Analytical Properties of Solutions of Three-Dimensional Conservative Systems with Two or Four Quadratic Nonlinearities
https://doi.org/10.1134/S2304487X2104012X
Abstract
The analytical properties of solutions of three families of three-dimensional autonomous conservative systems with two or four quadratic nonlinearities are studied. The conservative systems of the first family have two quadratic nonlinearities and two linear components. The conservative systems of the second family have two quadratic nonlinearities, one linear component, and one constant term. The conservative systems of the third family have four quadratic nonlinearities. To analyze the solutions of the systems under consideration, the Painlevé test, the reduction of the systems to second- or third-order equations equivalent to them, and the comparison of the latter with the known nonlinear P-type equations are used. Three systems whose general solutions have the Painlevé property are separated. The solutions of one of these systems are expressed in terms of elementary functions, and the other two are expressed in terms of solutions of the first or second Painlevé equation. It is shown that there are some non-Painlevé -type systems, one of the components of which does not have any moving singular points. For each of the systems of the third family, exact one-parameter families of solutions are constructed. In addition, it is shown that one of the systems of the third family has a two-parameter family of meromorphic solutions. A common qualitative property of thesesystems, except for one, is the absence of chaos in them.
About the Author
V. V. Tsegel’nik
Belarusian State University of Informatics and Radioelectronics
Belarus
Belarus
220013
Minsk
References
1. Tsegel’nik V. V. Analiticheskie svoistva reshenii trekhmernikh autonomnykh konservativnikh system s odnoi ili tremya kvadratichnimi nelineinostyami bes chaoticheskogo povedeniya [Analytical properties of solutions of three-dimensional autonomous conservative systems with one or three quadratic nonlinearities without chaotic behavior]. Vestnik NIYaU MIFI. 2020, vol. 9, no. 4. pp. 338–344 (in Russian).
2. Heidel J., Zhang Fu. Nonchaotic behaviour in three – dimensional quadratic systems. II. The conservative case. Nonlinearity. 1999. Vol. 12. P. 617–633.
3. Kudryashov N. A. Analyticheskaya teoriya nelineynykh differentsial’nykh uravnenii [Analytical theory of non-linear differential equations] Moskau-Izhevsk, Institute computernykh issledovaniy, 2004. 360 p.
4. Gromak V. I., Gritsuk E. V. K theorii nelineynykh differentsial’nykh uravnenii so svoistvom Penleve [On the theory nonlinear differential equations with Painleve’ property]. Izvestiya NAN Belarusi. Seria fiz.-mat. Nauk, 2010, no. 3. Pp. 25–30 (in Russian).
5. Tsegel’nik V. V. Analiticheskie svoistva reshenii autonomnykh system nelineinykh differentsial’nykh uravnenii tret’ego i chetvertogo poryadkov s chaoticheskim povedeniem [Analytical properties of solutions to autonomous systems of nonlinear third and fourth-order differential equations with chaotic behavior]. Vestnik NIYaU MIFI, 2015. vol. 4, no. 2. Pp. 101–106 (in Russian).
6. Ince E. L. Obiknovennye differentsial’nye uravneniya [Ordinary differential equations] Kharkov. ONTI, 1939. 720 p.
7. Cosgrove C. M. Chazy classes IX–XI of third-order differential equations. Stud. Appl. Math. 2001. Vol. 104, no. 3. Pp. 171–228.
8. Katke E. Spravochnik po obiknovennim differentsial’nim uravneniyam [Handbook of ordinary differentsial equations]. Moscow, Nauka Publ. 1971. 576 р.
9. Wittich H. Noveishie issledovaniya po odnoznachnym analiticheskim funktsiyam [The latest research on the unique analytic functions] Moscow, GIFML Publ. 1960. 319 p.
10. Yablonskii A. I. Ob odnoi systeme differential’nykh uravnenii bez podvizhnykh kriticheskikh tochek [On one system differential equations without movable critical points] Differentsial’nye uravneniya. 1966. vol. 2, no. 6. pp. 752–762 (in Russian).
Review
For citations:
Tsegel’nik V.V. Analytical Properties of Solutions of Three-Dimensional Conservative Systems with Two or Four Quadratic Nonlinearities. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2021;10(4):295-301. (In Russ.) https://doi.org/10.1134/S2304487X2104012X
Views:
95
Email this article 