On the Solutions of Ordinary Differential Equations in the Form of Dulac Series

• V. S. Samovol ^[1]
  1. [1] HSE University
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 2, 2022
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We consider a nonlinear ordinary differential equation of arbitrary order with coefficients in the form of power series that converge in a neighborhood of the origin. The methods created in power geometry in recent years make it possible to compute formal solutions to that equation in the form of Dulac series. We describe the corresponding algorithm and prove a sufficient convergence condition for such formal solutions.

• Referencias bibliográficas
  • 1. Bruno, A.D.: Asymptotic behaviour and expansions of solutions of an ordinary differential equation. Russian Math. Surveys 59(3), 429–480...
  • 2. Bruno, A.D., Parusnikova, A.V.: Local expansions of solutions to the fifth Painlevé equation. Doklady Math. 83(3), 348–352 (2011)
  • 3. Samovol, V.S.: On Convergent Series Expansions of Solutions of the Riccati Equation. Math. Notes 105(4), 592–603 (2019)
  • 4. Bruno, A.D., Goryuchkina, I.V.: Convergence of power expansions of solutions to an ODE. Preprint of the Keldysh Institute of Applied Mathematics...
  • 5. Gontsov, R.R., Goryuchkina, I.V.: Convergence of formal Dulac series satisfying an algebraic ordinary differential equation. Sb. Math....
  • 6. Ilyashenko, Yu.S.: Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane. Funct. Anal. Appl. 18(3),...
  • 7. Kamke, E.: Gewohnliche Differentialgleichungen. Akademie-Verlag, Leipzig (1959)