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Martinet-Ramis modulus for one Quadratic System

Name
Mikhail
Surname
Turov
Scientific organization
Chelyabisk State University
Academic degree
-
Position
Researcher
Scientific discipline
Mathematics & Mechanics
Topic
Martinet-Ramis modulus for one Quadratic System
Abstract
For quadratic vector field dp/dv=(p(1-v))/(v(p-v)) was determinate coefficients of the modulus Martinet-Ramis.
Keywords
Martinet-Ramis modulus, saddlenode
Summary

There are consider a quadratic system \(\left\{ \begin{array}{lll}\ \dot{p}=p(1-v) \\ \dot{v}=v(p-v) \end{array}\right.\)(1)

This system is in a some sense, limit system for well known Jouanolou system. The system (1) has a saddlenode singularity at the origin.
In this work we calculate first coefficients of Martinet-Ramis' modulus .

Martinet-Ramis' modulus $(C, \phi)$ (for saddlenode singular point) are constructed by transformations reducing initial system to its (orbital) formal normal form. Solutions of the system (1) with given initial conditions can be found as a series (with respect to initial condition). Using these solutions it is possible ti find coefficients of the normalizing transformations, and then to determinate coefficients of modulus. 

As a result we get
Theorem: Let \((C,\phi)\) be Martinet-Ramis modulus for (1). Then: \(C=0, \phi(z)=z+2\pi iz^2+(2\pi i-4\pi^2)z^3+. . .\)

Corollary: The system (1) is not analytically orbital equivalent to its formal normal form.