Quasi-periodic Orbits Research Articles

Periodic and Quasi-Periodic Orbits in the Collinear Four-Body Problem: A Variational Analysis

This paper investigated the periodic and quasi-periodic orbits in the symmetric collinear four-body problem through a variational approach. We analyze the conditions under which homographic solutions minimize the action functional. We compute the minimal value of the action functional for these solutions and show that, for four equal masses organized in a linear configuration, these solutions are the minimizers of the action functional. Additionally, we employ numerical experiments using Poincaré sections to explore the existence and stability of periodic and quasi-periodic solutions within this dynamical system. Our results provide a deeper understanding of the variational principles in celestial mechanics and reveal complex dynamical behaviors, crucial for further studies in multi-body problems.

Optimization Over Families of Quasi-Periodic Orbits

Quasi-periodic orbit families in astrodynamics are usually studied from a global standpoint without much attention to the specific orbits which are computed. Instead, we focus on the computation of particular quasi-periodic orbits and develop tools to do so. These tools leverage the parametric structure of families of quasi-periodic orbits to treat orbits only as a set of orbit frequencies instead of states in phase space. We develop a retraction on the family of quasi-periodic orbits to precisely navigate through frequency space, allowing us to compute orbits with specific frequencies. The retraction allows for movements in arbitrary directions. To combat the effects of resonances which slice through frequency space we develop resonance avoidance methods which detect resonances during continuation procedures and change the step size accordingly. We also develop an augmented Newton’s method for root-finding and an augmented gradient descent method for unconstrained optimization over a family of quasi-periodic orbits. Lastly, we implement an augmented Lagrangian method to solve constrained optimization problems. We note that many of the tools developed here are applicable to a wider range of solutions defined implicitly by a system of equations, but focus on quasi-periodic orbits.

Family of 2:1 resonant quasi-periodic distant retrograde orbits in cislunar space

Given the current enthusiasm for lunar exploration, the 2:1 resonant distant retrograde orbit (DRO) in Earth-Moon space is of interest. To gain an in-depth understanding of the complex dynamic environment in cislunar space and provide more options for parking orbits, this paper investigates the existence of quasi-periodic orbits near the 2:1 resonant DRO in the circular restricted three-body problem (CR3BP). Firstly, the numerical computation approach, continuation strategy, and stability analysis method of quasi-periodic orbits are introduced. Then, addressing the primary challenges in the continuation progress, we have developed an adaptive continuation algorithm with automatic adjustment of the step size and the number of discrete points that characterize the invariant torus. Finally, two types of 2D quasi-DROs and their linear stability properties are explored. Using Poincaré sections, we investigated the boundaries of the maximum extent attainable by both 2D quasi-DRO families in the CR3BP at a specific Jacobi energy, confirming that both types of quasi-periodic families have reached their respective boundaries. The algorithm described in this paper is beneficial for facilitating the computation of quasi-periodic families and aids in discovering additional potential dynamical structures.

Gauss Equations for Local Action-Angle Orbital Elements in Cislunar Space

To track orbits in cislunar space, predict where they will naturally move over time, and identify where unbounded trajectories come from, one may consider local action-angle orbital elements—coordinates that relate a spacecraft state to specific trajectories and exist in approximately integrable regions of the Earth–moon circular restricted three-body problem (CR3BP). As local action-angle elements are a semi-analytical analog to two-body orbital elements, the theory is extended to allow for the study of arbitrary perturbations to the dynamics. Namely, we derive Gauss equations for an arbitrary perturbing acceleration—continuous or discrete—to the CR3BP. Examples for continuous thrust and impulsive ΔV are provided in cislunar space, i.e., around Earth–moon L1,2 in the CR3BP. Strategies for instantaneous maneuver design and transfers between quasi-periodic orbits and manifolds are developed.

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.

Strong nonlinear mixing evolutions within phononic frequency combs

Phononic frequency combs (PFCs) represent an emerging attractive nonlinear vibrational phenomenon characterized by equidistant spectral lines. Despite the extensive experimental studies, the complex nonlinear mixing nature of PFCs continues to present significant challenges in solving and investigating their complete dynamics, which is difficult to achieve by existing computational approaches. In this paper, the entire solution space within a representative PFC induced by a 1:2 internal resonance is elucidated by conducting continuation computations and numerical long-time integrations. The proposed continuation approach is achieved by integrating our developed semi-analytical residue-regulating homotopy method (RRHM) with a pseudo arc-length continuation technique. In this solution space, we unearth wide-range nonlinear evolutions including overlapping intervals between the periodic and quasi-periodic branches, abundant multivalued sub-intervals, cyclic-fold (CF) bifurcations, and torus-doubling (TD) routes to chaos. In addition, multiple coexistences of a chaotic attractor and a periodic orbit, a chaotic attractor and a quasi-periodic orbit, as well as a periodic orbit and three quasi-periodic orbits are identified. Furthermore, we meticulously dissect and distinguish non-smooth variations in PFC morphology, which manifest as multiple jumps in comb spacing as the excitation frequency is swept across. This study could serve as a general guide for a comprehensive exploration of PFC dynamics and can offer insights to inform and inspire related experimental studies.

A Multistep Method for Integration of Perturbed and Damped Second-Order ODE Systems

Based on the Ψ-functions series method, a new numerical integration method for perturbed and damped second-order systems of differential equations is presented. This multistep method is defined for variable step and variable order (VSVO) and maintains the good properties of the Ψ-functions series method. In addition, it incorporates a recurring algebraic procedure to calculate the algorithm’s coefficients, which facilitates its implementation on the computer. The construction of Ψ-functions and the Ψ-functions series method are presented to address the construction of both explicit and implicit multistep methods and a predictor–corrector method. Three problems analogous to those solved by the Ψ-functions series method are analyzed, contrasting the results obtained with the exact solution of the problem or with its first integral. The first example is the integration of a quasi-periodic orbit. The second example is a Structural Dynamics problem associated with an earthquake, and the third example studies an equatorial satellite with perturbation J2. This allows us to compare the good behavior of the new code with other prestige codes.

Data-driven model reduction for pipes conveying fluid via spectral submanifolds

Pipes conveying fluid have non-linearizable dynamics when the flow velocity is large enough. Developing analytical models that accurately capture these highly nonlinear and complex dynamics is challenging. Efficient analysis and predictions of these nonlinear dynamics are essential in the design and control of pipe systems in engineering applications. To address these challenges, here we construct low-dimensional reduced-order models in a data-driven setting. In particular, we perform model reduction on spectral submanifolds (SSMs) via SSMLearn. This open-source package automates the identification of low-dimensional SSMs and their associated reduced-order models (ROMs). Our reduction via SSMs treats pipes with various boundary conditions and flow velocities in a unified way. It achieves a significant reduction because the ROMs are two-dimensional, which enables efficient and even analytical predictions on the nonlinear vibration of the pipes, paving the way for real-time prediction of the non-linearizable dynamics. In addition, our SSM-based reduction shows remarkable extrapolation capability because it is built on invariant manifolds and normal form theory. We demonstrate that the ROMs trained on unforced responses can be directly extrapolated to make reliable predictions on forced vibrations under various excitation frequencies and amplitudes, including periodic and quasi-periodic orbits and their bifurcations.

Applications of knot theory to the detection of heteroclinic connections between quasi-periodic orbits

Heteroclinic connections represent unique opportunities for spacecraft to transfer between isoenergetic libration point orbits for zero deterministic ΔV expenditure. However, methods of detecting them can be limited, typically relying on human-in-the-loop or computationally intensive processes. In this paper we present a rapid and fully systematic method of detecting heteroclinic connections between quasi-periodic invariant tori by exploiting topological invariants found in knot theory. The approach is applied to the Earth–Moon, Sun–Earth, and Jupiter–Ganymede circular restricted three-body problems to demonstrate the robustness of this method in detecting heteroclinic connections between various quasi-periodic orbit families in restricted astrodynamical problems.

Similarity signature curves for forming periodic orbits in the Lorenz system

In this paper, the short-periodic orbits of the Lorenz system are systematically investigated by the aid of the similarity signature curve, and a novel method to find the short-period orbits of the Lorenz system is proposed. We derive the similarity invariants by the equivariant moving frame theory, and then the similarity signature curve occurs along with them. Our analysis shows that the trajectory of the Lorenz system can be described by two completely different states. One is a stable state where the trajectory rotates around an equilibrium point. The other is a mutation state where the trajectory transitions to another equilibrium point. In particular, the similarity signature curve of the Lorenz system presents a more regular behavior than its trajectories. Additionally, with the assistance of the sliding window method, the quasi-periodic orbits can be detected numerically. Furthermore, all periodic orbits with period p⩽8 in the Lorenz system are found, and their period lengths and symbol sequences are calculated.

Minimal Quantization Model in Pilot-Wave Hydrodynamics.

Investigating how classical systems may manifest dynamics analogous to those of quantum systems is a broad subject of fundamental interest. Walking droplets, which self-propel through a resonant interaction with their own wave field, provide a unique macroscopic realization of wave-particle duality that exhibits behaviors previously thought exclusive to quantum particles. Despite significant efforts, elucidating the precise origin and form of the wave-mediated forces responsible for the walker's quantumlike behavior remained elusive. Here, we demonstrate that, owing to wave interference, the force responsible for orbital quantization originates from waves excited near stationary points on the walker's past trajectory. Moreover, we derive a minimal model with the essential ingredients to capture quantized orbital dynamics, including quasiperiodic and chaotic orbits. Notably, this minimal model provides an explicit distinction between local forces, which account for the walker's preferred speed and wave-induced added mass, and spatiotemporal nonlocal forces responsible for quantization. The quantization mechanism revealed here is generic, and will thus play a role in other hydrodynamic quantum analogs.

Quasi-periodic orbits of small solar sails with time-varying attitude around Earth–Moon libration points

This paper proposes new quasi-periodic orbits around Earth–Moon collinear libration points using solar sails. By including the time-varying sail orientation in the linearized equations of motion for the circular restricted three-body problem (CR3BP), four types of quasi-periodic orbits (two types around L1 and two types around L2) were formulated. Among them, one type of orbit around L2 realizes a considerably small geometry variation while ensuring visibility from the Earth if (and only if) the sail acceleration due to solar radiation pressure is approximately of a certain magnitude, which is much smaller than that assumed in several previous studies. This means that only small solar sails can remain in the vicinity of L2 for a long time without propellant consumption. The orbits designed in the linearized CR3BP can be translated into nonlinear CR3BP and high-fidelity ephemeris models without losing geometrical characteristics. In this study, new quasi-periodic orbits are formulated, and their characteristics are discussed. Furthermore, their extendibility to higher-fidelity dynamic models was verified using numerical examples.

Non-linear dynamics of a test particle near the Lagrange points L4 and L5 (Earth-Moon and Sun-Earth case)

The two-bodies problem can be fully solved, and was solved by Kepler (1609) and Newton (1687). The general three-body problem is often given as an example of a mathematical problem that ‘can’t be solved’. So, there is no general analytical solution. This problem can be significant and a special case of this problem is the Circular Restricted Three-Body Problem (CRTBP), which can be applied to the Earth-Moon system with a spacecraft, the Sun-Earth system with an asteroid, etc. In this paper, let’s focus on the motion of a test particle near the triangular Lagrange points L4 and L5 in the Earth-Moon and the Sun-Earth systems. Studying the movement of an object around these points is especially important for space mission design. To generate a trajectory around these points, the non-linear equations of motion for the circular restricted three-body problem were numerically integrated into MATLAB® 2023 software and the results are presented in the plane (x, y) and the phase plane (x, vx) and (y, vy). By numerical orbit integration, it is possible to investigate what happens when the displacement is relatively large or short from the Lagrange points. Then the small astronomical body may vibrate around these points. The results in this paper are shown in the rotating and inertia axes. Various initial positions near the Lagrange points and velocities are used to produce various paths the test particle can take. The same examples of numerical studies of trajectories associated with Lagrange points are shown in the inertial and the rotating coordinates system and are discussed. From the results of the numerical tests performed in MATLAB® 2023, it is possible to saw that there are different types of periodic, quasi-periodic, and chaotic orbits

Study on Displaced Orbits Below the Moon’s South Pole Near L2 Point Based on Solar Sail

This work explores displaced orbits of solar sails below the Moon’s south pole, near the L2 libration point in the Earth-Moon system. Light pressure provides acceleration for displaced orbits. These orbits enable continuous communication and observation of the Moon’s south polar region, where a lunar base is planned. Linearized dynamic equations yield analytical solutions of displaced orbits, which are either quasi-periodic or periodic. Quasi-periodic orbits have varying altitudes of hundreds of kilometers and a period of about a year, while periodic orbits have fixed altitudes аnd the same period. A sliding mode controller maintains the orbits using sail attitude angles and reflectivity as control variables. With a reflective area to mass ratio of 18 m2/kg, the displaced heights of quasiperiodic and periodic orbits near L2 are 2010.38 km and 2210.06 km, respectively. Numerical simulations confirm the controller’s effectiveness for both orbit types

Dynamics investigation and chaos-based application of a novel no-equilibrium system with coexisting hidden attractors

This work presents a novel three-dimensional system with multiple types of coexisting attractors (including chaotic, periodic, quasi-periodic, and unbound divergent orbit) that are categorized as hidden attractors and investigates its dynamics via the Lyapunov exponent spectrum, phase diagrams, and basins of attraction. The mechanism of the emergence of chaos is explored through numerical simulation. Under some conditions, the periodic orbits embedded in the new system without equilibria are studied using the variational method, and effective symbolic coding with four letters is successfully established to classify all short cycles. The symbols proposed by the method are highly consistent with the calculated results. Furthermore, the analogous circuit implementation is executed to demonstrate the flexibility and validity of the new system that possesses coexisting attractors. Finally, chaos-based applications of the new system, including synchronization, offset boosting control, and image encryption, are presented to show system’s feasibility.

The stability around Chariklo and the confinement of its rings

Context. Chariklo has two narrow and dense rings, C1R and C2R, located at 391 km and 405 km, respectively. Aims. In the light of new stellar occultation data, we study the stability around Chariklo. We also analyse three confinement mechanisms that prevent the spreading of the rings, based on shepherd satellites in resonance with the edges of the rings. Methods. This study was performed through a set of numerical simulations and the Poincaré surface of section technique. Results. From the numerical simulation results, and from the current parameters referring to the shape of Chariklo, we verify that the inner edge of the stable region is much closer to Chariklo than the rings. The Poincaré surface of sections allows us to identify periodic and quasi-periodic orbits of the first kind, and also the resonant islands corresponding to the 1:2, 2:5, and 1:3 resonances. We construct a map of aeq versus eeq space that gives the location and width of the stable region and the 1:2, 2:5, and 1:3 resonances. Conclusions. We find that the first kind periodic orbit family can be responsible for a stable region whose location and size meet that of C1R, for specific values of the ring particle eccentricities. However, C2R is located in an unstable region if the width of the ring is assumed to be about 120 m. After analysing different systems, we propose that the best confinement mechanism is composed of three satellites: two satellites shepherding the inner edge of C1R and the outer edge of C2R, and the third satellite trapped in the 1:3 resonance.

Embedding phase reduction for fast-slow systems with noise-induced stochastic quasiperiodic orbits

Noise is ubiquitous in natural systems and can induce coherent stochastic oscillations which cannot occur in the absence of noise, or stochastic quasiperiodic orbits which are markedly different from the deterministic limit cycle. For fast-slow systems, noise acting on the fast subsystem can induce quasi-periodic orbits tuned by the slow variable, which were termed as self-induced stochastic resonance (SISR). In our previous work [Zhu et al., 2022 [46]], we proposed a phase reduction framework on the SISR-oscillator by the hybrid approximation on the stochastic quasiperiodic orbit and the estimation of the effective noise. In this work, we remove the restriction of the additive Gaussian form assumption on the effective noise, and propose a more natural dimensionality-reduction method which we refer to as embedding phase reduction. Different SISR-oscillators induced by varying noise intensities, which can be embedded into a limit cycle, can have a unified form of the reduced phase equation. The efficiency of the embedding phase reduction is validated by Monte Carlo simulations of the case of periodic forcing, where excellent agreement is achieved for the entrainment and phase drifting dynamics. The simplicity and generality of the proposed embedding phase reduction approach indicate its potential applications in practical experiments and relevance in biological systems.

Bifurcation and chaos in a discrete Holling–Tanner model with Beddington–DeAngelis functional response

The dynamics of a discrete Holling–Tanner model with Beddington–DeAngelis functional response is studied. The permanence and local stability of fixed points for the model are derived. The center manifold theorem and bifurcation theory are used to show that the model can undergo flip and Hopf bifurcations. Codimension-two bifurcation associated with 1:2 resonance is analyzed by applying the bifurcation theory. Numerical simulations are performed not only to verify the correctness of theoretical analysis but to explore complex dynamical behaviors such as period-6, 7, 10, 12 orbits, a cascade of period-doubling, quasi-periodic orbits, and the chaotic sets. The maximum Lyapunov exponents validate the chaotic dynamical behaviors of the system. The feedback control method is considered to stabilize the chaotic orbits. These complex dynamical behaviors imply that the coexistence of predator and prey may produce very complex patterns.

MODELLING THE EVOLUTION OF THE TWO-PLANETARY THREE-BODY SYSTEM OF VARIABLE MASSES

A classical non-stationary three-body problem with two bodies of variable mass moving around the third body on quasi-periodic orbits is considered. In addition to the Newtonian gravitational attraction, the bodies are acted on by the reactive forces arising due to anisotropic variation of the masses. We show that Newtonian’s formalism may be generalized to the case of variable masses and equations of motion are derived in terms of the osculating elements of aperiodic motion on quasiconic sections. As equations of motion are not integrable the perturbative method is applied with the perturbing forces expanded into power series in terms of eccentricities and inclinations which are assumed to be small. Averaging these equations over the mean longitudes of the bodies in the absence of a mean-motion resonances, we obtain the differential equations describing the evolution of orbital parameters over long period of time. We solve the evolution equations numerically and demonstrate that the mass change modify essentially the system evolution.

Local Orbital Elements for the Circular Restricted Three-Body Problem

The Keplerian orbital elements make up a set of parameters that uniquely describe a two-body trajectory. Once perturbations are imposed on two-body dynamics, one often studies the time evolution of the orbital elements. Moving in complexity beyond two-body perturbations [that is, studying the dynamics in the circular restricted three-body problem (CR3BP)], the Keplerian orbital elements are no longer well defined in certain regions of phase space, especially when the gravitational attractions of both the primary and secondary bodies have similar magnitudes, as occurs in the vicinity of the libration points. In this work, we define a generalization of orbit elements that can be applied in these regions and others. Specifically, we define a set of semi-analytical action-angle orbital elements that are defined locally about any bounded special solution in the CR3BP: equilibria, periodic orbits, and quasi-periodic orbits. Local action-angle orbital elements are defined using action-angle coordinates in the Birkhoff–Gustavson normal form about the bounded invariant manifold. We include detailed examples around the five equilibria in the Earth–Moon CR3BP.