Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T05:28:32.941Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Poincaré problem and Liouvillian integrability of quadratic Liénard differential equations

Part of: Qualitative theory

Published online by Cambridge University Press:  09 January 2020

Maria V. Demina  and
Claudia Valls
Maria V. Demina
Affiliation:
National Research University Higher School of Economics, 34 Tallinskaya Street, 123458, Moscow, Russian Federation (maria_dem@mail.ru)
Claudia Valls
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001Lisboa, Portugal (cvalls@math.tecnico.ulisboa.pt)
Get access Rights & Permissions [Opens in a new window]

Abstract

We present the complete classification of irreducible invariant algebraic curves of quadratic Liénard differential equations. We prove that these equations have irreducible invariant algebraic curves of unbounded degrees, in contrast with what is wrongly claimed in the literature. In addition, we classify all the quadratic Liénard differential equations that admit a Liouvillian first integral.

Keywords

invariant algebraic curvesDarboux polynomialsLiouvillian first integralsLiénard systemsPoincaré problem

MSC classification

Primary: 34C05: Location of integral curves, singular points, limit cycles
Secondary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) 34C08: Connections with real algebraic geometry (fewnomials, desingularization, zeros of Abelian integrals, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3231 - 3251
Check for updates
Copyright
Copyright © 2020 The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bruno, A. D.. Power geometry in algebraic and differential equations (North-Holland: Elsevier Science, 2000).Google Scholar
2Bruno, A. D.. Asymptotic behaviour and expansions of solutions of an ordinary differential equation. Russ. Math. Surv. 59 (2004), 429–481.CrossRefGoogle Scholar
3Chavarriga, J., García, I. A. and Sorolla, J.. Resolution of the Poincaré problem and nonexistence of algebraic limit cycles in family (I) of Chinese classification. Chaos Soliton. Fract. 24 (2005), 491–499.CrossRefGoogle Scholar
4Christopher, C. J.. Invariant algebraic curves and conditions for a centre. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 1209–1229.CrossRefGoogle Scholar
5Demina, M. V.. Invariant algebraic curves for Liénard dynamical systems revisited. Appl. Math. Lett. 84 (2018), 42–48.CrossRefGoogle Scholar
6Demina, M. V.. Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems. Phys. Lett. A 382 (2018), 1353–1360.CrossRefGoogle Scholar
7Dumortier, F., Llibre, J. and Artés, J. C.. Qualitative theory of planar differential systems (New York: Springer Verlag, 2006).Google Scholar
8Fischer, G.. Plane algebraic curves (American Mathematical Society, 2001).Google Scholar
9Poincaré, H.. Sur l’integration algébrique des équations differentielles du 1-er ordre. Rend. Circ. Mat. Palermo 11 (1891), 193–239.CrossRefGoogle Scholar
10Singer, M.. Liouvillian first integrals of differential systems. Trans. Am. Math. Soc. 333 (1992), 673–688.CrossRefGoogle Scholar
11Valls, C.. Liouvillian integrability of some quadratic Liénard polynomial differential systems. Rend. Circ. Mat. Palermo 68 (2019), 499–519.CrossRefGoogle Scholar
12Z̆ola̧dek, H.. Algebraic invariant curves for the Liénard equation. Trans. Am. Math. Soc. 350 (1998), 1681–1701.CrossRefGoogle Scholar