The rectilinear three-body problem as a basis for studying highly eccentric systems

We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic orbits of the system given in a suitable rotating frame. The stability of periodic orbits characterize the evolution of any planetary system with initial conditions in their vicinity. Stable periodic orbits are associated with long term regular evolution, while unstable periodic orbits are surrounded by regions of chaotic motion. We compute many families of symmetric periodic orbits by applying two schemes of analytical continuation. In the first scheme, we start from the 2/1 (or 1/2) resonant periodic orbits of the restricted problem and in the second scheme, we start from vertical critical periodic orbits of the general planar problem. Most of the periodic orbits are unstable, but many stable periodic orbits have been, also, found with mutual inclination up to $50^\circ$ - $60^\circ$, which may be related with the existence of real planetary systems.

The long-term stability of the evolution of two-planet systems is considered by using the general three body problem (GTBP). Our study is focused on the stability of systems with adjacent orbits when at least one of them is highly eccentric. In these cases, in order for close encounters, which destabilize the planetary systems, to be avoided, phase protection mechanisms should be considered. Additionally, since the GTBP is a non-integrable system, chaos may also cause the destabilization of the system after a long time interval. By computing dynamical maps, based on Fast Lyapunov Indicator, we reveal regions in phase space with stable orbits even for very high eccentricities (e>0.5). Such regions are present in mean motion resonances (MMR). We can determine the position of the exact MMR through the computation of families of periodic orbits in a rotating frame. Elliptic periodic orbits are associated with the presence of apsidal corotation resonances (ACR). When such solutions are stable, they are associated with neighbouring domains of initial conditions that provide long-term stability. We apply our methodology so that the evolution of planetary systems of highly eccentric orbits is assigned to the existence of such stable domains. Particularly, we study the orbital evolution of the extrasolar systems HD 82943, HD 3651, HD 7449, HD 89744 and HD 102272 and discuss the consistency between the orbital elements provided by the observations and the dynamical stability.

We consider a planetary system consisting of two primaries, namely a star and a giant planet, and a massless secondary, say a terrestrial planet or an asteroid, which moves under their gravitational attraction. We study the dynamics of this system in the framework of the circular and elliptic restricted TBP, when the motion of the giant planet describes circular and elliptic orbits, respectively. Originating from the circular family, families of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion resonances are continued in the circular and the elliptic problems. New bifurcation points from the circular to the elliptic problem are found for each of the above resonances and thus, new families, continued from these points are herein presented. Stable segments of periodic orbits were found at high eccentricity values of the already known families considered as whole unstable previously. Moreover, new isolated (not continued from bifurcation points) families are computed in the elliptic restricted problem. The majority of the new families mainly consist of stable periodic orbits at high eccentricities. The families of the 5/1 resonance are investigated for the first time in the restricted three-body problems. We highlight the effect of stable periodic orbits on the formation of stable regions in their vicinity and unveil the boundaries of such domains in phase space by computing maps of dynamical stability. The long-term stable evolution of the terrestrial planets or asteroids is dependent on the existence of regular domains in their dynamical neighbourhood in phase space, which could host them for long time spans. This study, besides other celestial architectures that can be efficiently modelled by the circular and elliptic restricted problems, is particularly appropriate for the discovery of terrestrial companions among the single-giant planet systems discovered so far.

The dynamic binary neural network is characterized by ternary connection parameters and can generate various binary periodic orbits. This paper considers two interesting problems based on typical examples. First, effect of connection sparsity on stability of target periodic orbits is considered: adding branches adequately to the most sparse network, stability of the periodic orbits can be reinforced. Second, fault tolerance of the network is considered: cutting one branch from the network, storage and stability of the periodic orbits are preserved in high probability.

We study the mechanism of bifurcation of three-dimensional periodic orbits from the plane of motion of the primaries. The branchings of simple and double periodic three-dimensional orbits occur at the plane periodic orbits of indifferent vertical stability. These vertical-critical plane orbits are doubly important. Not only are they necessary in order to specify the vertically stable segments of the families of plane periodic orbits, but they are also the starting points of the families of the simplest possible three-dimensional periodic orbits, namely, the simple and double periodic. We describe here suitable numerical techniques for their accurate determination and for their continuation through the range of the mass parameter μ of the problem. We use these techniques to determine the first few vertical-critical orbits of the basic families of periodic orbits of the problem, namely, families l, m, i, h, a, b and c, for the entire range of μ. The initial conditions of over three hundred vertical-critical orbits are given, classified in twenty-one series. Their horizontal stability is also examined and this leads to predictions of where in the range of μ stable three-dimensional periodic orbits may not branch from these basic families.

Many of exoplanetary systems consist of more than one planet and the study of planetary orbits with respect to their long-term stability is very interesting. Furthermore, many exoplanets seem to be locked in a mean-motion resonance (MMR), which offers a phase protection mechanism, so that, even highly eccentric planets can avoid close encounters. However, the present estimation of their initial conditions, which may change significantly after obtaining additional observational data in the future, locate most of the systems in chaotic regions and consequently, they are destabilized. Hence, dynamical analysis is imperative for the derivation of proper planetary orbital elements. We utilize the model of spatial general three body problem, in order to simulate such resonant systems through the computation of families periodic orbits. In this way, we can figure out regions in phase space, where the planets in resonances should be ideally hosted in favour of long-term stability and therefore, survival. In this review, we summarize our methodology and showcase the fact that stable resonant planetary systems evolve being exactly centered at stable periodic orbits. We apply this process to co-orbital motion and systems HD 82943, HD 73526, HD 128311, HD 60532, HD 45364 and HD 108874.

We study the dynamics of 3:1 resonant motion for planetary systems with two planets, based on the model of the general planar three body problem. The exact mean motion resonance corresponds to periodic motion (in a rotating frame) and the basic families of symmetric and asymmetric periodic orbits are computed. Four symmetric families bifurcate from the family of circular orbits of the two planets. Asymmetric families bifurcate from the symmetric families, at the critical points, where the stability character changes. There exist also asymmetric families that are independent of the above mentioned families. Bounded librations exist close to the stable periodic orbits. Therefore, such periodic orbits (symmetric or asymmetric) determine the possible stable configurations of a 3:1 resonant planetary system, even if the orbits of the two planets intersect. For the masses of the system 55Cnc most of the periodic orbits are unstable and they are associated with chaotic motion. There exist however stable symmetric and asymmetric orbits, corresponding to regular trajectories along which the critical angles librate. The 55Cnc extra-solar system is located in a stable domain of the phase space, centered at an asymmetric periodic orbit.

In this work, the perturbed equations of motion of the infinitesimal body are constructed in the framework of the circular restricted three-body problem when the main two bodies are oblate and radiating. Under the perturbations effects of the oblateness and the radiation pressure the positions of collinear Lagrange points are evaluated, the interior and exterior first-order resonant periodic orbits are also studied. In addition, the initial positions of the periodic orbits and the size of loops have been estimated under these effects. Thus, the characteristics of periodic orbits have been studied under the combine effects of two, three and four perturbations for all the possible combinations of the perturbed parameters. The different order of resonant periodic orbits have been also analysed under the effects of Jacobi constant, mass factor, order of resonance and number of loops.

We study the dynamics of a two-planet system, which evolves being in a 1/1 mean motion resonance (co-orbital motion) with non-zero mutual inclination. In particular, we examine the existence of bifurcations of periodic orbits from the planar to the spatial case. We find that such bifurcations exist only for planetary mass ratios $\rho=\frac{m_2}{m_1}<0.0205$. For ρ in the interval 0<ρ<0.0205, we compute the generated families of spatial periodic orbits and their linear stability. These spatial families form bridges, which start and end at the same planar family. Along them the mutual planetary inclination varies. We construct maps of dynamical stability and show the existence of regions of regular orbits in phase space.

This paper considers two basic problems in artificial neural networks that can generate various binary periodic orbits. The first problem is relation between sparsity of network connection and stability of a target periodic orbit. The second problem is comparison between digital circuits and artificial neural networks in the orbit stability. We consider these problems in dynamic binary neural networks characterized by the signum activation function and ternary connection matrix. Performing basic numerical experiments, we give conjectures for the two problems. First, as the connection sparsity increases, the orbit stability varies. There exists suitable sparsity in which the orbit stability is very strong. Second, as the connection matrix approaches to the most sparse case, the dynamic binary neural network approaches to an equivalent system to the shift register that has no stable periodic orbit.

We study the orbits of photons (null geodesics) and particles (time-like geodesics) in the field of two fixed black holes. The orbits of photons terminate either at the black holes M 1 and M 2 (types (I) and (II)), or at infinity (type (III)). The limits of these three types of orbits are Cantor sets, defined by the unstable periodic orbits, that form a set of measure zero. In the case of particles with elliptic energy there are some stable periodic orbits that trap around them both quasiperiodic orbits (type (IV)) and chaotic orbits that do not reach the black holes (type (V)). The stable periodic orbits may become unstable, producing an infinity of period doubling bifurcations. The main periodic orbits are two almost circular satellite orbits around each black hole, M 1 and M 2, and almost elliptical orbits surrounding both black holes. Orbits crossing one of the satellite orbits inwards fall into the corresponding black hole. For certain values of the parameters one or both satellite orbits may not exist. Then the separation of the type (I) and/or type (II) orbits is done by some almost hyperbolic orbits. In the Newtonian case the satellite orbits around M 1 and M 2 never exist (except if M 1 = 0, or M 2 = 0 exactly). However we have many families of higher order periodic orbits that exist both in the relativistic and the Newtonian case. The differences between the corresponding orbits are large close to the black holes.

The restricted three-body problem has many important astronomical applications. We present a diverse collection of 28 symmetric and asymmetric, stable and unstable periodic orbits for the two- and three-dimensional problems and compare the performance of eight general purpose integrators on the orbits.

Bifurcations analysis of oscillating hypercycles

Birth of periodic and artificial halo orbits in the restricted three-body problem

We consider a chaotic switched arrival system with N buffers. We assume that both the input rate of work and the processing time per unit work are constant. In this article, we propose a method for stabilization of an unstable periodic orbit embedded in a chaotic attractor by controlling the limited continuous processing time.