Document Zbl 1453.37054

On the configurations of centers of planar Hamiltonian Kolmogorov cubic polynomial differential systems. (English) Zbl 1453.37054

Pac. J. Math. 306, No. 2, 611-644 (2020).

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)

Cited in 2 Documents

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 centers

Cite Review PDF

Full Text: DOI Link

References:

[1]	10.1090/trans2/008/07 · Zbl 0079.11301 · doi:10.1090/trans2/008/07
[2]	; Arnold, Mathematical methods of classical mechanics. Grad. Texts in Math., 60 (1978) · Zbl 0386.70001
[3]	; Bautin, Mat. Sb. (N.S.), 30, 181 (1952)
[4]	10.1007/s12346-008-0021-2 · Zbl 1311.34061 · doi:10.1007/s12346-008-0021-2
[5]	10.1016/j.jde.2014.05.024 · Zbl 1307.34060 · doi:10.1016/j.jde.2014.05.024
[6]	10.1016/j.aim.2014.04.002 · Zbl 1303.34027 · doi:10.1016/j.aim.2014.04.002
[7]	10.1016/j.jde.2014.10.006 · Zbl 1309.34045 · doi:10.1016/j.jde.2014.10.006
[8]	10.1016/j.jde.2017.02.001 · Zbl 1369.34057 · doi:10.1016/j.jde.2017.02.001
[9]	; Dulac, Bull. Sci. Math. (2), 32, 230 (1908) · JFM 39.0374.01
[10]	10.1007/978-3-540-32902-2 · doi:10.1007/978-3-540-32902-2
[11]	; Hilbert, Gött. Nachr., 1900, 253 (1900) · JFM 31.0068.03
[12]	; Hofbauer, The theory of evolution and dynamical systems: mathematical aspects of selection. Lond. Math. Soc. Student Texts, 7 (1988) · Zbl 0678.92010
[13]	10.1090/S0273-0979-02-00946-1 · Zbl 1004.34017 · doi:10.1090/S0273-0979-02-00946-1
[14]	; Kapteyn, Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 19, 1446 (1911)
[15]	; Kapteyn, Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 20, 1354 (1912)
[16]	; Kolmogorov, Giorn. Istituto Ital. Attuari, 7, 74 (1936) · Zbl 0013.31404
[17]	10.1142/S0218127403006352 · Zbl 1063.34026 · doi:10.1142/S0218127403006352
[18]	10.21042/AMNS.2016.1.00007 · Zbl 1386.34056 · doi:10.21042/AMNS.2016.1.00007
[19]	10.1016/j.jde.2016.10.042 · Zbl 1354.37061 · doi:10.1016/j.jde.2016.10.042
[20]	; Malkin, Volž. Mat. Sb. Vyp., 2, 87 (1964)
[21]	; Poincaré, J. Math. Pures Appl., 7, 375 (1881) · JFM 13.0591.01
[22]	10.2307/2154430 · Zbl 0777.58028 · doi:10.2307/2154430
[23]	10.1007/BF00307854 · Zbl 0344.92009 · doi:10.1007/BF00307854
[24]	; Volterra, Mem. Accad. Lencei, 2, 31 (1926)
[25]	; Vulpe, Differ. Uravn., 19, 371 (1983) · Zbl 0514.34030
[26]	; Vulpe, Dokl. Akad. Nauk SSSR, 301, 1297 (1988)
[27]	10.1090/mmono/101 · doi:10.1090/mmono/101
[28]	10.12775/TMNA.1994.024 · Zbl 0820.34016 · doi:10.12775/TMNA.1994.024
[29]	10.1088/0951-7715/7/1/013 · Zbl 0838.34035 · doi:10.1088/0951-7715/7/1/013
[30]	10.1006/jdeq.1994.1049 · Zbl 0797.34044 · doi:10.1006/jdeq.1994.1049

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.