Alexander D. Bruno

The Limit Problems for the Equation of Oscillations of a Satellite

AbstractWe 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

Chapter 7 - Self-similar solutions

e \to

Power Geometry and Elliptic Expansions of Solutions to the Painlevé Equations

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.

On Integrability of a Planar ODE System Near a Degenerate Stationary Point

Power Geometry develops Differential Calculus and aims at nonlinear problems. The algorithms of Power Geometry allow to simplify equations,to resolve their singularities, to isolate their first approximations, and to find either their solutions or the asymptotics of the solutions. This approach allows to compute also the asymptotic and the local expansions of solutions. Algorithms of Power Geometry are applicable to equations of various types: algebraic, ordinary differential and partial differential, and also to systems of such equations. Power Geometry is an alternative to Algebraic Geometry, Group Analysis, Nonstandard Analysis, Microlocal Analysis etc.

On New Integrals of the Algaba-Gamero-Garcia System

Publisher Summary This chapter presents a quasi-homogeneous partial differential equation, without considering parameters. It is shown how to find all its quasi-homogeneous (self-similar) solutions by the support of the equation with the help of Linear Algebra computations. The simplifications of such an equation are studied with the help of power and logarithmic transformations. It is shown that these transformations allow reducing the order of the quasi-homogeneous ordinary differential equation and that for such an equation the boundary value problems may be simplified. Generalizations of these results for a quasi-homogeneous system of differential equations are formulated. In examples, equations of combustion process without a source and with a source are considered. The solution presented is called “self-similar one,” if it is invariant under the changes of coordinates forming the Lie group.

Elements of Nonlinear Analysis

A description of a real algebraic variety in ℝ3 is given. This variety plays an important role in the investigation of the Einstein metrics whose evolution is studied using the normalized Ricci flow. To reveal the internal structure of this variety, a description of all its singular points is given. Due to the internal symmetry of this variety, a part of the investigation uses elementary symmetric polynomials. All the computations are performed using computer algebra algorithms (in particular, Gröbner bases) and algorithms for dealing with polynomial ideals. As an auxiliary result, a proposition about the structure of the discriminant surface of a cubic polynomial is proved.

On Possibility of Additional Solutions of the Degenerate System Near Double Degeneration at the Special Value of the Parameter

We consider an ordinary differential equation (ODE) which can be written as a polynomial in variables and derivatives. Several types of asymptotic expansions of its solutions can be found by algorithms of 2D Power Geometry. They are power, power-logarithmic, exotic, and complicated expansions. Here we develop 3D Power Geometry and apply it for calculation power-elliptic expansions of solutions to an ODE. Among them we select regular power-elliptic expansions and give a survey of all such expansions in solutions of the Painleve equations .

Calculation of normal forms of the euler---poisson equations

We consider an autonomous system of ordinary differential equations, which is solved with respect to derivatives. To study local integrability of the system near a degenerate stationary point, we use an approach based on Power Geometry method and on the computation of the resonant normal form. For a planar 5-parametric example of such system, we found the complete set of necessary and sufficient conditions on parameters of the system for which the system is locally integrable near a degenerate stationary point.