Periodic and Chaotic Trajectories of the Second Species for the n-Centre Problem

For the n-centre problem of one particle moving in the potential of attracting centres of small mass fixed in an arbitrary smooth potential and magnetic field, we prove the existence of periodic and chaotic trajectories shadowing sequences of collision orbits. In particular, we obtain large subshifts of solutions of this type for the circular restricted 3-body problem of celestial mechanics. Poincaré had conjectured existence of the periodic ones and given them the name ‘second species solutions’. This revised version was published online in July 2006 with corrections to the Cover Date.     

Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem

We analytically prove the existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise symmetric equal mass four-body problem. This is an extension of our previous proof of the analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar fully symmetric equal mass four-body problem. We then use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar pairwise symmetric 1, m, 1, m four-body problem for m between 0 and 1. Numerical estimates of the the characteristic multipliers show that these periodic orbits are linearly stability when 0.54 ≤ m ≤ 1, and are linearly unstable when 0 < m ≤ 0.53.     

New Periodic Solutions for 3-Body Problems

For Newtonian 3-body problems in ℝ², we prove the existence of new symmetric noncollision periodic solutions with some fixed winding numbers and masses.     

Periodic orbits accumulating onto elliptic tori for the (<Emphasis Type="Italic">N</Emphasis> + 1)-body problem

We prove the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori for (an “outer solar-system” model of) the planar (N + 1)-body problem.      

Periodic solutions with alternating singularities in the collinear four-body problem

This paper gives an analytic proof of the existence of Schubart-like orbit, a periodic orbit with singularities in the symmetric collinear four-body problem. In each period of the Schubart-like orbit, there is a binary collision (BC) between the inner two bodies and a simultaneous binary collision (SBC) of the two clusters on both sides of the origin. The system is regularized and the existence is proved by using a “turning point” technique and a continuity argument on differential equations of the regularized Hamiltonian.     

Some families of periodic solutions in the planarN-body problem

Recent results and existence proofs concerning periodic motions of circular-elliptic type forN=3 andN=4 are reviewed.     

EJECTION–COLLISION ORBITS AND INVARIANT PUNCTURED TORI IN A RESTRICTED FOUR‐BODY PROBLEM

We study the motion of an infinitesimal mass point under the gravitational action of three mass points of masses μ, 1–2μ and μ moving under Newton's gravitational law in circular periodic orbits around their center of masses. The three point masses form at any time a collinear central configuration. The body of mass 1–2μ is located at the center of mass. The paper has two main goals. First, to prove the existence of four transversal ejection–collision orbits, and second to show the existence of an uncountable number of invariant punctured tori. Both results are for a given large value of the Jacobi constant and for an arbitrary value of the mass parameter 0<μ≤1/2. This revised version was published online in July 2006 with corrections to the Cover Date.     

Periodic orbits under combined effects of oblateness and radiation in the restricted problem of three bodies

In this paper, we study the existence of libration points and their linear stability when the three participating bodies are axisymmetric and the primaries are radiating, we found that the collinear points remain unstable, it is further seen that the triangular points are stable for 0<μ<μ _c , and unstable for where , it is also observed that for these points the range of stability will decrease. In addition to this we have studied periodic orbits around these points in the range 0<μ<μ _c , we found that these orbits are elliptical; the frequencies of long and short orbits of the periodic motion are affected by the terms which involve parameters that characterize the oblateness and radiation repulsive forces. The implication is that the period of long periodic orbits adjusts with the change in its frequency while the period of short periodic orbit will decrease.     

Two‐Body Problems with Drag or Thrust: Qualitative Results

We consider two‐body problems in which the drag is proportional to the velocity divided by the square of the distance and whose radial and tangential components have distinct coefficients. For all parameters, we study the flow of the system obtained by suitable coordinate and time transformations and draw conclusions about the qualitative behavior of solutions. In each case, we examine the existence of collision–ejection, collision–escape, capture–collision, capture–escape, and oscillatory rectilinear orbits, study the motion near collision, and show that if periodic orbits exist they must be limit cycles. This revised version was published online in July 2006 with corrections to the Cover Date.     

Periodic orbits and bifurcations in the Sitnikov four-body problem

We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the “family parameter” ż₀ varies within a finite interval (while z ₀ tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the z-axis along its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincaré maps, we chart the motion near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of ordered motion surrounded by “sticky” and chaotic orbits as well as orbits which rapidly escape to infinity.     

THE HILL PROBLEM WITH OBLATE SECONDARY: NUMERICAL EXPLORATION

We introduce a three-dimensional version of Hill’s problem with oblate secondary, determine its equilibrium points and their stability and explore numerically its network of families of simple periodic orbits in the plane, paying special attention to the evolution of this network for increasing oblateness of the secondary. We obtain some interesting results that differentiate this from the classical problem. Among these is the eventual disappearance of the basic family g′ of the classical Hill problem and the existence of out-of-plane equilibrium points and a family of simple-periodic plane orbits non-symmetric with respect to the x-axis.     

The evolution of periodic orbits close to homoclinic points

For conservative dynamical systems having two degrees of freedom Birkhoff has established the existence of two classes of periodic orbits. The first consists of stable-unstable pairs close to periodic orbits of the stable type, and the second of orbits having fixed points (in a suitable surface of section) close to homoclinic points. In this paper orbits of the latter type are listed, and their evolution followed as a function of the energy. For the energy at which they were first computed, all were unstable; but they evolved, with diminishing energy, into one orbit of the stable type which appears to be a member of the first class of orbits mentioned above.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

Bi-periodic variation in the BL Lac AO 0235+164

High resolution VLBI hybrid map of the BL Lacertae object AO 0235+164 has been produced at a wavelength of 6 cm. The map shows that the object's radio structure is dominated by a strong, nearly unresolved core with a weak and clear component in northeast direction and a faint one in southwest direction. The positional angle of its jet component are equal to66.4^°, which is the biggest one in comparison with previous results. Based on the variation of its flux density with time at three different frequencies, we find that the flux density of AO 0235+164 shows bi-periodic variation, i.e., the shorter periodic variation of ∼ 1.81 years and a longer periodic variation of ∼ 3.63 years. The later is essentially in agreement with our earlier predicted results that the existence of the periodic variation of ∼ 3.63 years may be caused by the precession of its `central engine'. This bi-periodic variation is probably the results of the joint action of jet outbursts and jet rotation. With the binary black hole models of Kaastra and Roos, we get the minimum total mass of the binary system of 1.46 × 10⁸ M _⊙. This revised version was published online in July 2006 with corrections to the Cover Date.     

New doubly-symmetric families of comet-like periodic orbits in the spatial restricted (N + 1)-body problem

For any positive integer N ≥ 2 we prove the existence of a new family of periodic solutions for the spatial restricted (N +1)-body problem. In these solutions the infinitesimal particle is very far from the primaries. They have large inclinations and some symmetries. In fact we extend results of Howison and Meyer (J. Diff. Equ. 163:174–197, 2000) from N = 2 to any positive integer N ≥ 2.      

On the Nature of Modulation of Radio Emission During Solar Flares

The distribution of pauses between subsequent elements of a periodic process is symmetric, while a random process produces an asymmetric exponential distribution. The third moment of the pause distribution, which is sensitive to the asymmetry, can therefore be used to discriminate between perodic and random processes. With such a method we analyze the observations of 19 series of solar type III radio bursts and find with a confidence of 0.99 that, on average, the bursts are randomly distributed in time. Only one series can be considered to be periodic with a confidence 0.5. The bandwidth of the repetition frequency of most bursts corresponds to the quality of oscillations of Q¯ = 1.0±0.6 that does not indicates a resonance. Therefore, the modulation of particle beams and intensity of type III radio emission should be considered mainly as the result of random processes. Thus, these properties observed in the majority of radio type III bursts do not support the existence of any periodic or resonant oscillations in the solar corona during flares, although some periodic processes in active regions cannot entirely be ruled out.     

Fast Periodic Transfer Orbits in the Sun–Earth–Moon Quasi-Bicircular Problem

Starting from the identification and classification of a family of fast periodic transfer orbits in the Earth–Moon planar circular Restricted Three Body Problem (RTBP), and using analytic continuation techniques, we find two unstable periodic orbits in the Sun–Earth–Moon Quasi-Bicircular Problem (QBCP). The orbits found perform periodic Earth–Moon transfers with a period of approximately 29.5 days.     

On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries

The existence of new equilibrium points is established in the restricted three-body problem with equal prolate primaries. These are located on the Z-axis above and below the inner Eulerian equilibrium point L ₁ and give rise to a new type of straight-line periodic oscillations, different from the well known Sitnikov motions. Using the stability properties of these oscillations, bifurcation points are found at which new types of families of 3D periodic orbits branch out of the Z-axis consisting of orbits located entirely above or below the orbital plane of the primaries. Several of the bifurcating families are continued numerically and typical member orbits are illustrated.     

Chaos in Relativity and Cosmology

Chaos appears in various problems of Relativity and Cosmology. Here we discuss (a) the Mixmaster Universe model, and (b) the motions around two fixed black holes. (a) The Mixmaster equations have a general solution (i.e. a solution depending on 6 arbitrary constants) of Painlevé type, but there is a second general solution which is not Painlevé. Thus the system does not pass the Painlevé test, and cannot be integrable. The Mixmaster model is not ergodic and does not have any periodic orbits. This is due to the fact that the sum of the three variables of the system (α + β + γ) has only one maximum for τ = τ_m and decreases continuously for larger and for smaller τ. The various Kasner periods increase exponentially for large τ. Thus the Lyapunov Characteristic Number (LCN) is zero. The "finite time LCN" is positive for finite τ and tends to zero when τ → ∞. Chaos is introduced mainly near the maximum of (α + β + γ). No appreciable chaos is introduced at the successive Kasner periods, or eras. We conclude that in the Belinskii-Khalatnikov time, τ, the Mixmaster model has the basic characteristics of a chaotic scattering problem. (b) In the case of two fixed black holes M₁ and M₂ the orbits of photons are separated into three types: orbits falling into M₁ (type I), or M₂ (type II), or escaping to infinity (type III). Chaos appears because between any two orbits of different types there are orbits of the third type. This is a typical chaotic scattering problem. The various types of orbits are separated by orbits asymptotic to 3 simple unstable orbits. In the case of particles of nonzero rest mass we have intervals where some periodic orbits are stable. Near such orbits we have order. The transition from order to chaos is made through an infinite sequence of period doubling bifurcations. The bifurcation ratio is the same as in classical conservative systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the effect of the eccentricity of a planetary orbit on the stability of satellite orbits

The effect of the eccentricity of a planet’s orbit on the stability of the orbits of its satellites is studied. The model used is the elliptic Hill case of the planar restricted three-body problem. The linear stability of all the known families of periodic orbits of the problem is computed. No stable orbits are found, the majority of them possessing one or two pairs of real eigenvalues of the monodromy matrix, while a part of a family with complex instability is found. Two families of periodic orbits, bifurcating from the Lagrangian points L₁, L₂ of the corresponding circular case are found analytically. These orbits are very unstable and the determination of their stability coefficients is not accurate, so we compute the largest Liapunov exponent in their vicinity. In all cases these exponents are positive, indicating the existence of chaotic motions     

Existence of periodic orbits of the second kind in the elliptic restricted problem of three bodies

Hale's method is used to show the existence of symmetric periodic orbits of the second kind for the particular case of the elliptic restricted problem of three bodies. In this treatment we also obtain a new proof of the existence of periodic orbits of the first and second kinds in the circular restricted problem.