##
**Linearizability of linear systems perturbed by fifth degree homogeneous polynomials.**
*(English)*
Zbl 1127.34020

The authors study the linearizability problem of the planar complex system
\[
\dot x = x + P(x,y), \quad \dot y = -y + Q(x,y), \tag{1}
\]

where \(P(x,y)\) and \(Q(x,y)\) are homogeneous polynomials of 5-th degree, in a neighborhood of the singular point \(x=y=0\). They obtain necessary and sufficient conditions for linearizability of the germ of system (1) in form of explicit relations between the coefficients of the polynomials \(P(x,y)\) and \(Q(x,y)\). Using this result, the authors consider the isochronicity problem for the planar real system

\[ \dot u = -v + P(u,v), \quad \dot v = u + Q(u,v), \tag{2} \]

in a neighborhood of the centre \(u=v=0\), where all non-stationary trajectories are periodic. Here \(P(u,v)\) and \(Q(u,v)\) are also homogeneous polynomials of 5-th degree. If all non-stationary trajectories are periodic, the problem then arises to determine whether the period of oscillations is constant for all trajectories (a centre with such property is called isochronous centre).

From the results of Poincaré and Lyapunov it follows that the centre is isochronous if and only if the germ of system (2) is linearizable, i.e., if there exists an analytic transformation, which brings (2) into the linear system \(\dot u = -v\), \(\dot v = u\). Hence the authors obtain necessary and sufficient conditions of isochronicity for the germ of system (2). A new example of isochronous system of the given form is found.

where \(P(x,y)\) and \(Q(x,y)\) are homogeneous polynomials of 5-th degree, in a neighborhood of the singular point \(x=y=0\). They obtain necessary and sufficient conditions for linearizability of the germ of system (1) in form of explicit relations between the coefficients of the polynomials \(P(x,y)\) and \(Q(x,y)\). Using this result, the authors consider the isochronicity problem for the planar real system

\[ \dot u = -v + P(u,v), \quad \dot v = u + Q(u,v), \tag{2} \]

in a neighborhood of the centre \(u=v=0\), where all non-stationary trajectories are periodic. Here \(P(u,v)\) and \(Q(u,v)\) are also homogeneous polynomials of 5-th degree. If all non-stationary trajectories are periodic, the problem then arises to determine whether the period of oscillations is constant for all trajectories (a centre with such property is called isochronous centre).

From the results of Poincaré and Lyapunov it follows that the centre is isochronous if and only if the germ of system (2) is linearizable, i.e., if there exists an analytic transformation, which brings (2) into the linear system \(\dot u = -v\), \(\dot v = u\). Hence the authors obtain necessary and sufficient conditions of isochronicity for the germ of system (2). A new example of isochronous system of the given form is found.

Reviewer: Alexey Remizov (Porto)

### MSC:

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

34C25 | Periodic solutions to ordinary differential equations |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |