Restricted Problem Research Articles

Shapes and Mass Variation Effects of the Bodies in the Generalized Elliptic Restricted 3-Body Problem

Shapes and Mass Variation Effects of the Bodies in the Generalized Elliptic Restricted 3-Body Problem

A Study of the Axisymmetric Restricted Five-Body Problem within the Frame of Variable Mass: The Concave Case

A Study of the Axisymmetric Restricted Five-Body Problem within the Frame of Variable Mass: The Concave Case

Equilibrium dynamics of the restricted 3-body problem with prolate primaries

In this paper, the equilibrium dynamics of the circular restricted (2+1)-body problem with two elongated main bodies is investigated. More precisely, the influence of the mass parameter and the non-sphericity parameter on the position and stability of the libration points of this characteristic configuration is studied through numerical and semi-analytical methods. In addition, the effect of these two parameters of the system on the fractality of the basins of convergence is also investigated, for this, two quantitative indicators are used: the boundary basin entropy and the uncertainty dimension. It is shown that when the two objects have the same value of the non-sphericity parameter, the appearance of new collinear and non-collinear libration points takes place. Compared to the classical circular restricted three-body problem, our numerical study reveals that when both primaries are prolate, some of the collinear libration points may be stable.

On the basins of convergence in the beyond-Newtonian spatial collinear circular restricted four-body problem with spinning primaries

In this paper, we present the analysis of the Newton–Raphson basins of convergence (NR-BoCs) linked with the relative equilibria (LPs) in the Eulerian configuration of a pseudo-Newtonian planar circular restricted problem of four bodies (SCCR4BP) with spinning primaries. In this configuration of the SCCR4BP, the primaries are laid out in a symmetrical collinear shape in which the two peripheral primaries are of equal mass. We derive the equations of motion by assuming that the system consists of three Kerr-like primaries to examine how the geometry of the basins of convergence (BoCs) associated with the LPs of SCCR4BP are affected by the first-order non-Newtonian contribution to the gravitational field. Additionally, we analyze the correlation among the regions of convergence, the required number of iterations, and the associated probability distributions. Finally, for the quantitative study of the basins of convergence, we measure the uncertainty of the respective basins via the basin entropy and the basin boundary entropy.

Research on the impact of corporate social responsibility information disclosure and corporate innovation in China

China started to implement a new system in 2008, asking fractional companies to make public their corporate social responsibility information disclosure. The financial and corporate governance data of Chinese A-share listed businesses from 2007 to 2016 was gathered in order to determine any potential relationships between the mandated corporate social responsibility disclosure system and company patents. According to the two opposite hypotheses of the information hypothesis and reputation hypothesis, the results show that the number of patents of companies surveyed has increased significantly after the implement of the policy of mandatory CSR disclosure. Additionally, after companys patents are separated into invention type patents and utility models, industrial design (non-invention type) patents, invention type patents significantly increase in comparison. This means that the information hypothesis has been supported more. CSR disclosure can effectively alleviate the information asymmetry existed in the internal of the enterprise, increase shareholders confidence, and reduce the problems of corporate financial restrictions. Several robustness tests have proved that the model and conclusions are both steady. This may provide a reference for the government to improve the mandatory disclosure policy further and help improve the capital market development in China.

Covering almost all the layers of the hypercube with multiplicities

Given a hypercube Qn:={0,1}n in Rn and k∈{0,…,n}, the k-th layer Qkn of Qn denotes the set of all points in Qn whose coordinates contain exactly k many ones. For a fixed t∈N and k∈{0,…,n}, let P∈R[x1,…,xn] be a polynomial that has zeroes of multiplicity at least t at all points of Qn∖Qkn, and P has zeros of multiplicity exactly t−1 at all points of Qkn. In this short note, we show thatdeg(P)≥max⁡{k,n−k}+2t−2. Matching the above lower bound we give an explicit construction of a family of hyperplanes H1,…,Hm in Rn, where m=max⁡{k,n−k}+2t−2, such that every point of Qkn will be covered exactly t−1 times, and every other point of Qn will be covered at least t times. Note that putting k=0 and t=1, we recover the much celebrated covering result of Alon and Füredi (1993) [1]. Using the above family of hyperplanes we disprove a conjecture of Venkitesh (2022) [23] on exactly covering symmetric subsets of hypercube Qn with hyperplanes. To prove the above results we have introduced a new measure of complexity of a subset of the hypercube called index complexity which we believe will be of independent interest.We also study a new interesting variant of the restricted sumset problem motivated by the ideas behind the proof of the above result.

Approximation algorithms for the restricted k-Chinese postman problems with penalties

Approximation algorithms for the restricted k-Chinese postman problems with penalties

Bifurcations and enumeration of central configurations of some planar restricted problems

Bifurcations and enumeration of central configurations of some planar restricted problems

Finding regions of bounded motion in binary asteroid environment using Lagrangian descriptors

Trajectory design in highly-perturbed environments like binary asteroids is challenging. It typically requires using realistic, non-autonomous dynamical models in which periodic solutions derived in autonomous systems vanish. In this work, Lagrangian descriptors are employed in the perturbed planar bi-elliptic restricted four-body problem to find regions of bounded motion over a finite horizon about Dimorphos, the secondary body of the (65803) Didymos binary system. Results show that Lagrangian descriptors successfully reveal phase space organizing structures both in the unperturbed and perturbed planar bi-elliptic restricted four-body problem. With no solar radiation pressure, regions of bounded motion are visually identified, so granting access to a vast selection of bounded orbits about Dimorphos. Conversely, the presence of solar radiation pressure breaks down the majority of structures, leading to a large region of unstable motion with rare exceptions. Lagrangian descriptors are computationally inexpensive dynamical indicators that could be conveniently applied to astrodynamics.

Deep reinforcement learning method for turbofan engine acceleration optimization problem within full flight envelope

In order to solve the multidimensional restricted optimization problem of aeroengine acceleration process and improve the acceleration performance of full envelope, a design method of aeroengine full envelope acceleration controller based on deep reinforcement learning is proposed. Firstly, the thermo-aerodynamic principle of the engine acceleration process is analyzed, and the mathematical analytical model of the multidimensional restricted optimization problem of the turbofan engine acceleration process is established. On this basis, the design method of acceleration controller based on twin delayed deep deterministic policy gradient is studied. The proportional integral controller is designed to pursue the asymptotic stability at the acceleration destination and stable control is achieved through switching. Additionally, the flight envelope is divided into regions by clustering to reduce the span of data changes, so as to enhance the network convergence ability of deep reinforcement learning. Finally, based on the similarity conversion method of equal engine inlet total temperature, the acceleration controller of full flight envelope is realized. The digital simulation results demonstrate that compared with the traditional proportional integral differential controller, the acceleration controller based on twin delayed deep deterministic policy gradient shortens the acceleration time up to 48.33%, and the control effect in a large flight envelope (H=0∼10 km, Ma=0∼1.6) has been verified.

A Dynamical Systems Approach to the Theory of Circumbinary Orbits in the Circular Restricted Problem

To better understand the orbital dynamics of exoplanets around close binary stars, i.e., circumbinary planets (CBPs), we applied techniques from dynamical systems theory to a physically motivated set of solutions in the Circular Restricted Three-Body Problem (CR3BP). We applied Floquet theory to characterize the linear dynamical behavior—static, oscillatory, or exponential—surrounding planar circumbinary periodic trajectories (limit cycles). We computed prograde and retrograde limit cycles and analyzed their geometries, stability bifurcations, and dynamical structures. Orbit and stability calculations are exact computations in the CR3BP and reproducible through the open-source Python package pyraa. The periodic trajectories (doi.org/10.5281/zenodo.7532982) produce a set of noncrossing, dynamically cool circumbinary orbits conducive to planetesimal growth. For mass ratios μ ∈ [0.01, 0.50], we found recurring features in the prograde families. These features include (1) an innermost near-circular trajectory, inside which solutions have resonant geometries, (2) an innermost stable trajectory (a c ≈ 1.61 − 1.85 a bin) characterized by a tangent bifurcating limit cycle, and (3) a region of dynamical instability (a ≈ 2.1 a bin; Δa ≈ 0.1 a bin), the exclusion zone, bounded by a pair of critically stable trajectories bifurcating limit cycles. The exterior boundary of the exclusion zone is consistent with prior determinations of a c around a circular binary. We validate our analytic results with N-body simulations and apply them to the Pluto–Charon system. The absence of detected CBPs in the inner stable region, between the prograde exclusion zone and a c , suggests that the exclusion zone may inhibit the inward migration of CBPs.

The circular restricted eight-body problem

The circular restricted eight-body problem

Dynamics of the restricted vortex problem with a honeycomb configuration

In this paper, we have investigated the restricted 14-vortex problem with a honeycomb configuration similar to the football surface pattern. First, we give a sufficient condition for the existence of honeycomb configurations and obtain the instability of configurations. Then, we establish the equation of motion of the tracer particle for the restricted 14-vortex problem and analyze the stability and distribution of equilibrium points and singular points. As can be seen from the global phase diagram of system, there are only four types of orbits: equilibrium points, homoclinic orbits, heteroclinic orbits, and periodic orbits.

Periodic Orbits Around the Triangular Points with Prolate Primaries

ABSTRACT Periodic orbits play a fundamental role in the study and deep understanding of the behavior of dynamical systems. In the current work, we investigated the periodic orbits around the triangular libration points of the restricted three-body problem. The equations of motion of the restricted problem are presented when both primaries are prolate triaxial. Periodic orbits around the triangular points are obtained and then illustrated graphically for some selected initial conditions and for the entire domain of the mass ratio μ, as well. The eccentricities of the periodic orbits are obtained and then represented graphically. It is observed that the periodic orbits about the triangular stationary points are elliptical, and the frequencies of short and long orbits of the periodic motion are influenced by the shape of the primary bodies. Furthermore, we found that the perturbing forces influence the period, the orientation, and the eccentricities of the short and long periodic orbits.

A Comparative study on generalized Manev potential and Newtonian potential in perturbed restricted four-body problem

A Comparative study on generalized Manev potential and Newtonian potential in perturbed restricted four-body problem

The Photogravitational Restricted Problem of 2 + 2 Bodies with Straight Segment and Oblate Spheroid

The Photogravitational Restricted Problem of 2 + 2 Bodies with Straight Segment and Oblate Spheroid

Equilibrium points and their stability in a new generalized planar version of the collinear restricted four-body problem

We introduce a new generalized planar version of the collinear restricted four-body problem. In this version, all three main bodies are located on the horizontal x-axis, while they have unequal masses. The dynamical system contains two free parameters related to the masses and positions of the main bodies. We investigate the equilibrium dynamics of the system by determining how these two free parameters affect the dynamical properties of the equilibria, such as their position and linear stability. By deploying some standard numerical methods combined with rigorous and systematic analysis, we studied the equilibrium dynamics of the system and revealed its properties regarding the nature of the libration points and the connections with the parameters of the system.

Volume of convex polytopes equals mixed volume of simplices

qaaa We provide a simple proof for the equality between the normalized volume of a convex polytope with m vertices and the mixed volume of m simplices and thus show the seemingly restrictive problem of computing the mixed volume of simplices is still at least as hard as computing volumes of convex polytopes in general.

Solving infinite-domain CSPs using the patchwork property

The constraint satisfaction problem (CSP) has important applications in computer science and AI. In particular, infinite-domain CSPs have been intensively used in subareas of AI such as spatio-temporal reasoning. Since constraint satisfaction is a computationally hard problem, much work has been devoted to identifying restricted problems that are efficiently solvable. One way of doing this is to restrict the interactions of variables and constraints, and a highly successful approach is to bound the treewidth of the underlying primal graph. Bodirsky & Dalmau (2013) [14] and Huang et al. (2013) [47] proved that CSP(Γ) can be solved in nf(w) time (where n is the size of the instance, w is the treewidth of the primal graph and f is a computable function) for certain classes of constraint languages Γ. We improve this bound to f(w)⋅nO(1), where the function f only depends on the language Γ, for CSPs whose basic relations have the patchwork property. Hence, such problems are fixed-parameter tractable and our algorithm is asymptotically faster than the previous ones. Additionally, our approach is not restricted to binary constraints, so it is applicable to a strictly larger class of problems than that of Huang et al. However, there exist natural problems that are covered by Bodirsky & Dalmau's algorithm but not by ours, and we begin investigating ways of generalising our results to larger families of languages. We also analyse our algorithm with respect to its running time and show that it is optimal (under the Exponential Time Hypothesis) for certain languages such as Allen's Interval Algebra.

Design of a Low-Energy Earth-Moon Flight Trajectory Using a Planar Auxiliary Problem

The paper presents a sufficiently simple technique for designing a low-energy flight trajectory of a spacecraft (SC) from the Earth to the Moon with insertion into a low circular orbit of the latter. The proposed technique is based on the solution and subsequent analysis of a special model problem, which is a variant of the restricted circular four-body problem (RC4BP) Earth-Sun-SC-Moon; for which it is assumed that the planes of the orbits of all considered bodies coincides. The planar motion of the center of mass of the SC relative to the Earth is considered as perturbed (Sun, Moon). To describe it, equations in osculating elements are used, obtained by using the method of variation of constants based on the analytical solution of the planar circular restricted problem of two bodies (RC2BP)—Earth-SC, for which the rotation of the main axes of the coordinate system (the main plane) is synchronized with the motion of the Sun. The trajectory problem of designing a SC flight from a low circular near-Earth orbit to a low circular selenocentric one (“full” motion model—a restricted four-body problem (R4BP), an ephemeris model) is considered as an optimization one in the impulse formulation. The solution of the main problem is carried out in few (three) stages, on each the appropriate solution of the current variant of the auxiliary problem is determined, which is subsequently used as the basis of the initial approximation to the main one.