Manifolds Of Orbits Research Articles

Predictive Framework for Flow Reversals and Excursions in Turbulence.

We present a dynamical framework for intermittent reversals and excursions (R&Es) of large-scale circulations in turbulence. We show that R&Es can occur when turbulent trajectories in phase space shadow invariant manifolds of certain unstable periodic orbits (UPOs). Consequently, substantial flow reorganization and extreme fluctuations in flow metrics observed during R&Es can be reconstructed by splicing the unstable manifolds of such dynamically relevant UPOs. Using this geometrical framework, we predict imminent R&Es and preemptively avert these extreme events using closed-loop control.

Semi-analytical computation of bifurcation of orbits near collinear libration point in the restricted three-body problem

A unified semi-analytical solution is presented for constructing the phase space near collinear libration points in the Circular Restricted Three-body Problem (CRTBP), encompassing Lissajous orbits, quasihalo orbits, Axial orbits, and their invariant manifolds, as well as transit and non-transit orbits. Based on classical in-plane and out-of-plane frequency resonance mechanisms, the Lindstedt–Poincaré method could only derive separate analytical solutions for the invariant manifolds of Lissajous orbits and halo orbits, falling short for the invariant manifolds of quasihalo orbits. In this paper, by introducing a coupling coefficient η and a bifurcation equation, a unified series solution for these orbits is systematically developed using a coupling-induced bifurcation mechanism and Lindstedt–Poincaré method. Analyzing the bifurcation equation obtained from different coupling forms reveals bifurcation conditions for all kinds of orbits near collinear libration points. When η = 0, the series solution describes non-bifurcated orbits, while when η ≠ 0, the solution describes bifurcated orbits, including quasihalo orbits, Axial orbits, and their invariant manifolds, as well as newly bifurcated transit and non-transit orbits. This unified semi-analytical framework provides a more comprehensive understanding of the complex phase space structures near collinear libration points in the CRTBP.

A computational methodology for invariant manifold connections between quasi-periodic libration point orbits in non-autonomous problems

Semi-analytical and numerical techniques to systematically analyze and compute natural connections between quasi-periodic orbits associated to non-autonomous systems are considered. Focusing in the non-autonomous Sun–Earth+Moon coherent QBCP model, the center-unstable and center-stable invariant manifolds of Lyapunov quasi-periodic orbits are parameterized and a methodology to detect heteroclinic connections between manifolds is introduced. The methodology aims to decrease the dimensionality of the problem and to address the issue of searching good initial conditions inside the high dimensional state space. Then, connections are identified by searching for approximate patch points in a state-time Poincaré section. These points are subsequently refined to obtain smooth paths that are further continued inside their families.

Invariant Manifolds of Degenerate Tori and Double Parabolic Orbits to Infinity in the $$(n+2)$$-Body Problem

Invariant Manifolds of Degenerate Tori and Double Parabolic Orbits to Infinity in the $$(n+2)$$-Body Problem

Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

We study the relation between two special classes of Riemannian Lie groups [Formula: see text] with a left-invariant metric [Formula: see text]: The Einstein Lie groups, defined by the condition [Formula: see text], and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups [Formula: see text] are not geodesic orbit manifolds, thus providing large-scale answers to a relevant question of Nikonorov. Our approach involves studying and characterizing the [Formula: see text]-invariant geodesic orbit metrics on Lie groups [Formula: see text] for a wide class of subgroups [Formula: see text] that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds.

Unveiling the Electronic Structure of Pseudotetragonal WO3 Thin Films.

WO3 is a 5d compound that undergoes several structural transitions in its bulk form. Its versatility is well-documented, with a wide range of applications, such as flexopiezoelectricity, electrochromism, gating-induced phase transitions, and its ability to improve the performance of Li-based batteries. The synthesis of WO3 thin films holds promise in stabilizing electronic phases for practical applications. However, despite its potential, the electronic structure of this material remains experimentally unexplored. Furthermore, its thermal instability limits its use in certain technological devices. Here, we employ tensile strain to stabilize WO3 thin films, which we call the pseudotetragonal phase, and investigate its electronic structure using a combination of photoelectron spectroscopy and density functional theory calculations. This study reveals the Fermiology of the system, notably identifying significant energy splittings between different orbital manifolds arising from atomic distortions. These splittings, along with the system's thermal stability, offer a potential avenue for controlling inter- and intraband scattering for electronic applications.

Transfers between Jupiter–Ganymede and Jupiter–Europa resonant tori in a concentric circular restricted 4-body model

The phenomenon of mean-motion resonance overlapping, which generates heteroclinic connections between unstable resonant orbits, is known to be crucial for the generation of large-scale chaos and global instability in celestial systems. This instability in turn can be profitably leveraged for the purposes of low-energy space mission trajectory design. In most prior work leveraging such phenomena, though, connections were found between orbits resonant with the same moon; however, both Jupiter and Saturn have multiple moons, each having their own set of mean motion resonances. This study will present some results from a search for trajectories between orbits resonant with different Jovian moons, namely Ganymede and Europa. For this, the appropriate model to use is a concentric circular restricted 4-body model, where most unstable mean-motion resonant orbits correspond to 2D tori in a 5D extended phase space. We use GPU-assisted tools inspired by collision detection algorithms from computer graphics to rapidly find near-intersections of Jupiter–Ganymede and Jupiter–Europa resonant orbit manifolds, which correspond to potential transfers. We find the Jupiter–Ganymede 4:3 resonance to be the most promising candidate for trajectories to the key 3:4 Jupiter–Europa resonance, and thus further investigate the properties of 4:3 Jupiter–Ganymede orbits.

Photogenerated Ni(I)-Bipyridine Halide Complexes: Structure-Function Relationships for Competitive C(sp2)-Cl Oxidative Addition and Dimerization Reactivity Pathways.

We report the facile photochemical generation of a library of Ni(I)-bpy halide complexes (Ni(I)(Rbpy)X (R = t-Bu, H, MeOOC; X = Cl, Br, I) and benchmark their relative reactivity toward competitive oxidative addition and off-cycle dimerization pathways. Structure-function relationships between the ligand set and reactivity are developed, with particular emphasis on rationalizing previously uncharacterized ligand-controlled reactivity toward high energy and challenging C(sp2)-Cl bonds. Through a dual Hammett and computational analysis, the mechanism of the formal oxidative addition is found to proceed through an SNAr-type pathway, consisting of a nucleophilic two-electron transfer between the Ni(I) 3d(z2) orbital and the Caryl-Cl σ* orbital, which contrasts the mechanism previously observed for activation of weaker C(sp2)-Br/I bonds. The bpy substituent provides a strong influence on reactivity, ultimately determining whether oxidative addition or dimerization even occurs. Here, we elucidate the origin of this substituent influence as arising from perturbations to the effective nuclear charge (Zeff) of the Ni(I) center. Electron donation to the metal decreases Zeff, which leads to a significant destabilization of the entire 3d orbital manifold. Decreasing the 3d(z2) electron binding energies leads to a powerful two-electron donor to activate strong C(sp2)-Cl bonds. These changes also prove to have an analogous effect on dimerization, with decreases in Zeff leading to more rapid dimerization. Ligand-induced modulation of Zeff and the 3d(z2) orbital energy is thus a tunable target by which the reactivity of Ni(I) complexes can be altered, providing a direct route to stimulate reactivity with even stronger C-X bonds and potentially unveiling new ways to accomplish Ni-mediated photocatalytic cycles.

Transfer from Lunar Gateway to Sun-Earth Halo Orbits Using Solar Sails

To extend the usability of solar sails in the sun–Earth–moon system, we analyze the transfer trajectories from the 9:2 Earth–moon near-rectilinear halo orbit (NRHO) to halo orbits around the sun–Earth L1 and L2 points under the assumption of a future mission for a solar sail spacecraft equipped with a solar electric propulsion (SEP) system deployed from the Lunar Orbital Platform-Gateway. The dynamics are modeled using the bicircular restricted four-body problem, where the gravitational forces from the sun, Earth, and moon as well as solar radiation pressure (SRP) are considered. We propose a trajectory design method that utilizes both SRP and SEP. The method consists of initial guess generation and optimization steps. The initial guess generation comprises the forward propagation of the escape trajectory from the NRHO, the backward propagation of the stable manifold of the target halo orbits, and their apoapsis patching process. Optimization is conducted to minimize propellant consumption by effectively controlling SRP. We perform optimizations with various parameters, namely, the sail area-to-mass ratio (), specifications of SEP, target sun–Earth halo orbit, and departure direction. The results validate the proposed trajectory design method and verify that solar sail acceleration can reduce the necessary amount of propellant, which indicates that such missions can be realized by small CubeSats.

Dressed jeff-1/2 objects in mixed-valence lacunar spinel molybdates

The lacunar-spinel chalcogenides exhibit magnetic centers in the form of transition-metal tetrahedra. On the basis of density-functional computations, the electronic ground state of an Mo413+ tetrahedron has been postulated as single-configuration a12 e4 t25, where a1, e, and t2 are symmetry-adapted linear combinations of single-site Mo t2g atomic orbitals. Here we unveil the many-body tetramer wave-function: we show that sizable correlations yield a weight of only 62% for the a12 e4 t25 configuration. While spin–orbit coupling within the peculiar valence orbital manifold is still effective, the expectation value of the spin–orbit operator and the g factors deviate from figures describing nominal t5jeff = 1/2 moments. As such, our data documents the dressing of a spin–orbit jeff = 1/2 object with intra-tetramer excitations. Our results on the internal degrees of freedom of these magnetic moments provide a solid theoretical starting point in addressing the intriguing phase transitions observed at low temperatures in these materials.

Heteroclinic bifurcation in a quasi-periodically excited rigid rocking block with two frequencies

Heteroclinic bifurcation in a nonlinear rigid rocking block under external quasi-periodic excitation with two frequencies is investigated. By using the method of Melnikov type, we derive sufficient conditions under which the perturbed stable and unstable manifolds of the heteroclinic orbits intersect transversally. Then a complete description of the bifurcation sets and the chaotic zones in the parameter space are presented. Numerical simulations are performed for parameters chosen from the chaotic zones. The chaotic motions of the system are verified by computing the largest Lyapunov exponent of the system.

Analytical Nonlinear Observability Analysis for Spacecraft Autonomous Relative Navigation

Spacecraft autonomous relative navigation is widely required in the field of on-orbit services. In previous works, it has been shown that the monocular sequential images (MSIs) is a cost-effective autonomous relative navigation method. However, given the lack of size and initial orbit of noncooperative targets, the relative navigation system (RNS) with MSIs cannot ensure to estimate the entire relative motion states during the whole observation process. It is critical to perform an online observability analysis for RNS with MSIs to ensure autonomous relative motion estimation validity and accuracy. An analytical nonlinear observability analysis method is proposed in this article to improve the computational efficiency. We study the attributes of the relative orbit dynamics model and Lie algebra operations, and find that the observability matrix is only related to the limited states. The relationship among those states, rank, and the Frobenius norm of observability matrix is demonstrated, which simplify the observability criteria conditions and the degree measurements in analytical forms. In addition, based on the proposed method, we analyze the influence of the orbital manifolds on the observability degree of RNS with MSIs. Several suggestions are proposed for orbital manifolds detecting. The conclusion is a theoretical supplement to the past numerical research results. Finally, we implement the numerical simulations, and the current study’s findings verified the effectiveness of the proposed observability analysis method.

Identifying topological superconductivity in two-dimensional transition-metal dichalcogenides

We study the superconducting pairing instabilities and gap functions for prototypical two-dimensional (2D) transition-metal dichalcogenides (TMDCs) WS$_2$, MoTe$_2$, and MoS$_2$ in the 2H phase under both hole and electron doping at 10 K. Our first-principles quantum many-body Green's function approach allows us to treat the full $d$ and $p$ manifold of orbitals with strong spin-orbit coupling, yielding pairing predictions with material specific detail. The resulting gap functions exhibit a variety of mixed-parity superconducting states, including $s$, $p$, $d$, $f$, $d\pm id$, and $p\pm ip$ pairing modes. In particular, we predict 3% and 4% hole-doped WS$_2$ to be a chiral $p\pm ip$ topological superconductor. For 1% hole-doped MoS$_2$, we find a competition between three doubly degenerate chiral and non-chiral instabilities. Overall, the relative pairing strengths are found to follow the Fermi surface topology, due to nesting between the Fermi surface sheets. Finally, we discuss our predictions in relation to available experimental data and classify the topology of the predicted superconducting pairing symmetries.

Regulating Singlet-Triplet Energy Gaps through Substituent-Driven Modulation of the Exchange and Coulomb Interactions.

Control of the singlet-triplet energy gap (ΔEST) is central to realizing productive energy conversion reactions, photochemical reaction trajectories, and emergent applications that exploit molecular spin physics. Despite this, no systematic methods have been defined to tune ΔEST in simple molecular frameworks, let alone by an approach that also holds chromophore size and electronic structural parameters (such as the HOMO-LUMO gap) constant. Using a combination of molecular design, photophysical and potentiometric experiments, and quantum chemical analyses, we show that the degree of electron-electron repulsion in excited singlet and triplet states may be finely controlled through the substitution pattern of a simple porphyrin absorber, enabling regulation of relative electronically excited singlet and triplet state energies by the designed restriction of the electron-electron Coulomb (J) and exchange (K) interaction magnitudes. This approach modulates the ΔEST magnitude by controlling the densities of state in the occupied and virtual molecular orbital manifolds, natural transition orbital polarization, and the relative contributions of one electron transitions involving select natural transition orbital pairs. This road map, which regulates electron density overlaps in the occupied and virtual states that define the singlet and triplet wave functions of these chromophores, enables new approaches to preserve excitation energy despite intersystem crossing.

Orbital Stability and Invariant Manifolds on Distant Retrograde Orbits around Ganymede and Nearby Higher-Period Orbits

In the past few years, distant retrograde orbits (DROs) have become increasingly popular due to their conspicuous stability. Nevertheless, it is this characteristic that results in the challenge to the design of transfer orbits into/out of DROs. This paper investigates the DROs around Ganymede in order to utilize their dynamical characteristics for Jupiter system exploration. In particular, the DRO family is calculated by numerical integration and numerical continuation, higher-period orbits near the DROs are detected using bifurcation theory, and characteristics including orbital stability and invariant manifolds of these orbits are investigated through stability indices and manifold theory. The stability of DROs and the higher-period orbits are first investigated in the circular restricted three-body problem and are then verified in a third-body gravitation perturbation model. The results show that the strong stability of DROs makes it possible to observe the Galilean moons for long periods and that the higher-period orbits that bifurcate from the DROs offer additional insight into the motion of probes approaching/departing from the vicinities of the DROs. Further investigation of the invariant manifolds around higher-period orbits reveals the feasibility of utilizing the DRO family and the nearby unstable structures for multi-target exploration and low-energy transfer between the Galilean moons.

Real- and momentum-space description of the excitons in bulk and monolayer chromium tri-halides

Excitons with large binding energies ~2–3 eV in CrX3 have been characterized as being localized (Frenkel) excitons that emerge from the atomic d − d transitions between the Cr-3d-t2g and eg orbitals. The argument has gathered strength in recent years as the excitons in recently made monolayers are found at almost the same energies as the bulk. The Laporte rule, which restricts such parity forbidden atomic transitions, can relax if a symmetry-breaking mechanism is present. While what can be classified as a purely Frenkel exciton is a matter of definition, we show using an advanced first principles parameter-free approach that these excitons in CrX3, in both its bulk and monolayer variants, have band origin and it is the dp hybridization between Cr and X that primarily acts as the symmetry-breaking mechanism that relaxes the Laporte rule. We show that the character of these excitons is mostly determined by the Cr-d orbital manifold, nevertheless, the fractions of the spectral weight shared with the ligand halogen states increases as the dp hybridization enhances. The hybridization enhances as the halogen atom becomes heavier, bringing the X-p states closer to the Cr-d states in the sequence Cl → Br → I, with an attendant increase in exciton intensity and a decrease in binding energy. By applying a range of different kinds of perturbations that qualitatively mimics the effects originating from the missing vertex in self-energy, we show that moderate changes to the two-particle Hamiltonian that essentially modifies the Cr-d-X-p hybridization, can alter both the intensities and positions of the exciton peaks. A detailed analysis of several deep-lying excitons, with and without strain, elucidates the fact that the exciton is most Frenkel-like in CrCl3 and CrBr3 and acquires mixed Frenkel–Wannier character in CrI3, making the excitons in CrI3 most susceptible to environmental screening and spin–orbit coupling.

Electronic versatility of vanadium in tris-chelates with redox-active ligands.

Spectroscopic and computational examination of the neutral tris-dioxolene complex [V(dbcat)3] (dbcat2- = 3,6-di-tert-butylcatecholate) reveals a Class III mixed-valent ground state. The radical is stabilised by delocalisation across the ligands mediated by the energy matched d orbital manifold of the V(V) centre. This electronic structure is compared to the tris-dithiolene and tris-diimine analogues that possess V(IV) and V(II) ions, respectively.

Impact of chaos on precursors of quantum criticality.

Excited-state quantum phase transitions (ESQPTs) are critical phenomena that generate singularities in the spectrum of quantum systems. For systems with a classical counterpart, these phenomena have their origin in the classical limit when the separatrix of an unstable periodic orbit divides phase space into different regions. Using a semiclassical theory of wave propagation based on the manifolds of unstable periodic orbits, we describe the quantum states associated with an ESQPT for the quantum standard map: a paradigmatic example of a kicked quantum system. Moreover, we show that finite-size precursors of ESQPTs shrink as chaos increases due to the disturbance of the system. This phenomenon is explained through destructive interference between principal homoclinic orbits.

Transfers from the Earth to L_2\xa0Halo orbits in the Earth\u2013Moon bicircular problem

This paper deals with direct transfers from the Earth to Halo orbits related to the translunar point. The gravitational influence of the Sun as a fourth body is taken under consideration by means of the Bicircular Problem (BCP), which is a periodic time dependent perturbation of the Restricted Three Body Problem (RTBP) that includes the direct effect of the Sun on the spacecraft. In this model, the Halo family is quasi-periodic. Here we show how the effect of the Sun bends the stable manifolds of the quasi-periodic Halo orbits in a way that allows for direct transfers.

Support vector machines for learning reactive islands.

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 the support vector machines are 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.