> = (<i>C</i><sub>11</sub>–<i>C</i><sub>12</sub>)/2) also turns soft, indicating that the magnetic disorderingis conducive to the lattice tetragonal deformation in the three alloys from both the energetic view and the mechanical view. The electronic origination of the magnetic disordering effect on the stabilities of the L2<sub>1</sub> and D0<sub>22</sub> phases can be ascribed to the Jahn-Teller instability effect. In the FM L2<sub>1</sub>-Co<sub>2</sub>CrGa<sub>1–<i>x</i></sub>Si<sub><i>x</i></sub> and L2<sub>1</sub>-Co<sub>2</sub>CrGa<sub>1–<i>x</i></sub>Ge<sub><i>x</i></sub> quaternary alloys, with <i>x</i> increasing, the total magnetic moment increases monotonically according to the Slater-Pauling rule, and <i>C</i><i>'</i> also stiffens, reflecting that the adding of Si and Ge can promote the mechanical stability of L2<sub>1</sub>-Co<sub>2</sub>CrGa alloy, thereby depressing the lattice tetragonal deformation.

Anomalous Sun Flyby of 1I/2017 U1 (`Oumuamua)

The findings of Micheli et al. (Nature2018, 559, 223–226) that 1I/2017 U1 (`Oumuamua) showed anomalous orbital accelerations have motivated us to apply an impact model of gravity in search for an explanation. A small deviation from the 1/r potential, where r is the heliocentric distance, is expected for the gravitational interaction of extended bodies as a consequence of this model. This modification of the potential results from an offset of the effective gravitational centre from the geometric centre of a spherically symmetric body. Applied to anomalous Earth flybys, the model accounts for energy gains relative to an exact Kepler orbit and an increased speed of several spacecraft. In addition, the flat rotation profiles of eight disk galaxies could be explained, as well as the anomalous perihelion advances of the inner planets and the asteroid Icarus. The solution in the case of `Oumuamua is also based on the proposal that the offset leads to an approach and flyby trajectory different from a Kepler orbit without postulating cometary activity. As a consequence, an adjustment of the potential and centrifugal orbital energies can be envisaged outside the narrow uncertainty ranges of the published post-perihelion data without a need to re-analyse the original data. The observed anomalous acceleration has been modelled with respect to the orbit solutions JPL 16 and “Pseudo-MPEC” for 1I/`Oumuamua.

Invariant states in inclined layer convection. Part 2. Bifurcations and connections between branches of invariant states

Convection in a layer inclined against gravity is a thermally driven non-equilibrium system, in which both buoyancy and shear forces drive spatio-temporally complex flow. As a function of the strength of thermal driving and the angle of inclination, a multitude of convection patterns is observed in experiments and numerical simulations. Several observed patterns have been linked to exact invariant states of the fully nonlinear 3D Oberbeck-Boussinesq equations. These exact equilibria, traveling waves and periodic orbits reside in state space and, depending on their stability properties, are transiently visited by the dynamics or act as attractors. To explain the dependence of observed convection patterns on control parameters, we study the parameter dependence of the state space structure. Specifically, we identify the bifurcations that modify the existence, stability and connectivity of invariant states. We numerically continue exact invariant states underlying spatially periodic convection patterns at Pr=1.07 under changing control parameters for temperature difference between the walls and inclination angle. The resulting state branches cover various inclinations from horizontal layer convection to vertical layer convection and beyond. The collection of all computed branches represents an extensive bifurcation network connecting 16 different invariant states across control parameter values. Individual bifurcation structures are discussed in detail and related to the observed complex dynamics of individual convection patterns. Together, the bifurcations and associated state branches indicate at what control parameter values which invariant states coexist. This provides a nonlinear framework to explain the multitude of complex flow dynamics arising in inclined layer convection.

Invariant states in inclined layer convection. Part 1. Temporal transitions along dynamical connections between invariant states

Thermal convection in an inclined layer between two parallel walls kept at different fixed temperatures is studied for fixed Prandtl number Pr=1.07. Depending on the angle of inclination and the imposed temperature difference, the flow exhibits a large variety of self-organized spatio-temporal convection patterns. Close to onset, these patterns have been explained in terms of linear stability analysis of primary and secondary flow states. At larger temperature difference, far beyond onset, experiments and simulations show complex, dynamically evolving patterns that are not described by stability analysis and remain to be explained. Here we employ a dynamical systems approach. We construct stable and unstable exact invariant states, including equilibria and periodic orbits of the fully nonlinear three-dimensional Oberbeck-Boussinesq equations. These invariant states underlie the observed convection patterns beyond their onset. We identify state-space trajectories that, starting from the unstable laminar flow, follow a sequence of dynamical connections between unstable invariant states until the dynamics approaches a stable attractor. Together, the network of dynamically connected invariant states mediates temporal transitions between coexisting invariant states and thereby supports the observed complex time-dependent dynamics in inclined layer convection.

Bounded flight and collision avoidance control for satellite clusters using intersatellite flight bounds

Abstract A novel satellite cluster control method by means of a set of artificial potential functions is proposed in this paper. To avoid the huge difficulty of designing target configuration for a large-scale satellite cluster, the proposed cluster control method relies on intersatellite flight bounds so that satellites can autonomously converge to a given boundary range, rather than a set of exact orbit configuration. The analytic relative motion bounds in terms of relative orbital elements are deduced and used as variables in artificial potential functions. The accumulated control effort is minute, that shows great potential in micro satellite cluster control. On the other hand, collision avoidance as a key requirement for satellite cluster, also gets attention in this paper. For a long-term perspective, the minimum intersatellite flight bound rather than the real-time relative distance between satellites is introduced into the repulsive artificial potential function. Satellite members can adjust their control effort in real time according to nearby satellites, which reflect the characteristic of swarm intelligence. Monte Carlo simulation reveals the superiority in fuel consumption of the proposed collision avoidance control.

A Terrestrial Validation of ICESat Elevation Measurements and Implications for Global Reanalyses

The primary goal of NASA's Ice, Cloud, and land Elevation Satellite (ICESat) mission was to detect centimeter-level changes in global ice sheet elevations at the spatial scale of individual ice streams. Confidence in detecting these small signals requires careful validation over time to characterize the uncertainty and stability of measured elevations. A common validation approach compares altimeter elevations to an independently characterized and stable reference surface. Using a digital elevation model (DEM) from geodetic surveys of one such surface, the salar de Uyuni in Bolivia, we show that ICESat elevations at this location have a 0.0-cm bias relative to the WGS84 ellipsoid, 4.0-cm (1-sigma) uncertainty overall, and 1.8-cm uncertainty under ideal conditions over short (50 km) profiles. We observe no elevation bias between ascending and descending orbits, but we do find that elevations measured immediately after transitions from low to high surface albedo may be negatively biased. Previous studies have reported intercampaign biases (ICBs) between various ICESat observation campaigns, but we find no statistically significant ICBs or ICB trends in our data. We do find a previously unreported 3.1-cm bias between ICESat's Laser 2 and Laser 3, and we find even larger interlaser biases in reanalyzed data from other studies. For an altimeter with an exact repeat orbit like ICESat, we also demonstrate that validation results with respect to averaged elevation profiles along a single ground track are comparable to results obtained using reference elevations from an in situ survey.

Evolution of relativistic thin discs with a finite ISCO stress – I. Stalled accretion

ABSTRACT We present solutions to the relativistic thin disc evolutionary equation using an α-model for the turbulent stress tensor. Solutions with a finite stress at the innermost stable circular orbit (ISCO) give rise to bolometric light curves with a shallow power-law time dependence, in good agreement with those observed in tidal disruption events. A self-similar model based on electron scattering opacity, for example, yields a power-law index of −11/14, as opposed to −19/16 for the case of zero ISCO stress. These solutions correspond to an extended period of relaxation of the evolving disc which, like the light curves they produce, is not sustainable indefinitely. Cumulative departures from the approximation of exact circular orbits cause the power-law index to evolve slowly with time, leading eventually to the steeper fall-off associated with traditional zero ISCO stress models. These modified solutions are discussed in detail in a companion paper.

The orbital Lense–Thirring precession in a strong field

We study the exact evolution of the orbital angular momentum of a massive particle in the gravitational field of a Kerr black hole. We show analytically that, for a wide class of orbits, the angular momentum's hodograph is always close to a circle. This applies to both bounded and unbounded orbits that do not end up in the black hole. Deviations from the circular shape do not exceed $\approx10\%$ and $\approx7\%$ for bounded and unbounded orbits, respectively. We also find that nutation provides an accurate approximation for those deviations, which fits the exact curve within $\sim 0.01\%$ for the orbits of maximal deviation. Remarkably, the more the deviation, the better the nutation approximates it. Thus, we demonstrate that the orbital Lense-Thirring precession, originally obtained in the weak-field limit, is also a valid description in the general case of (almost) arbitrary exact orbits. As a by-product, we also derive the parameters of unstable spherical timelike orbits as a function of their radii and arbitrary rotation parameter $a$ and Carter's constant $Q$. We verify our results numerically for all the kinds of orbits studied.

Relative periodic solutions in conducting liquid films flowing down vertical fibres

The dynamics of a conducting liquid film flowing down a cylindrical fibre, subjected to a radial electric field, is investigated using a long-wave model (Ding et al., J. Fluid Mech., vol. 752, 2014, p. 66). In this study, to account for the complicated interactions between droplets, we consider two large droplets in a periodic computational domain and find two distinct types of travelling wave solutions, which consist of either two identical droplets (type I) or two slightly different droplets (type II). Both are ‘relative’ equilibria, i.e. steady in a frame moving at their phase speed, and are stable in smaller domains when the electric field is weak. We also study relative periodic orbits, i.e. temporally recurrent dynamic solutions of the system. In the presence of the electric field, we show how these invariant solutions are linked to the dynamics, where the system can evolve into one of the steady travelling wave states, into an oscillatory state, or into a ‘singular structure’ (Wray et al., J. Fluid Mech., vol. 735, 2013, pp. 427–456; Ding et al., J. Fluid Mech., vol. 752, 2014, p. 66). We find that the oscillation between two similarly sized large droplets in the oscillatory state is well represented by relative periodic orbits. Varying the electric field strength, we demonstrate that relative periodic solutions arise as the dynamically important solution once the type-I or type-II travelling wave solutions lose stability. Oscillation can be either enhanced or impeded as the electric field’s strength increases. When the electric field is strong, no relative periodic solutions are found and a spike-like singular structure is observed. For the case where the electric field is not present, the oscillation is instead caused by the interaction between a large droplet and a nearby much smaller droplet. We show that this oscillation phenomenon originates from the instability of the type-I travelling wave solution in larger domains, and that the oscillatory state can again be represented by an exact relative periodic orbit. The relative periodic orbit solution is also compared with experimental study for this case. The present study demonstrates that the relative periodic solutions are better at capturing the wave speed and oscillatory dynamics than the travelling wave solutions in the unsteady flow regime.

Periodic and quasi-periodic orbit design based on the center manifold theory

This paper proposes a new numerical method for finding libration point orbits in the vicinity of collinear libration points in the circular restricted three-body problem. The main advantage of this method is that it requires neither an initial guess nor complex algebraic manipulations for finding both quasi-periodic and periodic orbits. The proposed method consists of two steps: center manifold design and differential correction. The first step provides a quasi-periodic orbit parametrized by a single parameter vector. In the second step, the parameter vector in the first step is used to obtain an exact periodic orbit. This method is applied to find periodic and quasi-periodic orbits in the Sun-Earth restricted three-body problem around the L1 and L2 libration points.

Periodic orbits of a conservative 2-DOF vibro-impact system by piecewise continuation: bifurcations and fractals

The exact periodic orbits of a conservative 2-degree-of-freedom vibro-impact system with stereo-mechanical impact model is studied using a piecewise continuation method. Feasible initial guesses are extracted from grazing solution points so that each branch of solution can be initiated smoothly from previously solved ones. Frequency-energy plots (FEPs) are produced where an emphasis is placed on enumerating potential bifurcations. Extra critical points are discovered on multiple-period duplicates of existing stable solution branches and lead to cascades of period-multiplying bifurcations. The results indicate that the system’s complete FEP can be viewed through a process of infinite fractals toward zero frequency where pseudo-periodic or chaotic responses are approached. Finally, it is shown both mathematically and through the comparison of FEPs that the impacting system can be represented explicitly as the extreme case of nonlinear systems with an odd-order polynomial internal force. It is thus proposed that as the counterpart to the superposition of linear normal modes, the free responses of a general conservative nonlinear system can be tracked via bifurcations from its nonlinear normal modes.

Exact Torus Knot Periodic Orbits and Homoclinic Orbits in a Class of Three-Dimensional Flows Generated by a Planar Cubic System

This paper considers a class of three-dimensional systems constructed by a rotating planar symmetric cubic vector field. To study its periodic orbits including homoclinic orbits, which may be knotted in space, we classify the types of periodic orbits and then calculate their exact parametric representations. Our study shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on three families of invariant tori. Numerical examples of [Formula: see text]-torus knot periodic orbits have also been provided to illustrate our theoretical results.

Gravitational radiation driven capture in unequal mass black hole encounters

The gravitational radiation driven capture (GR capture) between unequal mass black holes without spins has been investigated with numerical relativistic simulations. We adopt the parabolic approximation which assumes that the gravitational wave radiation from a weakly hyperbolic orbit is the same as that from the parabolic orbit having the same pericenter distance. Using the radiated energies from the parabolic orbit simulations, we have obtained the critical impact parameter ($b_{\rm crit}$) for the GR capture for weakly hyperbolic orbit as a function of initial energy. The most energetic encounters occur around the boundary between the direct merging and the fly-by orbits, and can emit several percent of initial total ADM energy at the peak. When the total mass is fixed, energy and angular momentum radiated in the case of unequal mass black holes are smaller than those of equal mass black holes having the same initial orbital angular momentum for the fly-by orbits. We have compared our results with two different Post-Newtonian (PN) approximations, the exact parabolic orbit (EPO) and PN corrected orbit (PNCO). We find that the agreement between the EPO and the numerical relativity breaks down for very close encounters ($\it{e.g.}$, $b_{\rm crit} \lesssim 100$ M), and it becomes worse for higher mass ratios. For instance, the critical impact parameters can differ by more than $50\%$ from those obtained in EPO if the relative velocity at infinity $v_{\infty}$ is larger than 0.1 for the mass ratio of $m_{1}/m_{2}=16$. The PNCO gives more consistent results than EPO, but it also underestimates the critical impact parameter for the GR capture at $b_{\rm crit} \lesssim40$ M.