Families Of Periodic Orbits Research Articles

The dud canard: Existence of strong canard cycles in [formula omitted

In this paper, we provide a rigorous description of the birth of canard limit cycles in slow-fast systems in R3 through the folded saddle-node of type II and the singular Hopf bifurcation. In particular, we prove – in the analytic case only – that for all 0<ϵ≪1 there is a family of periodic orbits, born in the (singular) Hopf bifurcation and extending to O(1) cycles that follow the strong canard of the folded saddle-node. Our results can be seen as an extension of the canard explosion in R2, but in contrast to the planar case, the family of periodic orbits in R3 is not explosive. For this reason, we have chosen to call the phenomena in R3, the “dud canard”. The main difficulty of the proof lies in connecting the Hopf cycles with the canard cycles, since these are described in different scalings. As in R2, we use blowup to overcome this, but we also have to compensate for the lack of uniformity near the Hopf bifurcation, due to its singular nature; it is a zero-Hopf bifurcation in the limit ϵ=0. In the present paper, we do so by imposing analyticity of the vector-field. This allows us to prove existence of an invariant slow manifold, that is not normally hyperbolic.

Generating Symmetric Periodic Orbits and Their Optimal Maintenance in a Binary Asteroid System

In this paper, generating symmetric periodic orbits around a binary asteroid and optimal maintenance of spacecraft on these periodic orbits are investigated. Due to the irregular shape of asteroids, numerical methods are needed to generate periodic orbits. In this regard, the 32 families of symmetric periodic orbits were calculated by grid search and shooting methods utilizing the restricted three-body problem as a basic model. The irregularity of the shape of asteroids and solar radiation pressure are considered perturbations. These perturbations should be compensated by a thruster with a continuous thrust through the continuous control strategy, i.e., the linear quadratic regulator (LQR) method. The effect of different matrix coefficients in the performance of the LQR method is examined. The results show that the LQR method could appropriate control spacecraft on periodic orbits with specular shapes. Also, the practical feasibility of a continuous thrust control strategy is discussed.

The concentric Sitnikov problem: Circular case

A new configuration in the sense of concentric Sitnikov problem is introduced in this paper. This configuration consists of four primaries Pi(i=1,2,3,4) of masses mi, respectively. Here, all the primaries are placed on a straight line and moving in concentric circular orbits around their common center of mass under the restrictions of masses as m1=m2=m, m3=m4=ḿ and m>ḿ. The objective of the present manuscript is to study the existence of equilibrium points and their linear stability, first return map, periodic orbits, and N-R BoA in the proposed model. Under the restriction of mass parameter 0<μ́<1/4, it is found that there exist three equilibrium points and all the equilibrium points are linearly unstable for all values of mass parameter 0<μ́<1/4. Graphically, the nature of orbits have been examined with the help of first return map. The families of periodic orbits of infinitesimal mass m5 around the primaries and equilibrium points have been obtained for different values of mass parameter μ́. Finally, we explore the Newton–Raphson Basins of Attraction (N-R BoA), related to the equilibrium points in the concentric problem of Sitnikov.

Stability and Periodic Motions for a System Coupled with an Encapsulated Nonsmooth Dynamic Vibration Absorber

The dynamic vibration absorber (DVA) is widely used in engineering models with complex vibration modes. The research on the stability and periodic motions of the DVA model plays an important role in revealing its complex vibration modes and energy transfer. The aim of this paper is to study the stability and periodic motions of a two-degrees-of-freedom system coupled with an encapsulated nonsmooth dynamic vibration absorber under low-frequency forced excitation. Based on the slow–fast method, the model is transformed into a six-dimensional piecewise smooth system coupling two time scales. The existence and stability of the admissible equilibrium points for the model are discussed under different parameter conditions. Based on the first integrals, the Melnikov vector function of the nonsmooth dynamic vibration absorber model is calculated. The existence and number of periodic orbits bifurcated from a family of periodic orbits under different parameters are discussed. The phase diagram configuration of periodic orbits is given based on numerical simulation. The results obtained in this paper offer a new perspective for vibration analysis and parameter control for nonsmooth dynamic vibration absorbers.

Bifurcation cascade, self-similarity, and duality in the three-rotor problem.

The three-rotor system concerns equally massive point particles moving on a circle subject to attractive cosine potentials of strength g. The quantum theory models chains of coupled Josephson junctions. Classically, it displays order-chaos-order behavior with increasing energy E along with a seemingly globally chaotic phase for 5.33g≲E≲5.6g. It is also known to admit pendulum and isosceles breather families of periodic orbits at all energies. While pendula display a doubly infinite sequence of stability transitions accumulating at their libration to rotation threshold at E=4g, breathers undergo only one stability transition. Here, we show that these stability transitions are associated with forward and reverse fork-like isochronous and period-doubling bifurcations. The new family of periodic orbits born at each of these bifurcations is found using an efficient search algorithm starting from a transverse perturbation to the parent orbit. The graphs of stability indices of various classes of orbits born at pendulum bifurcations meet at E=4g forming "fans." The transitions in the librational and rotational phases are related by an asymptotic duality between bifurcation energies and shapes of newly born periodic orbits. The latter are captured by solutions to a Lamé equation. We also find and numerically validate values of scaling constants for self-similarity in (a) stability indices of librational and rotational pendula and (b) shapes of newly born orbits as E→4g. Finally, we argue that none of the infinitely many families of periodic orbits we have found is stable for 5.33g≲E≲5.6g, providing further evidence for global chaos in this energy band.

From Babylonian lunar observations to Floquet multipliers and Conley–Zehnder indices

In 1878, Hill found numerically, in his limiting case of the restricted three-body problem, the so-called Hill’s lunar problem, a planar direct periodic orbit with a period of one synodic month. By virtue of the spatial system’s invariance under a symplectic involution, whose fixed point set corresponds to the planar problem, we can assign to Hill’s orbit planar and spatial Floquet multipliers and planar and spatial Conley–Zehnder indices. We show that these have deep astronomical significance because, on the one hand, we relate the anomalistic month to the planar Floquet multipliers and the planar Conley–Zehnder index. On the other hand, we relate the draconitic month to the spatial Floquet multipliers and the spatial Conley–Zehnder index. Knowledge of this lunar month dates back to the Babylonians, who lived until around 500 BCE. In order to determine the indices, we analyze analytically the bifurcation procedure of the fundamental families of planar direct and retrograde periodic orbits (traditionally known as families g and f) from the rotating Kepler problem for very low energies.

Equilibrium points and Lyapunov families in the circular restricted three-body problem with an oblate primary and a synchronous rotating dipole secondary: Application to Luhman-16 binary system

We numerically study a version of the synchronous circular restricted three-body problem, where an infinitesimal mass body is moving under the Newtonian gravitational forces of two massive bodies. The primary body is an oblate spheroid while the secondary is an elongated asteroid of a combination of two equal masses forming a rotating dipole which is synchronous to the rotation of the primaries of the classic circular restricted three-body problem. In this paper, we systematically examine the existence, positions, and linear stability of the equilibrium points for various combinations of the model's parameters. We observe that the perturbing forces have significant effects on the positions and stability of the equilibrium points as well as the regions where the motion of the particle is allowed. The allowed regions of motion as determined by the zero-velocity surface and the corresponding isoenergetic curves as well as the positions of the equilibrium points are given. Finally, we numerically study the binary system Luhman-16 by computing the positions of the equilibria and their stability as well as the allowed regions of motion of the particle. The corresponding families of periodic orbits emanating from the collinear equilibrium points are computed along with their stability properties.

Time-frequency high-resolution for weak signal detection using chaotic intermittence

BackgroundA non-linear Liènard-type system based on trigonometric functions is designed and used as a highly sensitive signal detector. Based on the intermittence mechanism presented in some chaotic systems, an adaptive detection array built by five chaotic oscillators is fully tuned and tested. In the array, each oscillator identifies changes in frequency for non-stationary weak signals and gives an estimate of the timing at which intermittence occurred, allowing the representation of a time-frequency spectrogram even when high-level noise is present. MethodsThis work shows that the trigonometric-based system has an asymptotically stable limit cycle centered at the origin through two methods: first under the fixed-point analysis and later by the Melnikov method, which uses a family of periodic orbits. Once the limit cycle is assured, through an iterative method for searching and tuning parameters, the system is configured to drive itself into the route to chaos named intermittence. Then, the new system can be used as a non-stationary signal detector that simultaneously provides high-resolution time-frequency representation. ResultsComputer experiments showed that our trigonometric oscillator when working in the chaotic intermittence regime and configured as an adaptive array, overcomes the SNR threshold present in other second-order non-linear oscillators such as Duffing, Van der Pol, and Van der Pol-Duffing, achieving high detection rates under noisy conditions, and maintaining a relative error of around 1% with -35 dB SNR. ConclusionsOur oscillator provides better sensitivity to small signals as well as noise rejection, in comparison with traditional oscillators. Furthermore, our entire system formed by a Liènard-type five-oscillator array promises to detect low-power signals and their TF reconstruction with a minimum error.

The restricted three-body problem with logarithm potential

We present a study of the restricted three body problem with logarithm potential. We discuss equilibria, stability, Hill’s regions of motion and the families of periodic orbits near equilibria. Moreover, we show that equilibria and some periodic orbits continue in the logarithm three body problem.

A parametric design method of nanosatellite close-range formation for on-orbit target inspection

This paper proposes an efficient design method for nano satellites formation flying near a large space target to perform ultra-close inspection missions. A parametric model for periodic relative motion between two satellites is firstly proposed through a detailed analysis of the relative orbital dynamics. It is proved that the existing periodic solutions of satellite relative motion such as in-plane 2:1 elliptic and circular periodic relative orbits both belong to the ellipsoid family of periodic relative orbits. The motion planes and their locations and orientations of the general periodic relative orbits are then determined as the analytic functions of the initial relative states. The maximal and minimal distances from the relative orbit to the origin are further analytically calculated too. A formation design algorithm is then proposed for optimal observation of feature points of the target considering various requirements of collision avoidance and observable distance by using this parametric model. Numerical examples about target inspection are introduced to quantitively evaluate and verify the models and methods. The simulation results are well consistent with the theoretical predictions, showing that the design proposed can be potentially applied for future practical on-orbit service missions.

The Conley–Zehnder indices of the spatial Hill three-body problem

We explore the interaction between the Conley–Zehnder index and bifurcation points of symmetric planar as well as spatial periodic orbits in the spatial Hill three-body problem. We start with the fundamental families of planar periodic orbits which are those of direct (family g) and retrograde periodic orbits (family f). Since the spatial system is invariant under a symplectic involution, whose fixed point set corresponds to the planar problem, planar orbits have planar and spatial Floquet multipliers, and planar and spatial Conley–Zehnder indices. When the Floquet multipliers move through a root of unity, new families of periodic orbits bifurcate and the index jumps. For very low energies, the families g and f arise dynamically from the rotating Kepler problem, and in a recent work (Aydin From Babylonian lunar observations to Floquet multipliers and Conley-Zehnder Indices) we determined analytically their indices. By their numerical continuations for higher energies, we determine the index of various families of planar and spatial periodic orbits bifurcating from g and f. Since these families can bifurcate again and meet each other, this procedure can get complicated. This index leads to a grading on local Floer homology. Since the local Floer homology and its Euler characteristic stay invariant under bifurcation, the index provides important information about the interconnectedness of such families, which we illustrate in form of bifurcation graphs. Since the solutions of Hill’s system may serve as orbits for space mission design or astronomical observations, our results promote the interaction between Symplectic Geometry and practical problems.

The wave trace and Birkhoff billiards

The purpose of this article is to develop a Hadamard–Riesz-type parametrix for the wave propagator in bounded planar domains with smooth, strictly convex boundary. This parametrix then allows us to rederive an oscillatory integral representation for the wave trace appearing in the work by Marvizi and Melrose (1982) and compute its principal symbol explicitly in terms of geometric data associated to the billiard map. This results in new formulas for the wave invariants. The order of the principal symbol, which appears to be inconsistent in the works by Marvizi and Melrose (1982) and Popov (1994), is also corrected. In those papers, the principal symbol was never actually computed and to our knowledge, this paper contains the first explicit formulas for the principal symbol of the wave trace. The wave trace formulas we provide are localized near both simple lengths corresponding to nondegenerate periodic orbits and degenerate lengths associated to one parameter families of periodic orbits tangent to a single rational caustic. Existence of a Hadamard–Riesz-type parametrix with explicit symbol and phase calculations in the interior appears to be new in the literature, with the exception of the author’s previous work [J. Geom. Anal. 31 (2021), 2238–2295] in the special case of elliptical domains. This allows us to circumvent the symbol calculus used by Duistermaat and Guillemin (1975) and Hezari and Zelditch (2012) when computing trace formulas, which are instead derived from integrating our explicit parametrix over the diagonal.

Computer assisted proofs for transverse collision and near collision orbits in the restricted three body problem

This paper considers two point boundary value problems for conservative systems defined in multiple coordinate systems, and develops a flexible a posteriori framework for computer assisted existence proofs. Our framework is applied to the study collision and near collision orbits in the circular restricted three body problem. In this case the coordinate systems are the standard rotating coordinates, and the two Levi-Civita coordinate systems regularizing collisions with each of the massive primaries. The proposed framework is used to prove the existence of a number of orbits which have long been studied numerically in the celestial mechanics literature, but for which there are no existing analytical proofs at the mass and energy values considered here. These include transverse ejection/collisions from one primary body to the other, Strömgren's asymptotic periodic orbits (transverse homoclinics for L4,5), families of periodic orbits passing through collision, and orbits connecting L4 to ejection or collision.

NUMERICAL INVESTIGATIONS OF THE ORBITAL DYNAMICS AROUND A SYNCHRONOUS BINARY SYSTEM OF ASTEROIDS

In this article, equilibrium points and families of periodic orbits in the vicinity of the collinear equilibrium points of a binary asteroid system are investigated with respect to the angular velocity of the secondary body, the mass ratio of the system and the size of the secondary. We assume that the gravitational fields of the bodies are modeled considering the primary as a mass point and the secondary as a rotating mass dipole. This model allows to compute families of planar and halo periodic orbits that emanate from the equilibrium points L1 and L2. The stability and bifurcations of these families are analyzed and the results are compared with the results obtained with the restricted three-body problem (RTBP). The results provide an overview of the dynamical behavior in the vicinity of a binary asteroid system.

Differential algebra methods applied to continuous abacus generation and bifurcation detection: application to periodic families of the Earth–Moon system

The return of human space missions to the Moon puts the Earth–Moon system (EMS) at the center of attention. Hence, studying the periodic solutions to the circular restricted three-body problem (CR3BP) is crucial to ease transfer computations, find new solutions, or to better understand these orbits. This work proposes a novel continuation method of periodic families using differential algebra (DA) mapping. We exploit DA with automatic control of the truncation error to represent each family of periodic orbits as a set 2D Taylor polynomial maps. These maps guarantee the access to any point of the family without any numerical propagation, providing a continuous abacus. When applied to the halo family at $$L_1$$ and $$L_2$$ , the planar Lyapunov at $$L_1$$ and $$L_2$$ , the distant retrograde orbit (DRO) family, and the butterfly family, we show that the DA-based 2D mapping is asymptotically more efficient than point-wise methods by at least two orders of magnitude, with controlled accuracy. To assist the computation of family of periodic orbits, we propose a novel DA-based automatic bifurcation detection algorithm that enables the continuous mapping of the family’s bifurcation criteria. A bifurcation study on the halo $$L_2$$ shows identical results as point-wise methods while highlighting two undocumented bifurcations.

On the restricted (N + 1)-body problem on surfaces of constant curvature

In this paper, we consider a restricted (N+1)-body problem on surfaces Mκ2, where the constant κ≠0 is the Gaussian curvature, which by means of a rescaling can be reduced to κ=±1. This problem consists in the study of the dynamics of an infinitesimal mass particle attracted by N primaries of identical masses describing elliptic relative equilibria of the N-body problem on Mκ2, i.e., a solution where the primaries are rotating uniformly with constant angular velocity ω on a fixed parallel of S2 or H2 and placed at the vertices of a regular polygon. In a rotating frame, this problem yields a two degrees of freedom Hamiltonian system. The goal of this paper is to describe analytically some dynamics features for κ=±1. Precisely, we study the relative location of equilibria, obtaining, in particular, that the poles of S2 and vertex of H2 represent equilibrium points for any value of the parameters. Thus, analysis of the nonlinear stability of these equilibria is carried out, as well as two types of bifurcations are detected: Hamiltonian-Hopf and N-bifurcation. Additionally, we prove the existence of a family of Hill's orbits and a family of periodic orbits when the primaries are located near the poles of S2 or the vertex of H2. Finally we prove the existence of KAM 2-tori related to these periodic orbits.

Evolution of the characteristic curves in the restricted three-body problem in terms of the mass parameter

In this work, we study the evolution of the families of simple symmetric periodic orbits in the restricted three-body problem whatever the value of the mass parameter mu . To classify these characteristic curves, we introduce a topological characterization of both orbits and families. Starting from the work of Strömgren for the Copenhagen case, we analyze the evolution of these families, when the mass parameter mu varies in (0, 1/2], focusing on their topological characterization, the existence of asymptotic points and the appearance of certain types of orbits such as horseshoe orbits. Lastly, we consider two samples, the Earth–Moon and Sun–Jupiter systems and classify the different types of orbits for these systems.

Model-free continuation of periodic orbits in certain nonlinear systems using continuous-time adaptive control

This paper generalizes recent results by the authors on noninvasive model-reference adaptive control designs for control-based continuation of periodic orbits in periodically excited linear systems with matched uncertainties to a larger class of periodically excited nonlinear systems with matched uncertainties and known structure. A candidate adaptive feedback design is also proposed in the case of scalar problems with unmodeled nonlinearities. In the former case, rigorous analysis shows guaranteed performance bounds for the associated prediction and estimation errors. Together with an assumption of persistent excitation, there follows asymptotic convergence to periodic responses determined uniquely by an a priori unknown periodic reference input and independent of initial conditions, as required by the control-based continuation paradigm. In particular, when the reference input equals the sought periodic response, the steady-state control input vanishes. Identical conclusions follow for the case of scalar dynamics with unmodeled nonlinearities, albeit with slow rates of convergence. Numerical simulations validate the theoretical predictions for individual parameter values. Integration with the software package coco demonstrates successful continuation along families of stable and unstable periodic orbits with a minimum of parameter tuning. The results expand the envelope of known noninvasive feedback strategies for use in experimental model validation and engineering design.

On the retrograde planar co-orbital asteroid motion with Jupiter

AbstractWe study the motion of an asteroid being in retrograde 1/1 resonance with Jupiter (co-orbital motion). We consider the planar case (i=180°) and Jupiter is on a circular or elliptic orbit (e′ = 0.048). In the cirular model we compute families of symmetric periodic orbits and their stability type. In the elliptic model we have isolated periodic orbits which affect the orbital modes of motion as it is shown by the FLI dynamical maps.

The phase space structure of retrograde mean motion resonances with Neptune: the 4/5, 7/9, 5/8 and 8/13 cases

We compute planar and three-dimensional retrograde periodic orbits in the vicinity of the restricted three-body problem (RTBP) with the Sun and Neptune as primaries and we concentrate on the dynamics of higher-order exterior mean motion resonances with Neptune. By using the circular planar model as the basic model, families of retrograde symmetric periodic orbits are computed at the 4/5, 7/9, 5/8 and 8/13 resonances. We determine the bifurcation points from the planar circular to the planar elliptic problem and we find all the corresponding families. In order to obtain a global view of the families of periodic orbits, the eccentricity of the primaries takes values in the whole interval 0<e'<1. Then, we find all the possible vertical critical orbits (v.c.o) of the planar circular problem and we proceed to the three-dimensional circular restricted 3-body problem. In this model, retrograde periodic orbits are generated mainly from the retrograde v.c.o. Also, if we continue families of direct orbits for i>90^circ , then we can obtain families of 3D symmetric retrograde periodic orbits. The linear stability is examined too. Stable periodic orbits are associated with phase space domains of resonant motion where TNOs can be captured. In order to study the phase space structure of the above resonances, we construct dynamical stability maps for the whole inclination interval (0<i<180^circ ) by using the well-known “MEGNO Chaos Indicator”. Finally, we discuss about TNOs which are currently located at these resonances.