Analysis of singularities and of integrability of ODE's by algorithms of Power Geometry

Alexander D. Bruno. "Analysis of singularities and of integrability of ODE's by algorithms of Power Geometry." Banach Center Publications 94.1 (2011): 83-98. <http://eudml.org/doc/286493>.

@article{AlexanderD2011,
abstract = {Here we present basic ideas and algorithms of Power Geometry and give a survey of some of its applications. In Section 2, we consider one generic ordinary differential equation and demonstrate how to find asymptotic forms and asymptotic expansions of its solutions. In Section 3, we demonstrate how to find expansions of solutions to Painlevé equations by this method, and we analyze singularities of plane oscillations of a satellite on an elliptic orbit. In Section 4, we consider the problem of local integrability of a planar ODE system. In Section 5, we expound the spacial generalizations of planar constructions. Power Geometry gives alternatives to some methods of Algebraic Geometry, Differential Algebra, Nonstandard Analysis, Microlocal Analysis, Group Analysis and to other algebraic methods in Dynamical Systems.},
author = {Alexander D. Bruno},
journal = {Banach Center Publications},
keywords = {power geometry; nonlinear analysis; asymptotic forms; asymptotic expansions; integrability; normal forms},
language = {eng},
number = {1},
pages = {83-98},
title = {Analysis of singularities and of integrability of ODE's by algorithms of Power Geometry},
url = {http://eudml.org/doc/286493},
volume = {94},
year = {2011},
}

TY - JOUR
AU - Alexander D. Bruno
TI - Analysis of singularities and of integrability of ODE's by algorithms of Power Geometry
JO - Banach Center Publications
PY - 2011
VL - 94
IS - 1
SP - 83
EP - 98
AB - Here we present basic ideas and algorithms of Power Geometry and give a survey of some of its applications. In Section 2, we consider one generic ordinary differential equation and demonstrate how to find asymptotic forms and asymptotic expansions of its solutions. In Section 3, we demonstrate how to find expansions of solutions to Painlevé equations by this method, and we analyze singularities of plane oscillations of a satellite on an elliptic orbit. In Section 4, we consider the problem of local integrability of a planar ODE system. In Section 5, we expound the spacial generalizations of planar constructions. Power Geometry gives alternatives to some methods of Algebraic Geometry, Differential Algebra, Nonstandard Analysis, Microlocal Analysis, Group Analysis and to other algebraic methods in Dynamical Systems.
LA - eng
KW - power geometry; nonlinear analysis; asymptotic forms; asymptotic expansions; integrability; normal forms
UR - http://eudml.org/doc/286493
ER -