×

Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form \(H = H_1(x)+H_2(y)\). (English) Zbl 1412.37059

The phase portraits in the Poincaré disc of the linear-type centers of the polynomial Hamiltonian system \[ \dot{x}=-y(1+b_3y+b_4y^2+b_5y^3)~,~~\dot{y}=x(1+a_3x+a_4x^2+a_5x^3), \] are analyzed.
The main result gives topological equivalences of the phase portraits in the Poincaré disc of the linear-type centers of the considered system. This result provides the global phase portraits in the Poincaré disc as well as some local phase portraits. The results depend on the roots of the polynomials \(\hat p(y)=1+b_3y+b_4y^2+b_5y^3\) and \(\hat q(x)=1+a_3x+a_4x^2+a_5x^3.\) A characterization of the polynomials \(\hat p\) and \(\hat q\) according to their roots is given. The local phase portraits of all the possible finite equilibrium points in relation to their eigenvalues and linear matrix are analyzed. Moreover the infinite equilibrium points and their stability are characterized.

MSC:

37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37C10 Dynamics induced by flows and semiflows
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. A. Andronov, A. A. Vitt and S. E. Khaikin, Theory of Oscillators, Dover Publications Inc., New York, 1987. Translated form the Russian by F. Immirzi, Reprint of the 1966 translation. · Zbl 0188.56304
[2] V. I. Arnold and Y. S. Ilyashenko, Dynamical Systems I, Ordinary Differential Equation, Encyclopaedia of Mathematical Sciences, Vols 1-2, Springer-Verlag, Heidelberg, 1988.
[3] J. C. Artés; J. Llibre, Quadratic Hamiltonian vector fields, J. Differential Equations, 107, 80-95 (1994) · Zbl 0791.34048
[4] N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sb., 30 (1952), 181-196; Mer. Math. Soc. Transl., 100 (1954), 1-19. · Zbl 0059.08201
[5] J. Chavarriga; J. Giné, Integrability of a linear center perturbed by a fourth degree homogeneous polynomial, Publ. Mat., 40, 21-39 (1996) · Zbl 0851.34001
[6] J. Chavarriga; J. Giné, Integrability of a linear center perturbed by a fifth degree homogeneous polynomial, Publ. Mat., 41, 335-356 (1997) · Zbl 0897.34030
[7] I. Colak; J. Llibre; C. Valls, Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, Adv. Math., 259, 655-687 (2014) · Zbl 1303.34027
[8] I. Colak; J. Llibre; C. Valls, Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 257, 1623-1661 (2014) · Zbl 1307.34060
[9] I. Colak; J. Llibre; C. Valls, Bifurcations diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 258, 846-879 (2015) · Zbl 1309.34045
[10] I. Colak; J. Llibre; C. Valls, Bifurcations diagrams for nilpotent centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 262, 5518-5533 (2017) · Zbl 1369.34057
[11] H. Dulac, Détermination et integration dâ une certaine classe dâ équations différentielle ayant par point singulier un centre, Bull. Sci. Math. Sér., 32, 230-252 (1908) · JFM 39.0374.01
[12] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, 2006. · Zbl 1110.34002
[13] J. García-Saldana; A. Gasull; H. Giacomini, Bifurcation values for a family of planar vector fields of degree five, Discrete Contin. Dyn. Syst., 35, 669-701 (2015) · Zbl 1319.34066
[14] A. Garijo; A. Gasull; X. Jarque, Local and global phase portrait of equation \(\begin{document} \dot{z} = f(z)\end{document} \), Discrete Contin. Dyn. Syst., 17, 309-329 (2007) · Zbl 1125.34025
[15] H. Goldstein, Classical Mechanics, Addison-Wesley Press, Inc., Cambridge, Mass, 1951. · Zbl 0043.18001
[16] A. Guillamon; C. Pantazi, Phase portraits of separable Hamiltonian systems, Nonl. Analysis, 74, 4012-4035 (2011) · Zbl 1223.37066
[17] W. Kapteyn, On the midpoints of integral curves of differential equations of the first Degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk. Konikl., Nederland, (1911), 1446-1457 (in Dutch).
[18] W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk, 20 (1912), 1354-1365; Nederl. Akad. Wetensch. Verslag Afd. Natuurk, 21 (1913), 27-33 (in Dutch).
[19] J. H. Kim; S. W. Lee; H. Massen; H. W. Lee, Relativistic oscillator of constant period, Phys. Rev. A, 53, 2991-2997 (1996)
[20] K. Lan; Ch. Zhu, Phase portraits of predator-prey systems with harvesting rates, Discrete Contin. Dyn. Syst., 32, 901-933 (2012) · Zbl 1252.34052
[21] K. E. Malkin, Criteria for the center for a certain differential equation, Vols. Mat. Sb. Vyp., 2 (1964), 87-91 (in Russian).
[22] L. Markus, Global structure of ordinary differential equations in the plane, Trans. Amer. Math Soc., 76, 127-148 (1954) · Zbl 0055.08102
[23] Y. P. Martínez; C. Vidal, Classification of global phase portraits and bifurcation diagrams of Hamiltonian systems with rational potential, J. Differential Equations, 261, 5923-5948 (2016) · Zbl 1360.34072
[24] D. A. Neumann, Classification of continuous flows on 2-manifolds, Proc. Amer. Math. Soc., 48, 73-81 (1975) · Zbl 0307.34044
[25] M. M. Peixoto, Dynamical Systems. Proccedings of a Symposium held at the University of Bahia, Acad. Press, New York, (1973), 389-420.
[26] H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, Journal de Mathématiques, 37 (1881), 375-422; Oeuvres de Henri Poincaré, vol. Ⅰ, Gauthier-Villars, Paris, 1951, 3-84.
[27] H. Stommel, Trajectories of small bodies sinking slowly through convection cells, J. Mar. Res., 8, 24-29 (1949)
[28] N. I. Vulpe, Affine-invariant conditions for the topological discrimination of quadratic systems with a center, Differential Equations, 19, 273-280 (1983) · Zbl 0556.34019
[29] N. I. Vulpe and K. S. Sibirskii, Centro-affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities, Dokl. Akad. Nauk. SSSR, 301 (1988), 1297-1301 (in Russian); translation in: Soviet Math. Dokl., 38 (1989), 198-201.
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.