Application of Gauss's method in the formation of periodic solutions in the relativistic restricted problem of three bodies

The paper discusses the existence of periodic and quasi-periodic solutions in the space relativistic problem of three bodies with the help of Poincaré's small parameter method starting from non-Keplerian generating solutions, i.e., using Gauss's method. The main peculiarity of these periodic orbits is the fact that they close, in general, after many revolutions. It is worth noticing that these periodic orbits give a new class of periodic solutions of the classical circular problem of three bodies, if relativistic effects are neglected.     

Conditionally periodic solutions near the libration points of the restricted circular three-body problem

Families of conditionally periodic solutions have been found by a slightly modified Lyapunov method of determining periodic solutions near the libration points of the restricted three-body problem. When the frequencies of free oscillations are commensurable, the solutions found are transformed into planar or spatial periodic solutions. The results are confirmed by numerically integrating the starting nonlinear differential equations of motion.     

On stationary solutions of some canonical systems of differential equations

In our article (Zhuravlev, 1979) a formal method of constructing conditionally periodic solutions of canonical systems of differential equations with a quick-rotating phase in the case of sharp commensurability was presented. The existence of stationary (or periodic) solutions of an averaged system of differential equations corresponding to the initial system of differential equations is necessary for an effective application of the method for different problems.Evidently, the stationary solutions do not always exist but in numerous papers on stationary solutions (oscillations or motions), the conditions of existence of such solutions are very often not considered at all. Usually a simple assumption is used that the stationary solutions do exist.Otherwise it is well known that Poincaré's theory of periodic solutions (Poincaré, 1892) let one set up conditions of existence of periodic solutions in different systems of differential equations. Particularly, in papers,Mah (1949, 1956), see alsoexmah (1971), the necessary and sufficient conditions of the existence of periodic solutions of (non-canonical) systems of differential equations which are close to arbitrary non-linear systems are given. For canonical autonomous systems of differential equations the conditions of existence of periodic solutions and a method of calculation are presented in the paperMepmah (1952).In our paper another approach is given and the conditions of existence of stationary solutions of canonical systems of differential equations with a quick-rotating phase are proved. For this purpose Delaunay-Zeipel's transformation and Poincaré's small parameter method are used.     

Periodic motions generated by Lagrangian solutions of the circular restricted three-body problem

The predictor-corrector method is described for numerically extending with respect to the parameters of the periodic solutions of a Lagrangian system, including recurrent solutions. The orbital stability in linear approximation is investigated simultaneously with its construction.The method is applied to the investigation of periodic motions, generated from Lagrangian solutions of the circular restricted three body problem. Small short-period motions are extended in the plane problem with respect to the parameters h, µ (h = energy constant, µ = mass ratio of the two doninant gravitators); small vertical oscillations are extended in the three-dimensional problem with respect to the parameters h, µ. For both problems in parameter's plane h, µ domaines of existince and stability of derived periodic motions are constructed, resonance curves of third and fourth orders are distinguished.     

Stability of the periodic solutions of the restricted three-body problem representing analytic continuations of Keplerian rectilinear periodic motions

The periodic solutions of the restricted three-body problem representing analytic continuations of Keplerian rectilinear periodic motions are well known (Kurcheeva, 1973). Here the stability of these solutions are examined by applying Poncaré's characteristic equation for periodic solutions. It is found that the isoperiodic solutions are stable and all other solutions are unstable.     

Periodic solution of the nonlinear Sitnikov restricted three-body problem

The objective of this paper is to find periodic solutions of the circular Sitnikov problem by the multiple scales method which is used to remove the secular terms and find the periodic approximated solutions in closed forms. Comparisons among a numerical solution (NS), the first approximated solution (FA) and the second approximated solution (SA) via multiple scales method are investigated graphically under different initial conditions. We observe that the initial conditions play a vital role in the numerical and approximated solutions behaviour. The obtained motion is periodic, but the difference of its amplitude is directly proportional with the initial conditions. We prove that the obtained motion by the numerical or the second approximated solutions is a regular and periodic, when the infinitesimal body starts its motion from a nearer position to the common center of primaries. Otherwise when the start point distance of motion is far from this center, the numerical solution may not be represent a periodic motion for along time, while the second approximated solution may present a chaotic motion, however it is always periodic all time. But the obtained motion by the first approximated solution is periodic and has regularity in its periodicity all time. Finally we remark that the provided solutions by multiple scales methods reflect the true motion of the Sitnikov restricted three–body problem, and the second approximation has more accuracy than the first approximation. Moreover the solutions of multiple scales technique are more realistic than the numerical solution because there is always a warranty that the motion is periodic all time.     

基于Broyden方法的不对称与对称周期解延拓算法与应用

考虑周期解的数值延拓问题并提出基于Broyden拟牛顿法来延拓周期解的一种有效算法,先后以布鲁塞尔振子、平面圆型限制性三体问题(Planar Circular Restricted Three-Body Problem, PCRTBP)的周期解为例进行了验证.这里的Broyden方法包含线性搜索、正交三角分解求线性方程组的步骤.对一般的周期解,周期性条件方程组中含有周期作为待延拓参数,可用周期来决定积分时长,将解代入周期性条件得到积分型的非线性方程组,利用Broyden方法迭代延拓直至初值收敛.根据两次垂直通过一个超平面的轨道是对称周期轨道的性质,可采用插值的方法求得再次抵达超平面的解分量,得到周期性条件方程组,再用Broyden方法求解.结合哈密顿系统的对称性和PCRTBP周期轨道的一些分类,对2/1、3/1的内共振周期解族进行了数值研究.最后,对算法和计算结果做了总结和讨论.     

On the stability of periodic solutions in the problem of the motion of a star inside an elliptical galaxy

The problem of the spatial motion of a star inside an inhomogeneous rotating elliptical galaxy with a homothetic density distribution is considered. Periodic solutions are constructed by the method of a small Poincaré parameter. Linear variational equations with periodic coefficients are used to analyze the Lyapunov stability of these solutions.     

A note on large amplitude non-linear radial oscillation in polytropic stellar models

The periodic solutions of inharmonic oscillators describing the systematics of large amplitude single mode radial pulsator have been obtained using modified Lindstedt-Poincaré method for different values of the amplitude,a.     

Bifurcations of dust acoustic solitary waves and periodic waves in an unmagnetized plasma with nonextensive ions

Bifurcations of dust acoustic solitary waves and periodic waves in an unmagnetized plasma with q-nonextensive velocity distributed ions are studied through non-perturbative approach. Basic equations are reduced to an ordinary differential equation involving electrostatic potential. After that by applying the bifurcation theory of planar dynamical systems to this equation, we have proved the existence of solitary wave solutions and periodic wave solutions. Two exact solutions of the above waves are derived depending on the parameters. From the solitary wave solution and periodic wave solution, the effect of the parameter (q) is studied on characteristics of dust acoustic solitary waves and periodic waves. The parameter (q) significantly influence the characteristics of dust acoustic solitary and periodic structures.     

Deflation techniques for the determination of periodic solutions of a certain period

The computation of periodic orbits of nonlinear mappings or dynamical systems can be achieved by applying a root-finding method. To determine a periodic solution, an initial guess should be located within a proper area of the mapping or a surface of section of the phase space of the dynamical system. In the case of Newton or Newton-like methods these areas are the basins of convergence corresponding to the considered solution. When several solutions of the same period exist in a particular region, then the deflation technique is suitable for the calculation of all these solutions. This technique is applied here to the Hénon's mapping and the driven conservative Duffing's oscillator.     

Bifurcations of ion acoustic solitary waves and periodic waves in an unmagnetized plasma with kappa distributed multi-temperature electrons

Ion acoustic solitary waves and periodic waves in an unmagnetized plasma with superthermal (kappa distributed) cool and hot electrons have been investigated using non-perturbative approach. We have transformed basic model equations to an ordinary differential equation involving electrostatic potential. Then we have applied the bifurcation theory of planar dynamical systems to the obtained equation and we have proved the existence of solitary wave solutions and periodic wave solutions. We have derived two exact solutions of solitary and periodic waves depending on the parameters. From the solitary wave solution and periodic wave solution, we have shown the effects of density ratio p of cool electrons and ions, spectral index κ, and temperature ratio σ of cool electrons and hot electrons on characteristics of ion acoustic solitary and periodic waves.     

Numerical exploration of commensurable periodic solutions of the restricted problem of three bodies and their stability

The long period problem provides the initial conditions for numerical computation of close periodic solutions separated in three categories. For each type of commensurability a number of periodic solutions are computed and their stability is studied by computing the characteristic exponents of the matrizant. The Runge-Kutta method for the solution of differential equations of motion was used in all cases. The results obtained are presented for a four cases of commensurability.     

Three-dimensional periodic solutions around equilibrium points in Hill's problem

The three-dimensional periodic solutions originating at the equilibrium points of Hill's limiting case of the Restricted Three Body Problem, are studied. Fourth-order parametric expansions by the Lindstedt-Poincaré method are constructed for them. The two equilibrium points of the problem give rise to two exactly symmetrical families of three-dimensional periodic solutions. The family_HL _2v ^e originating at L₂ is continued numerically and is found to extend to infinity. The family originating at L₁ behaves in exactly the same way and is not presented. All orbits of the two families are unstable.     

Hierarchy of periodic solutions families of spatial Hill’s problem

Many modern space projects require the knowledge of orbits with certain properties. Most of these projects assume the motion of a space vehicle in the neighborhood of a celestial body, which in turn moves in the field of the Sun or another massive celestial body. A good approximation of this situation is Hill’s problem. This paper is devoted to the investigation of the families of spatial periodic solutions to the three-dimensional Hill’s problem. This problem is nonintegrable; therefore, periodic solutions are studied numerically. The Poincare theory of periodic solutions of the second kind is applied; either planar or vertical impact orbits are used as generating solutions.     

Periodic solutions of the restricted problem that are analytic continuations of periodic solutions of Hill's problem for small μ>0

This paper establishes the existence and first order perturbation approximation of an infinite number of one-parameter families of symmetric periodic solutions of the restricted three body problem that are analytic continuations of symmetric periodic solutions of Hill's problem for small values of the mass ratio μ>0.     

Periodic orbits near infinity in the restrictedN-body problem

This paper shows that there exist two families of periodic solutions of the restrictedN-body problem which are close to large circular orbits of the Kepler problem. These solutions are shown to be of general elliptic type and hence are stable. If the restricted problem admits a symmetry, then there are symmetric periodic solutions which are close to large elliptic orbits of the Kepler problem.     

Numerical investigation of symmetric and asymmetric periodic oscillations

Global information for the periodic solutions — symmetric and asymmetric — of the ‘gravitational’ spring-pendulum problem is given for the first time. For two different sets of the parameters of this problem, the families of symmetric periodic solutions which emanate from the equilibrium point have been determined. Further families of asymmetric and symmetric solutions which bifurcate from them have also been examined and interesting results for their behaviour have been pointed out.     

Conditionally periodic solutions of the canonical systems of differential equations in non-autonomous resonant case

A formal method of constructing of conditionally periodic solutions of canonical systems of differential equations in the vicinity of a commemsurability of frequencies is proposed. The method is a union of the rapid convergence method and (well-known in celestial mechanics) Delaunay-Zeipel's method of canonical transformations. For a successful application of the method an existence of stationary resonant solutions of an averaged system of the differential equations is necessary.