Integrable Cases of the Polynomial Liénard-type Equation with Resonance in the Linear Part

Edneral, V.F.

Автор: Edneral Victor F.
Журнал: Mathematics in Computer Science
Том: 17
Номер: 3-4
Год издания: 2023
Издательство: Birkhauser Verlag
Местоположение издательства: Switzerland
Номер статьи: 19
DOI: 10.1007/s11786-023-00567-6
Аннотация: The paper considers the possible relationship between the local integrability of an autonomous two-dimensional ODE system with polynomial right hand sides and its global integrability. A hypothesis is put forward that local integrability in a neighborhood of each point of some region of the phase space is necessary for the existence of the first integral in this region. By integrability in some domain of the phase space it is meant the existence there of a differentiable function which is constant along the orbit of the system. Using the example of a polynomial resonance case of the Liénard-type equation with parameters, we have written out the conditions for local integrability near stationary points and found restrictions on the parameters under which these conditions are satisfied. The resulting constraint is written as a system of algebraic equations for the ODE parameters. It is shown that for parameter values that are solutions of such an algebraic system, the ODE turns out to be integrable. In this way we have found several cases of integrability. We propose a heuristic method that allows one to a priori determine the cases of integrability of an autonomous ODE with a polynomial right-hand side. The paper has an experimental character.
Добавил в систему: Еднерал Виктор Федорович

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Integrable Cases of the Polynomial Liénard-type Equation with Resonance in the Linear Partстатья