The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses $1-μ$ and $μ$ that circle each other with period equal to $2π$. For small $μ$, a resonant periodic motion of the massless particle in the rotating frame can be described by relatively prime integers $p$ and $q$, if its period around the heavier primary is approximately $2πp/q$, and by its approximate eccentricity $e$. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions. We prove the validity of this formal development and the existence of homoclinic points in the resonant region.
In the study of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with $q/p$ equal to 2/1, 7/3, 5/2, 3/1, and 4/1.
Another application is to the existence of asymmetric librations in the exterior resonances. We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2 resonances for any $μ$ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist for $μ$ small enough in the unaveraged restricted three-body problem.
The set of transverse homoclinic intersections for a saddle-focus equilibrium in the planar equilateral restricted four-body problem admit certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection -- or homoclinic web. In the present work, the planar restricted four-body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view. Starting from the vertical Lyapunov families emanating from saddle focus equilibria, we compute the s…
The present work studies the robustness of certain basic homoclinic motions in an equilateral restricted four body problem. The problem can be viewed as a two parameter family of conservative autonomous vector fields. The main tools are numerical continuation techniques for homoclinic and periodic orbits, as well as formal series methods for computing normal forms and center stable/unstable manifold parameterizations. After careful numerical study of a number of special cases we formulate several conjectures about the global bifurcations of the homoclinic families.
We describe a method for computing an atlas for the stable or unstable manifold attached to an equilibrium point, and implement the method for the saddle-focus libration points of the planar equilateral restricted four body problem. We employ the method at the maximally symmetric case of equal masses, where we compute atlases for both the stable and unstable manifolds. The resulting atlases are comprised of thousands of individual chart maps, with each chart represented by a two variable Taylor polynomial. Post-processing the atlas data yields approximate intersections of the invariant manifol…
We prove the existence of chaotic motions in a planar restricted four body problem, establishing that the system is not integrable. The idea of the proof is to verify the hypotheses of a topological forcing theorem. The forcing theorem applies to two freedom Hamiltonian systems where the stable and unstable manifolds of a saddle-focus equilibrium intersect transversally in the energy level set of the equilibrium. We develop a mathematically rigorous computer assisted argument which verifies the hypotheses of the forcing theorem, and we implement our approach for the restricted four body proble…
We study the continuation of periodic orbits from various compound of homoclinics in classical system. Together with the homoclinics, the periodic orbits make up a $C^1$-smooth, normally hyperbolic invariant cylinder with holes. It plays a key role to cross multiple resonant point.
We present a method for proving the existence of symmetric periodic, heteroclinic or homoclinic orbits in dynamical systems with the reversing symmetry. As an application we show that the Planar Restricted Circular Three Body Problem (PCR3BP) corresponding to the Sun-Jupiter-Oterma system possesses an infinite number of symmetric periodic orbits and homoclinic orbits to the Lyapunov orbits. Moreover, we show the existence of symbolic dynamics on six symbols for PCR3BP and the possibility of resonance transitions of the comet. This extends earlier results by Wilczak and Zgliczynski - Heteroclin…
We consider a homoclinic orbit to a saddle fixed point of an arbitrary $C^\infty$ map $f$ on $\mathbb{R}^2$ and study the phenomenon that $f$ has an infinite family of asymptotically stable, single-round periodic solutions. From classical theory, this requires $f$ to have a homoclinic tangency. We show it also necessary for $f$ to satisfy a `global resonance' condition and for the eigenvalues associated with the fixed point, $λ$ and $σ$, to satisfy $|λσ| = 1$. The phenomenon is codimension-three in the case $λσ= -1$, but codimension-four in the case $λσ= 1$ because here the coefficients of the…
In this paper, a dc-dc boost converter with sliding mode control and washout filter is analysed. This device is modelled as a three-dimensional Filippov system, characterized by the existence of sliding movement and restricted to the switching manifold. The operating point of the boost converter is a pseudo-equilibrium, and it, undergoes a subcritical Hopf bifurcation. Such a bifurcation occurs in the sliding vector field and creates, in this field, an unstable limit cycle. The limit cycle is confined to the switching manifold and disappears when it touches the visible-invisible two-fold point…
We study the orbit behavior of a four dimensional smooth symplectic diffeomorphism $f$ near a homoclinic orbit $Γ$ to an 1-elliptic fixed point under some natural genericity assumptions. 1-elliptic fixed point has two real eigenvalues out of unit circle and two others on the unit circle. Thus there is a smooth 2-dimensional center manifold $W^c$ where the restriction of the diffeomorphism has the elliptic fixed point supposed to be generic (no strong resonances and first Birkhoff coefficient is nonzero). Moser's theorem guarantees the existence of a positive measure set of KAM invariant curves…
We develop computer assisted arguments for proving the existence of transverse homoclinic connecting orbits, and apply these arguments for a number of non-perturbative parameter and energy values in the spatial equilateral circular restricted four body problem. The idea is to formulate the desired connecting orbits as solutions of certain two point boundary value problems for orbit segments which originate and terminate on the local stable/unstable manifolds attached to a periodic orbit. These boundary value problems are studied via a Newton-Kantorovich argument in an appropriate Cartesian pro…
The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries. If the primaries perform circular motions and the massless body is coplanar with them, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In synodic coordinates, it is a two degrees of freedom Hamiltonian system with five critical points, L1,..,L5, called the Lagrange points. The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and is located beyond the largest of the two. In this paper …
The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies called the primaries. If one assumes that the primaries perform circular motions and that all three bodies are coplanar, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In rotating coordinates, it can be modeled by a two degrees of freedom Hamiltonian, which has five critical points called the Lagrange points L1,.., L5. The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and beyond the largest of the two. …
In this paper we study the dynamics near the equilibrium point of a family of Hamiltonian systems in the neighborhood of a $0^2 iw$ resonance. The existence of a family of periodic orbits surrounding the equilibrium is well-known and we show here the existence of homoclinic connections with several loops for every periodic orbit close to the origin, except the origin itself. To prove this result, we first show a Hamiltonian normal form theorem inspired by the Elphick-Tirapegui-Brachet-Coullet-Iooss normal form. We then use a local Hamiltonian normalization relying on a result of Moser. We obta…
Chaotic phenomena in generic unfoldings of the Hamilton Hopf bifurcation with emphasis on the restricted planar circular 3-body problem beyond the Gascheau-Routh mass ratio
We present a method for establishing invariant manifolds for saddle--center fixed points. The method is based on cone conditions, suitably formulated to allow for application in computer assisted proofs, and does not require rigorous integration of the vector field in order to prove the existence of the invariant manifolds. We apply our method to the restricted three body problem and show that for a given choice of the mass parameter, there exists a homoclinic orbit to one of the libration points.
This paper introduces techniques of symplectic topology to the study of homoclinic orbits in Hamiltonian systems. The main result is a strong generalization of homoclinic existence results due to Sere and to Coti-Zelati, Ekeland and Sere, which were obtained by variational methods. Our existence result uses a modification of a construction due to Mohnke (originally in the context of Legendrian chords), and an energy--capacity inequality of Chekanov. In essence, we show the existence of a homoclinic orbit by showing a certain Lagrangian embedding cannot exist. We consider a (possibly time depen…
Global resonance is a mechanism by which a homoclinic tangency of a smooth map can have infinitely many asymptotically stable, single-round periodic solutions. To understand the bifurcation structure one would expect to see near such a tangency, in this paper we study one-parameter perturbations of typical globally resonant homoclinic tangencies. We assume the tangencies are formed by the stable and unstable manifolds of saddle fixed points of two-dimensional maps. We show the perturbations display two infinite sequences of bifurcations, one saddle-node the other period-doubling, between which…
We consider the planar circular restricted three-body problem (PCRTBP), as a model for the motion of a spacecraft relative to the Earth-Moon system. We focus on the Lagrange equilibrium points $L_1$ and $L_2$. There are families of Lyapunov periodic orbits around either $L_1$ or $L_2$, forming Lyapunov manifolds. There also exist homoclinic orbits to the Lyapunov manifolds around either $L_1$ or $L_2$, as well as heteroclinic orbits between the Lyapunov manifold around $L_1$ and the one around $L_2$. The motion along the homoclinic/heteroclinic orbits can be described via the scattering map, w…
We consider a Hamiltonian system with 2 degrees of freedom, with a hyperbolic equilibrium point having a loop or homoclinic orbit (or, alternatively, two hyperbolic equilibrium points connected by a heteroclinic orbit), as a step towards understanding the behavior of nearly-integrable Hamiltonians near double resonances. We provide a constructive approach to study whether the unstable and stable invariant manifolds of the hyperbolic point intersect transversely along the loop, inside their common energy level. For the system considered, we establish a necessary and sufficient condition for the…
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 be…
The attractors of a dynamical system govern its typical long-term behaviour. The presence of many attractors is significant as it means the behaviour is heavily dependent on the initial conditions. To understand how large numbers of attractors can coexist, in this thesis we study the occurrence of infinitely many stable single-round periodic solutions associated with homoclinic connections in two-dimensional maps. We show this phenomenon has a relatively high codimension requiring a homoclinic tangency and `global resonance', as has been described previously in the area-preserving setting. How…
We consider a restricted four-body problem on the dynamics of a massless particle under the gravitational force produced by three mass points forming an equilateral triangle configuration. We assume that the mass m3 of one primary is very small compared with the other two, m1 and m2, and we study the Hamiltonian system describing the motion of the massless particle in a neighborhood of m3. In a similar way to Hill approximation of the lunar problem, we perform a symplectic scaling, sending the two massive bodies to infinity, expanding the potential as a power series in m3, and taking the limit…
The elliptic isosceles restricted three body problem (REI3BP) models the motion of a massless body under the influence of the Newtonian gravitational force caused by two other bodies called the primaries. The primaries of masses $m_{1}=m_{2}$ move along a degenerate Keplerian elliptic collision orbit (on a line) under their gravitational attraction, whereas the third, massless particle, moves on the plane perpendicular to their line of motion and passing through the center of mass of the primaries. By symmetry, the component of the angular momentum $G$ of the massless particle along the direct…
The planetary restricted three-body problem (RTBP) is considered. The primary mass M is much more than another masses mj, i=1..N, which revolve around M. The massless probe particle m moves on elliptic orbit, is perturbed by mj. It is well known, that equations of motion of restricted 3-body problem in rotating rectangular frame may be reduced to the second order differential equation with periodic coefficients (Hill equation). It gives the ability to estimate width and position of the unstable zones. Moreover, the dependence of the position of instable zones from orbital eccentricity of the p…
The restricted planar elliptic three body problem (RPETBP) describes the motion of a massless particle (a comet) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this paper is to show the existence of orbits whose angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive orbits, that is, with a large variation of angular momentum. The leading idea of the proof consists in analysing parabolic motions of the …
In this paper we consider the circular restricted three body problem which models the motion of a masless body under the influence of the Newtionan gravitational force caused by two other bodies, the primaries, which move along cicular planar Keplerian orbits. In a suitable system of coordinates, this system has two degrees of freedom and the conserved energy is usually called the Jacobi constant. In 1980, J. Llibre and C. Simó proved the existence of oscillatory motions for the restricted planar circular three body problem, that is, of orbits which leave every bounded region but which return …
Stochastic behavior along mean motion resonances in the restricted planar 3-body problem
Universitat de Barcelona（巴塞罗那大学）；Centre de Recerca Matemàtica（数学研究中心）；Institute of Science and Technology, Austria（奥地利科学技术研究所）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
We study the dynamics of the restricted planar three-body problem near mean motion resonances, i.e. a resonance involving the Keplerian periods of the two lighter bodies revolving around the most massive one. This problem is often used to model Sun--Jupiter--asteroid systems. For the primaries (Sun and Jupiter), we pick a realistic mass ratio $μ=10^{-3}$ and a small eccentricity $e_0>0$. The main result is a construction of a variety of non local diffusing orbits which show a drastic change of the osculating (instant) eccentricity of the asteroid, while the osculating semi major axis is kept a…
Consider the spatial three-body problem, in the regime where one body revolves far away around the other two, in space, the masses of the bodies being arbitrary but fixed; in this regime, there are no resonances in mean motions. The so-called secular dynamics governs the slow evolution of the Keplerian ellipses. We show that it contains a horseshoe and all the chaotic dynamics which goes along with it, corresponding to motions along which the eccentricity of the inner ellipse undergoes large, random excursions. The proof goes through the surprisingly explicit computation of the homoclinic solu…
We present a diffusion mechanism for time-dependent perturbations of autonomous Hamiltonian systems introduced in [25]. This mechanism is based on shadowing of pseudo-orbits generated by two dynamics: an `outer dynamics', given by homoclinic trajectories to a normally hyperbolic invariant manifold, and an `inner dynamics', given by the restriction to that manifold. On the inner dynamics the only assumption is that it preserves area. Unlike other approaches, [25] does not rely on the KAM theory and/or Aubry-Mather theory to establish the existence of diffusion. Moreover, it does not require to …
We propose a model for local dynamics of a perturbed convex real-analytic Liouville-integrable Hamiltonian system near a resonance of multiplicity $1+m, m\geq 0$. Physically, the model represents a toroidal pendulum, coupled with a Liouville-integrable system of $n$ non-linear rotators via a small analytic potential. The global bifurcation problem is set-up for the $n$-dimensional isotropic manifold, corresponding to a specific homoclinic orbit of the toroidal pendulum. The splitting of this manifold can be described by a scalar function on an $n$-torus, whose $k$th Fourier coefficient satisfi…
Many unstable periodic orbits of the planar circular restricted 3-body problem (PCRTBP) persist as invariant tori when a periodic forcing is added to the equations of motion. In this study, we compute tori corresponding to exterior Jupiter-Europa and interior Jupiter-Ganymede PCRTBP resonant periodic orbits in a concentric circular restricted 4-body problem (CCR4BP). Motivated by the 2:1 Laplace resonance between Europa and Ganymede's orbits, we then attempt the continuation of a Jupiter-Europa 3:4 resonant orbit from the CCR4BP into the Jupiter-Ganymede PCRTBP. We strongly believe that the re…
In the mathematical framework of a restricted, slightly dissipative spin-orbit model, we prove the existence of periodic orbits for astronomical parameter values corresponding to all satellites of the Solar system observed in exact spin-orbit resonance.
When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, families of unstable periodic orbits break up into whiskered tori, with most tori persisting into the perturbed system. In this study, we 1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable directions; 2) implement continuation by both perturbation as well as rotation numbers; 3) compute Fourier-Taylor parameterizations of the stable and unstable manifolds; 4) regularize the equations of motion; and 5) globalize these manifolds. Our methodology impro…
In this work, we revisit the planar restricted four-body problem to study the dynamics of an infinitesimal mass under the gravitational force produced by three heavy bodies with unequal masses, forming an equilateral triangle configuration. We unify known results about the existence and linear stability of equilibrium points of this problem which have been obtained earlier, either as relative equilibria or a central configuration of the planar restricted $(3 + 1)$-body problem. It is the first attempt in this direction. A systematic numerical investigation is performed to obtain the resonance …
In recent years, stable and unstable manifolds of invariant objects (such as libration points and periodic orbits) have been increasingly recognized as an efficient tool for designing transfer trajectories in space missions. However, most methods currently used in mission design rely on using eigenvectors of the linearized dynamics as local approximations of the manifolds. Since such approximations are not accurate except very close to the base invariant object, this requires large amounts of numerical integration to globalize the manifolds and locate intersections. In this paper, we study hyp…
We study the fourth-order stability of the triangular libration points in the absence of resonance for the three-body problem when the infinitesimal mass is affected not only by gravitation but also by light pressure from both primaries. A comprehensive summary of previous results is given, with some inaccuracies being corrected. The Lie triangle method is used to obtain the fourth-order Birkhoff normal form of the Hamiltonian, and the corresponding complex transformation to pre-normal form is given explicitly. We obtain an explicit expression for the determinant required by the Arnold-Moser t…
