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

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Abstract

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.

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References

  1. Bruno, A.D.: Asymptotic behaviour and expansions of solutions of an ordinary differential equation. Russian Math. Surveys 59(3), 429–480 (2004)

    Article  MathSciNet  Google Scholar 

  2. Bruno, A.D., Parusnikova, A.V.: Local expansions of solutions to the fifth Painlevé equation. Doklady Math. 83(3), 348–352 (2011)

    Article  MathSciNet  Google Scholar 

  3. Samovol, V.S.: On Convergent Series Expansions of Solutions of the Riccati Equation. Math. Notes 105(4), 592–603 (2019)

    Article  MathSciNet  Google Scholar 

  4. Bruno, A.D., Goryuchkina, I.V.: Convergence of power expansions of solutions to an ODE. Preprint of the Keldysh Institute of Applied Mathematics of RAS, Moscow (2013)

    Google Scholar 

  5. Gontsov, R.R., Goryuchkina, I.V.: Convergence of formal Dulac series satisfying an algebraic ordinary differential equation. Sb. Math. 210(9), 1207–1221 (2019)

    Article  MathSciNet  Google Scholar 

  6. Ilyashenko, Yu.S.: Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane. Funct. Anal. Appl. 18(3), 199–209 (1984)

    Article  MathSciNet  Google Scholar 

  7. Kamke, E.: Gewohnliche Differentialgleichungen. Akademie-Verlag, Leipzig (1959)

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  1. HSE University, Moscow, Russia

    V. S. Samovol

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  1. V. S. Samovol
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Samovol, V.S. On the Solutions of Ordinary Differential Equations in the Form of Dulac Series. Qual. Theory Dyn. Syst. 21, 47 (2022). https://doi.org/10.1007/s12346-022-00579-w

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  • DOI: https://doi.org/10.1007/s12346-022-00579-w

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