Normal Form Method and Approximated ODEs Solutions
till the 4-th Order Dimension

Victor F. Edneral
Institute for Nuclear Physics of Moscow State University
Vorobievy Gory, Moscow, 119899, Russia
edneral@theory.npi.msu.su

Abstract The report describes a usage of high order resonant normal forms for analytical approximation of families of solutions systems nonlinear autonomous ODE near stationary points. This method supplies a uniform approach to approximation wide class ODE and gives results which are very close to the results of Carleman linearisation method for non-periodic families and to results of Poincare-Lindstedt methods for periodic families of solutions. Method provides also enough conditions of periodicity (or conditional periodicity) as a system of equations in power series. Method is very flexible and demonstrates a quantitatively good agreement with results of numerical integration of ODEs. The report is illustrated by several examples including a treatment of the van der Pol equation, Henon-Heiles and Contopoulos systems.