The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

FROM the beginning of civilization one need must make itself felt, namely, for some form of calendar. In turn, this gives rise to the primitive study of astronomy, involving some knowledge of the sun and the moon. When the planets are added in the next stage, the astronomer needs an ephemeris or almanac ; the modern Nautical Almanac, with its perfection within narrow limits, is the final outcome. There have been many stages on the road, and some apparent interruptions. But the urge has always been the same, essentially a practical one. The Analytical Foundations of Celestial Mechanics By Aurel Wintner. (Princeton Mathematical Series, No. 5.) Pp. xii + 448. (Princeton, N.J.: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

The Analytical Foundations of Celestial Mechanics

We obtain all asymptotic expansions of solutions of the sixth Painlevé equation near all three singular points x = 0, x = 1, and x = ∞ for all values of four complex parameters of this equation. The expansions are obtained for solutions of five types: power, power-logarithmic, complicated, semiexotic, and exotic. They form 117 families. These expansions may contain complex powers of the independent variable x. First we use methods of two-dimensional power algebraic geometry to obtain those asymptotic expansions of all five types near the singular point x = 0 for which the order of the leading term is less than 1. These expansions are called basic expansions. They form 21 families. All other asymptotic equations near three singular points are obtained from basic ones using symmetries of the equation. The majority of these expansions are new. Also, we present examples and compare our results with previously known ones. Introduction In 1884–1885 L. Fuchs [47] and H. Poincaré [70, 71, 72] suggested looking for differential equations whose solutions do not have movable critical points and cannot be expressed in terms of previously known functions. In 1889 S. Kovalevskaya [36] had shown that the absence of movable critical points of solutions allows one to construct solutions analytically. A singular point x = x0 of a function y(x) of a complex variable x is called a critical singular point if the value of the function y(x) changes as x moves along the path surrounding x0. A movable singular point of a solution of a differential equation is a singular point such that its position depends on initial conditions of the problem. For example for the solution y = 1/ √ x− x0, where x0 is an arbitrary constant, the point x = x0 is a movable critical point. By a meromorphic function we mean a function whose only singularities in the finite part of the complex place are poles. In 1887 E. Picard [68] suggested studying the following class of second order equations: (1) y′′ = F (x, y, y′), where the function F is rational in y and y′ and meromorphic in x, and to find, among equations (1), those that have only immovable critical singular points. At the beginning of the 20th century P. Painlevé [65, 66, 67], and his students B. Gambier [49] and R. Garnier [50, 51] solved the problem formulated by Fuchs and Picard. They have found 50 canonical equations of the form (1) whose solutions have no movable critical singular 2010 Mathematics Subject Classification. Primary 34E05; Secondary 34M55. The work was supported by the Russian Foundation of Fundamental Research (Project 08–01–00082) and the Foundation for the Assistance to Russian Science. Editorial Note: The following text incorporates changes and corrections submitted by the authors for the English translation. c ©2010 American Mathematical Society 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2 A. D. BRUNO AND I. V. GORYUCHKINA points. Solutions of 44 of these equations could be expressed in terms of known (elementary or special) functions, and solutions of the remaining six equations determine new special functions, which are now called Painlevé transcendents. The sixth Painlevé equation first appeared in the paper by R. Fuchs [48]. In has the form y′′ = (y′)2 2 ( 1 y + 1 y − 1 + 1 y − x ) − y′ ( 1 x + 1 x− 1 + 1 y − x ) + y(y − 1)(y − x) x2(x− 1)2 [ a+ b x y2 + c x− 1 (y − 1)2 + d x(x− 1) (y − x)2 ] , (2) where a, b, c, d are complex parameters, x and y are complex variables, y′ = dy dx . It has three singular points x = 0, x = 1 and x = ∞, and is usually denoted P6. E. Picard [69] has found solutions of this equation in an explicit form for special values of four parameters: a = b = c = 0, d = 12 . R. Garnier [51] was studying solutions of this equation without any restrictions on parameters. A new wave of interest to Painlevé equations occurred in the 1970s after M. Ablowitz, A. Ramani, and H. Segur [1, 42, 43] discovered that integrable nonlinear partial differential equations are related to Painlevé equations (see also [35, 37]). For example, the sixth Painlevé equation is a reduction of the Ernst equation in general relativity. Nowadays, the followng problems for the Painlevé equations are being studied: asymptotic behavior of solutions near singular points, local and global properties of solutions, rational and algebraic solutions, discretization, applications of Painlevé equations (mainly in physics). In the present paper we study asymptotic expansions of solutions of the sixth Painlevé equation at the singular points x = 0, 1, ∞. Expansions in nonsingular points were described in [54, § 46], and, using power geometry, in [13, 24]. Similar studies were performed by many authors. S. Shimomura [73]–[76], M. Jimbo [61], H. Kimura [62], K. Okamoto [64] proved, using a variety of methods, existence and convergence of twoparameter families of expansions for solutions of the sixth Painlevé equation. In the book [54, § 46] by I. V. Gromak, I. Laine, and S. Shimomura the authors describe asymptotic expansions of solutions in integer powers of the independent variable. For some special values of parameters of the sixth Painlevé equation, B. Dubrovin and M. Mazzocco [46, 63], and also D. Guzzetti [55] obtained several initial terms of nonpower and exotic expansions. A comparison of these results with ours is presented at the end of the Introduction and in Section 2 of Chapter 4. The study of asymptotic extensions and asymptotic properties of solutions of Painlevé equations near singular points consists of the following three main steps. Step 1. To find formal solutions in the form of asymptotic expansions

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Asymptotic expansions of solutions of the sixth Painlevé equation

1. Exposition 2. Kinematics: wavenumbers 3. Kinematics: resonance clusters 4. Dynamics 5. Mechanical playthings 6. Wave turbulent regimes 7. Epilogue Appendix: software Index.

Nonlinear Resonance Analysis: Theory, Computation, Applications

The 2/1 resonant dynamics of a two-planet planar system is studied within the framework of the three-body problem by computing families of periodic orbits and their linear stability. The continuation of resonant periodic orbits from the restricted to the general problem is studied in a systematic way. Starting from the Keplerian unperturbed system we obtain the resonant families of the circular restricted problem. Then we nd all the families of the resonant elliptic restricted three body problem, which bifurcate from the circular model. All these families are continued to the general three body problem, and in this way we can obtain a global picture of all the families of periodic orbits of a two-planet resonant system. The parametric continuation, within the framework of the general problem, takes place by varying the planetary mass ratio . We obtain bifurcations which are caused either due to collisions of the families in the space of initial conditions or due to the vanishing of bifurcation points. Our study refers to the whole range of planetary mass ratio values ( 2 (0;1)) and, therefore we include the passage from external to internal resonances. Thus we can obtain all possible stable congurations in a systematic way. Finally, we study whether the dynamics of the four known planetary systems, whose currently observed periods show a 2/1 resonance, are associated with a stable periodic orbit.

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On the 2/1 resonant planetary dynamics – periodic orbits and dynamical stability

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Асимптотики и разложения решений обыкновенного дифференциального уравнения@@@Asymptotic behaviour and expansions of solutions of an ordinary differential equation

We consider the ordinary differential equation of the second order, which describes oscillations of a satellite with respect to its mass center moving along an elliptic orbit with eccentricity e. The equation has two parameters: e and µ. It is regular for 0 ≤ e < 1 and singular when e = 1. For \(e \to \) 1 we obtain three limit problems. Their bounded solution to the first limit problem form a two-dimensional (2D) continuous invariant set with a periodic structure. Solutions to the second limit problem form 2D and 3D manifolds. The µ-depending families of odd bounded solutions are singled out. One of the families is twisted into a self-similar spiral. To obtain the limit families of the periodic solutions to the original problem match together the odd bounded solutions to the first and the second limit problem. The point of conjunction is described by the third (the basic) limit problem. The limit families are very close to prelimit ones computed in earlier studies.

Alexander Dmitrievich Bruno

Papers

Asymptotic expansions of solutions of the sixth Painlevé equation

Асимптотики и разложения решений обыкновенного дифференциального уравнения@@@Asymptotic behaviour and expansions of solutions of an ordinary differential equation

The Limit Problems for the Equation of Oscillations of a Satellite

Stability sets of multiparameter Hamiltonian systems

Periodic solutions of the restricted three-body problem for a small mass ratio