Algebraic and analytical tools for the study of the period function. (English) Zbl 1305.34050
The authors study the period function of the system
\[
\dot x=p(x,y), \quad \dot y=q(x,y),
\]
which has a center at the origin and a first integral of the form \(H(x,y)=A(x)+B(x) y+C(x) y^2\) with \(A(0)=0\). It is also assumed that the system has an integrating factor, which depends only on \(x\). First, they give a criterion for estimation the number of critical periods in such systems and a necessary condition for the period function to be monotone. Then, the study of the number of critical periods bifurcating from the period annulus of an isochrone perturbed linearly inside a family of centers is performed. Four examples illustrating the obtained results are presented.
Reviewer: Valery Romanovski (Maribor)
MSC:
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
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