Liouvillian integrability and the Poincaré problem for nonlinear oscillators with quadratic damping and polynomial forces. (English) Zbl 1482.34003
The upper bound on the degrees of irreducible Darboux polynomials associated to the ordinary differential equations
\[
x_{tt}+x_{t}^2+x_{t}+f(x)=0\text{ with }f(x)\in\mathbb{C}[x]\backslash\mathbb{C}\text{ and }\neq 0
\]
is derived. The Poincaré problem for dynamical systems is solved. Two useful tables on irreducible Darboux poynomials and on Liouvillian first integrals of dynamical systems are presented.
Reviewer: Ioannis P. Stavroulakis (Ioannina)
MSC:
| 34A05 | Explicit solutions, first integrals of ordinary differential equations |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 34C45 | Invariant manifolds for ordinary differential equations |
Keywords:
Darboux polynomials; invariant algebraic curves; Poincaré problem; Liouvillian integrability; nonlinear oscillators with quadratic dampingReferences:
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