Orbital normal forms of two-dimensional analytic systems with zero linear part and nonzero quadratic part.

*(English. Russian original)*Zbl 0944.34031
Differ. Equations 35, No. 1, 50-56 (1999); translation from Differ. Uravn. 35, No. 1, 51-57 (1999).

The authors classify all possible local phase portraits of the planar systems
\[
\begin{aligned} dx/dt &= A_1x^2+ A_2xy+ A_3y^2+ X(x,y),\\ dy/dt &= -B_1x^2- B_2xy- B_3y^2- Y(x,y),\end{aligned}\tag{\(*\)}
\]
where \(X\), \(Y\) are real-analytic in a neighborhood of the origin whose Taylor series start with third-order terms. Using affine transformations, K. S. Sibirskij had previously given a classification of \((*)\) with respect to the quadratic terms only [Differ. Equations 22, 669-674 (1986); translation from Differ. Uravn. 22, No. 6, 954-961 (1986; Zbl 0618.34027) and Introduction to the algebraic theory of invariants of differential equations. Transl. from the Russian. Nonlinear Science: Theory and Applications. Manchester etc.: Manchester University Press (1988; Zbl 0691.34031)]. He thus obtained nine types of systems \((*)\).

The present authors apply the formal transformations \[ x'= x+ \sum_{i+j\geq 2} \alpha_{ij}x^iy^j,\quad y'= y+\sum_{i+j\geq 2} \beta_{ij}x^iy^j, \]

\[ dt'= \Biggl(1+\sum_{i+ j\geq 1} \gamma_{ij}x^iy^j\Biggr) dt \] to Sibirskij’s normal forms to classify the terms of order \(\geq 3\) as well. In this way, they obtain forty-nine formal “orbital normal forms” of \((*)\). For example, Sibirskij’s normal form \[ dx/dt= x^2+ p(x,y),\quad dy/dt= xy- q(x,y), \] is reduced to the system \[ dx/dt= x^2+ \sum_{k\geq 3} C_ky^k,\quad dy/dt= xy. \] (Here, \(x'\), \(y'\), \(t'\) are replaced by \(x\), \(y\), \(t\) again.) However, some of Sibirskij’s normal forms are transformed into a considerable number of different orbital normal forms. The local phase portraits of all these orbital normal forms and, hence, all possible local phase portraits of system \((*)\), can now be constructed.

The present authors apply the formal transformations \[ x'= x+ \sum_{i+j\geq 2} \alpha_{ij}x^iy^j,\quad y'= y+\sum_{i+j\geq 2} \beta_{ij}x^iy^j, \]

\[ dt'= \Biggl(1+\sum_{i+ j\geq 1} \gamma_{ij}x^iy^j\Biggr) dt \] to Sibirskij’s normal forms to classify the terms of order \(\geq 3\) as well. In this way, they obtain forty-nine formal “orbital normal forms” of \((*)\). For example, Sibirskij’s normal form \[ dx/dt= x^2+ p(x,y),\quad dy/dt= xy- q(x,y), \] is reduced to the system \[ dx/dt= x^2+ \sum_{k\geq 3} C_ky^k,\quad dy/dt= xy. \] (Here, \(x'\), \(y'\), \(t'\) are replaced by \(x\), \(y\), \(t\) again.) However, some of Sibirskij’s normal forms are transformed into a considerable number of different orbital normal forms. The local phase portraits of all these orbital normal forms and, hence, all possible local phase portraits of system \((*)\), can now be constructed.

Reviewer: J.Hainzl (Freiburg)

##### MSC:

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |

37C10 | Dynamics induced by flows and semiflows |