On the configurations of centers of planar Hamiltonian Kolmogorov cubic polynomial differential systems. (English) Zbl 1453.37054
This paper proposes a study of centers and their configurations for Hamiltonian cubic polynomial differential systems having two intersecting invariant straight lines. By applying an affine transformation these two invariant straight lines go through the axes of coordinates and these systems become Kolmogorov systems. The authors find that the real algebraic curve \(xy(a+bx+cy+dx^2+exy+fy^2)=h\) has at most four families of level ovals in the phase plane of such systems for all real parameters \(a\), \(b\), \(c\), \(d\), \(e\), \(f\) and \(h\).
Reviewer: Alexander Grin (Grodno)
MSC:
37J25 | Stability problems for finite-dimensional Hamiltonian and Lagrangian systems |
37C27 | Periodic orbits of vector fields and flows |
34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
Keywords:
Hamiltonian system; Kolmogorov systems; cubic polynomial differential systems; centers; configuration of centersReferences:
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