Periodic Trojan orbits in the elliptic restricted problem

1966 ◽  
Vol 25 ◽  
pp. 176-186
Author(s):  
E. Rabe
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Restricted Problem ◽  
Angular Motion ◽  
Basic Period ◽  
Short Period ◽  
Elliptic Restricted Problem ◽  
The Mean

As previously shown the generally non-periodic librations about the equilateral points of the plane elliptic restricted problem depend on three frequencies. After elimination of the short-period oscillations, the solutions depend on the libration frequencynand the mean angular motionNof Jupiter. Whenn/Nis commensurable the librations become periodic. One periodic solution corresponds to the actual Trojans, but additional planar solutions exist also with periods equal to multiples of the basic period. For some specific values of the eccentricityethese solutions are accompanied by an associated sequence of non-planar periodic solutions. The values ofedepend slightly on the inclinationi.

On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

1966 ◽  
Vol 25 ◽  
pp. 197-222 ◽  
Author(s):  
P. J. Message
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Large Amplitude ◽  
Restricted Problem ◽  
Small Amplitude ◽  
Minor Planets ◽  
Long Period ◽  
Numerical Integrations ◽  
Period Variations ◽  
The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

scholarly journals A Theory of the Trojan Asteroids

1978 ◽  
Vol 41 ◽  
pp. 145-145
Author(s):  
B. Garfinkel
Keyword(s):  
Periodic Solution ◽  
Restricted Problem ◽  
Elliptic Integral ◽  
Mass Parameter ◽  
Small Mass ◽  
Polar Coordinates ◽  
Lagrangian Point ◽  
Elementary Functions ◽  
The Family ◽  
The Mean

AbstractThe paper constructs a long-periodic solution for the case of 1:1 resonance in the restricted problem of three bodies. The polar coordinates r and 0 appear in the formHere λ is the mean synodic longitude, m is the small mass-parameter, k is the integer nearest to the ratio ω2/ω1 of the fundamental angular frequencies of the motion, and ck is a Fourier coefficient of a certain periodic function. Only elementary functions enter r(λ) and θ(λ), while the calculation of λ(t) requires the inversion of a hyper-elliptic integral t(λ).The internal resonant terms, carrying the critical divisor D, impart to the orbit an epicyclic character, in qualitative accord with the results of the numerical integration by Deprit and Henrard (1970). Our solution is valid except in the vicinity of the singularities at D = 0 and λ = 0.The presence of the resonant terms invalidates the Brown conjecture (1911) regarding the termination of the family of the tadpoleshaped orbits at the Lagrangian point L3. However, this conjecture holds for the mean orbits defined by r = r(λ), θ = θ(λ), and it also holds in the limit as m → 0.

scholarly journals New Results for the Linear Stability of the Triangular Points in the Elliptic Restricted Problem

1983 ◽  
Vol 74 ◽  
pp. 289-299
Author(s):  
R. Meire
Keyword(s):  
Periodic Solutions ◽  
Linear Stability ◽  
Restricted Problem ◽  
Triangular Points ◽  
Elliptic Restricted Problem ◽  
The Hill ◽  
Transition Curves

AbstractNew results are obtained for the linear stability of the triangular points in the elliptic restricted problem using the Hill equations which describe the infinitesimal motion around L4,L5. Also the shape of the 4Π-periodic solutions along the transition curves in the μ-e plane is investigated .

scholarly journals On Asymmetric Periodic Solutions of the Plane Restricted Problem of Three Bodies, and Bifurcations of Families

1978 ◽  
Vol 41 ◽  
pp. 319-323
Author(s):  
P.J. Message ◽  
D.B. Taylor
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Restricted Problem ◽  
Initial Conditions ◽  
Characteristic Exponent ◽  
Mean Value ◽  
Mirror Image ◽  
Mean Values ◽  
Major Semi Axis ◽  
The Mean

Previous work on the plane circular restricted problem of three bodies (Message 1953, 1959, 1970, and Fragakis 1973) has shown the existence, in association with each of the commensurabilities 2:1 and 3:1 of the orbital periods, of a pair of families of asymmetric periodic solutions, branching from the stable series of symmetric periodic solutions of Poincaré’s second sort associated with that commensurability. (Each solution of either family is the mirror image, in the line of the two finite bodies, of a member of the other family of solutions associated with the commensurability.) The stability is transferred at the bifurcation to the two series of asymmetric orbits, each of which is therefore stable. Recent numerical integrations carried out by one of us (P.J.M.) have found such asymmetric periodic orbits associated also with the 4:1 commensurability, and quantities describing orbits of one of the two series are given in Table 1, showing the run of such orbits up to a second bifurcation with the same series of symmetric periodic orbits from which it sprang. Quantities describing some members of this series of symmetric orbits are given in Table 2. It is seen that stability is transferred back to the symmetric series at the second bifurcation. (The unit of distance is the distance between the two finite bodies, the unit of speed is the speed, of their relative motion, and the initial conditions given (x°, ẋ°, ẏ°) are for a crossing of the line of the two finite bodies, this line being taken as axis of “x” in a rotating Cartesian frame in the usual way. The mean values of the major semi-axis and eccentricity are denoted by ā and ē, respectively, C is Jacobi’s constant, and ȳ2 is the mean value of the critical argument ȳ2 = 4λ – λ′ – 3ω. The mass ratio used is 0.000954927, T is the period of the solution in units of the period of the motion of the two finite bodies, and 2π c/T is the non-zero characteristic exponent.)

the effects of resonance with Jupiter in the system of short-period comets

1974 ◽  
Vol 22 ◽  
pp. 193-203
Author(s):  
L̆ubor Kresák
Keyword(s):  
Structural Effects ◽  
Order Resonance ◽  
Mean Motion ◽  
Short Period ◽  
The Mean ◽  
Compound Effect ◽  
Low Order

AbstractStructural effects of the resonance with the mean motion of Jupiter on the system of short-period comets are discussed. The distribution of mean motions, determined from sets of consecutive perihelion passages of all known periodic comets, reveals a number of gaps associated with low-order resonance; most pronounced are those corresponding to the simplest commensurabilities of 5/2, 2/1, 5/3, 3/2, 1/1 and 1/2. The formation of the gaps is explained by a compound effect of five possible types of behaviour of the comets set into an approximate resonance, ranging from quick passages through the gap to temporary librations avoiding closer approaches to Jupiter. In addition to the comets of almost asteroidal appearance, librating with small amplitudes around the lower resonance ratios (Marsden, 1970b), there is an interesting group of faint diffuse comets librating in characteristic periods of about 200 years, with large amplitudes of about±8% in μ and almost±180° in σ, around the 2/1 resonance gap. This transient type of motion appears to be nearly as frequent as a circulating motion with period of revolution of less than one half that of Jupiter. The temporary members of this group are characteristic not only by their appearance but also by rather peculiar discovery conditions.

scholarly journals Multiplicity of positive periodic solutions to second order differential equations

2006 ◽  
Vol 73 (2) ◽  
pp. 175-182 ◽  
Author(s):  
Jifeng Chu ◽  
Xiaoning Lin ◽  
Daqing Jiang ◽  
Donal O'Regan ◽  
R. P. Agarwal
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Positive Periodic Solution ◽  
Second Order ◽  
Positive Periodic Solutions ◽  
Second Order Differential Equations

In this paper, we study the existence of positive periodic solutions to the equation x″ = f (t, x). It is proved that such a equation has more than one positive periodic solution when the nonlinearity changes sign. The proof relies on a fixed point theorem in cones.

scholarly journals The lunar and solar diurnal variations of water-level in a well at Kew observatory, Richmond

1918 ◽  
Vol 94 (663) ◽  
pp. 476-478
Keyword(s):  
Water Level ◽  
Royal Society ◽  
Correlation Coefficients ◽  
Barometric Pressure ◽  
Diurnal Variations ◽  
Range Of Movement ◽  
Nearest Point ◽  
Similar Relation ◽  
Short Period ◽  
The Mean

In a paper communicated to the Royal Meteorological Society, it was shown that the experimental well at Kew Observatory responded to the lunar fortnightly oscillation of mean level in the River Thames, which is 300 yards from the Observatory at its nearest point. The sensitiveness of the water-level to barometric pressure has also been investigated, and the results have been given in a paper recently read before the Royal Society. The present paper deals with the effects of the short-period tides in the solar and lunar series, S 1 , S 2 , S 3 , S 4 , and M 1 , M 2 , M 3 , M 4 . Two-hourly measurements, both in lunar and solar time, were made on the traces obtained during the first two years, August, 1914-August, 1916, omitting days of very irregular movement. Monthly mean inequalities were then computed. Well marked solar and lunar diurnal variations were found in each month, taking the form of double oscillations with two maxima and two minima during the 24 hours. The range of movement was in each case found to be highly associated with the mean height of the water in the well, the correlation coefficients being 0·89 (lunar) and 0·90 (solar). A similar relation had been previously found to exist in the case of barometric pressure.

Bifurcations and Chaos in a Minimal Neural Network: Forced Coupled Neurons

2021 ◽  
Vol 31 (10) ◽  
pp. 2150147
Author(s):  
Yo Horikawa
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Period Doubling ◽  
Output Function ◽  
Piecewise Constant ◽  
Border Collision Bifurcations ◽  
Saddle Node ◽  
Periodic Input ◽  
Symmetric Periodic Solution ◽  
Period Doubling Bifurcations

The bifurcations and chaos in a system of two coupled sigmoidal neurons with periodic input are revisited. The system has no self-coupling and no inherent limit cycles in contrast to the previous studies and shows simple bifurcations qualitatively different from the previous results. A symmetric periodic solution generated by the periodic input underdoes a pitchfork bifurcation so that a pair of asymmetric periodic solutions is generated. A chaotic attractor is generated through a cascade of period-doubling bifurcations of the asymmetric periodic solutions. However, a symmetric periodic solution repeats saddle-node bifurcations many times and the bifurcations of periodic solutions become complicated as the output gain of neurons is increasing. Then, the analysis of border collision bifurcations is carried out by using a piecewise constant output function of neurons and a rectangular wave as periodic input. The saddle-node, the pitchfork and the period-doubling bifurcations in the coupled sigmoidal neurons are replaced by various kinds of border collision bifurcations in the coupled piecewise constant neurons. Qualitatively the same structure of the bifurcations of periodic solutions in the coupled sigmoidal neurons is derived analytically. Further, it is shown that another period-doubling route to chaos exists when the output function of neurons is asymmetric.

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