Mounting discoveries of debris discs orbiting newly-formed stars and white dwarfs (WDs) showcase the importance of modeling the long-term evolution of small bodies in exosystems. WD debris discs are in particular thought to form from very long-term (0.1-5.0 Gyr) instability between planets and asteroids. However, the time-consuming nature of N -body integrators which accurately simulate motion over Gyrs necessitates a judicious choice of initial conditions. The analytical tools known as periodic orbits can circumvent the guesswork. Here, we begin a comprehensive analysis directly linking periodic orbits with N -body integration outcomes with an extensive exploration of the planar circular restricted three-body problem (CRTBP) with an outer planet and inner asteroid near or inside of the 2:1 mean motion resonance. We run nearly 1000 focused simulations for the entire age of the Universe (14 Gyr) with initial conditions mapped to the phase space locations surrounding the unstable and stable periodic orbits for that commensurability. In none of our simulations did the planar CRTBP architecture yield a long-timescale ( 0.25% of the age of the Universe) asteroid-star collision. The pericentre distance of asteroids which survived beyond this timescale (≈ 35 Myr) varied by at most about 60%. These results help affirm that collisions occur too quickly to explain WD pollution in the planar CRTBP 2:1 regime, and highlight the need for further periodic orbit studies with the eccentric and inclined TBP architectures and other significant orbital period commensurabilities.

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.Comment: Typos corrected. Published in Celestial Mechanics and Dynamical Astronom

We consider a two-planet system, which migrates under the influence of dissipative forces that mimic the effects of gas-driven (Type II) migration. It has been shown that, in the planar case, migration leads to resonant capture after an evolution that forces the system to follow families of periodic orbits. Starting with planets that differ slightly from a coplanar configuration, capture can, also, occur and, additionally, excitation of planetary inclinations has been observed in some cases. We show that excitation of inclinations occurs, when the planar families of periodic orbits, which are followed during the initial stages of planetary migration, become vertically unstable. At these points, vertical critical orbits may give rise to generating stable families of 3D periodic orbits, which drive the evolution of the migrating planets to non-coplanar motion. We have computed and present here the vertical critical orbits of the 2/1 and 3/1 resonances, for various values of the planetary mass ratio. Moreover, we determine the limiting values of eccentricity for which the "inclination resonance" occurs.

The planetary dynamics of 4/3, 3/2, 5/2, 3/1 and 4/1 mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance. Both planar and spatial cases are examined. In the spatial problem, families of periodic orbits are obtained after analytical continuation of vertical critical orbits. The linear stability of orbits is also examined. Concerning initial conditions nearby stable periodic orbits, we obtain long-term planetary stability, while unstable orbits are associated with chaotic evolution that destabilizes the planetary system. Stable periodic orbits are of particular importance in planetary dynamics, since they can host real planetary systems. We found stable orbits up to 60 • of mutual planetary inclination, but in most families, the stability does not exceed 20 • -30 • , depending on the planetary mass ratio. Most of these orbits are very eccentric. Stable inclined circular orbits or orbits of low eccentricity were found in the 4/3 and 5/2 resonance, respectively.keywords Extrasolar planets, general three body problem, mean motion resonances, periodic orbits, planetary systems.

We consider the planar three body problem of planetary type and we study the generation and continuation of periodic orbits and mainly of asymmetric periodic orbits. Asymmetric orbits exist in the restricted circular three body problem only in particular resonances called "asymmetric resonances". However, numerical studies showed that in the general three body problem asymmetric orbits may exist not only for asymmetric resonances, but for other kinds, too. In this work, we show the existence of asymmetric periodic orbits in the elliptic restricted problem. These orbits are continued and clarify the origin of many asymmetric periodic orbits in the general problem. Also, we illustrate how the families of periodic orbits of the restricted circular problem and those of the elliptic one join smoothly and form families in the general problem, verifying in this way the scenario described firstly by Bozis and Hadjidemetriou (1976).keywords three body problem (TBP), periodic orbits, bifurcations.

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.

Librational motion in celestial mechanics is generally associated with the existence of stable resonant configurations and signified by the existence of stable periodic solutions and oscillation of critical (resonant) angles. When such an oscillation takes place around a value different than 0 or π, the libration is called asymmetric. In the context of the planar circular restricted three-body problem (CRTBP), asymmetric librations have been identified for the exterior meanmotion resonances (MMRs) 1 : 2, 1 : 3 etc. as well as for co-orbital motion (1 : 1). In exterior MMRs the massless body is the outer one. In this paper, we study asymmetric librations in the 3-dimensional space. We employ the computational approach of Markellos (1978) and compute families of asymmetric periodic orbits and their stability. Stable, asymmetric periodic orbits are surrounded in phase space by domains of initial conditions which correspond to stable evolution and librating resonant angles. Our computations were focused on the spatial circular restricted three-body model of the Sun-Neptune-TNO system (TNO = trans-Neptunian object). We compare our results with numerical integrations of observed TNOs, which reveal that some of them perform 1 : 2-resonant, inclined asymmetric librations. For the stable 1 : 2 TNOs librators, we find that their libration seems to be related with the vertically stable planar asymmetric orbits of our model, rather than the 3-dimensional ones found in the present study.keywords circular restricted TBP -exterior resonances -spatial asymmetric periodic orbits -trans-Neptunian object dynamics ) used an analytical approach to show the existence of asymmetric librations in 1 : p exterior MMRs and their absence in other exterior MMRs as e.g. the 2 : 3 and 3 : 4. We should remark that i) a.p.o are found also in the elliptic planar restricted and in the general planetary planar problem (Antoniadou et al., 2011) ii) 1 : 1 families of a.p.o. also exist and emanate from L 4 for the circular planar restricted (Zagouras et al., 1996) and these families continue in the general planetary problem (Giuppone et al., 2010;Hadjidemetriou and Voyatzis, 2011).Concerning spatial three-body models, families of symmetric periodic orbits (s.p.o.) have been mainly computed for asteroids and TNOs (Ichtiaroglou et al., 1989;Hadjidemetriou, 1993; or for exoplanetary systems (Antoniadou and Voyatzis, 2013;Antoniadou and Voyatzis, 2014). Generally, families of s.p.o. bifurcate from planar periodic orbits which are critical with respect to their vertical stability (Hénon, 1973) and are called vertical critical orbits (v.c.o.). These families extend up to some critical value of the inclination, i, or terminate at i = 180 • (planar retrograde orbit). With respect to their linear stability, the studies cited above showed that most prograde orbits of moderate or high inclination are unstable. Nevertheless, segments of stable orbits also exist providing restricted phase space domains, where resonant angles librate around 0 or π. Markellos (1978)...

Hundreds of giant planets have been discovered so far and the quest of exo-Earths in giant planet systems has become intriguing. In this work, we aim to address the question of the possible long-term coexistence of a terrestrial companion on an orbit interior to a giant planet, and explore the extent of the stability regions for both non-resonant and resonant configurations. Our study focuses on the restricted three-body problem, where an inner terrestrial planet (massless body) moves under the gravitational attraction of a star and an outer massive planet on a circular or elliptic orbit. Using the Detrended Fast Lyapunov Indicator as a chaotic indicator, we constructed maps of dynamical stability by varying both the eccentricity of the outer giant planet and the semi-major axis of the inner terrestrial planet, and identify the boundaries of the stability domains. Guided by the computation of families of periodic orbits, the phase space is unravelled by meticulously chosen stable periodic orbits, which buttress the stability domains. We provide all possible stability domains for coplanar symmetric configurations and show that a terrestrial planet, either in mean-motion resonance or not, can coexist with a giant planet, when the latter moves on either a circular or an (even highly) eccentric orbit. New families of symmetric and asymmetric periodic orbits are presented for the 2/1 resonance. It is shown that an inner terrestrial planet can survive long time spans with a giant eccentric outer planet on resonant symmetric orbits, even when both orbits are highly eccentric. For 22 detected singleplanet systems consisting of a giant planet with high eccentricity, we discuss the possible existence of a terrestrial planet. This study is particularly suitable for the research of companions among the detected systems with giant planets, and could assist with refining observational data. keywordscelestial mechanics -planetary and satellites: dynamical evolution and stability -minor planets, asteroids: general -planetary systems -methods: analytical -methods: numerical 1 See e.g. exoplanet.eu (Schneider et al. 2011) and exoplanets.org (Han et al. 2014)