Stability sets of multiparameter Hamiltonian systems

Batkhin, A. B.; Bruno, A D; Варин, В. П.

doi:10.1016/j.jappmathmech.2012.03.006

Cited by 22 publications

(13 citation statements)

References 3 publications

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“…j is the lower edge, then it corresponds to values of y near 0 and values of exp ϕ(x) near 0 corresponds to solutions of the initial equation, but values of exp ϕ(x) near infinity do not corresponds to solutions of the initial equation. Thus, expansion (28) gives only parts of solutions for sectors of complex plane x with Re αx β < 0 and it does not give information about solutions outside these sectors.…”

Section: Exponential Expansions Of Solutions [9 10]mentioning

confidence: 99%

Asymptotic solution of some nonlinear problems

Bruno

2018

KIAM Prepr.

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О р д е н а Л е н и н а ИНСТИТУТ ПРИКЛАДНОЙ МАТЕМАТИКИ имени М.В.КЕЛДЫША Р о с с и й с к о й а к а д е м и и н а у к A. D. Bruno Asymptotic solution of some nonlinear problems Moscow-2018 УДК 517.91 Alexander Dmitrievich Bruno Asymptotic solution of some nonlinear problems We propose algorithms that allow for nonlinear equations to obtain asymptotic expansions of solutions in the form of: (a) power series with constant coefficients, (b) power series with coefficients which are power series of logarithm and (c) power series of exponent of a power series with coefficients which are power series as well. These algorithms are applicable to nonlinear equations (A) algebraic, (B) ordinary differential and (C) partial differential, and to systems of such equations as well. We give the description of the method for one ordinary differential equation and we enumerate some applications of these algorithms.

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Section: Exponential Expansions Of Solutions [9 10]mentioning

confidence: 99%

Asymptotic solution of some nonlinear problems

Bruno

2018

KIAM Prepr.

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“…Implementation of the described algorithm see in [9]. Its application to computation of a set of stability of a certain ODE system depending on several parameters see in [10].…”

Section: Implementation and Applicationmentioning

confidence: 99%

“…Example 11 (cont. of Examples 6-10) [27] The 3D supportS of the third Painleve equation (10) consists of 6 points: Their convex hull Γ is a pentahedron (Fig. 12).…”

Section: Differential Approach [25-30]mentioning

confidence: 99%

Asymptotic Solving Essentially Nonlinear Problems

Bruno¹

2016

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Here we present a way of computation of asymptotic expansions of solutions to algebraic and differential equations and present a survey of some of its applications. The way is based on ideas and algorithms of Power Geometry. Power Geometry has applications in Algebraic Geometry, Differential Algebra, Nonstandard Analysis, Microlocal Analysis, Group Analysis, Tropical/Idempotent Mathematics and so on. We also discuss a connection of Power Geometry with Idempotent Mathematics.

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“…when → 0 or → ∞, then ( ) can be the asymptotic form of the solutions to the full equation (6). Here is a small real number.…”

Section: Power Transformationsmentioning

confidence: 99%

Power Geometry and Elliptic Expansions of Solutions to the Painlevé Equations

Bruno

2015

International Journal of Differential Equations

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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 Painlevé equationsP1,…,P6.

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Stability sets of multiparameter Hamiltonian systems

Cited by 22 publications

References 3 publications

Asymptotic solution of some nonlinear problems

Asymptotic solution of some nonlinear problems

Asymptotic Solving Essentially Nonlinear Problems

Power Geometry and Elliptic Expansions of Solutions to the Painlevé Equations

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