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On the structure of a neighborhood of an isolated equilibrium of a local planar dynamical system admiting the first approximation

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Abstract

We show that if all parabolic sectors in the space Z ∞ are stable, then neighborhoods of the point under study in the phase planes of the spaces Z and Z ∞ have the same structure; i.e., the number and order of sectors coincide. (Parabolic sectors may degenerate into a single trajectory.) If there is no hyperbolic sector in the space Z ∞, then the spaces Z and Z ∞ are isomorphic. We present examples showing that all conditions in these assertions are essential.

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References

  1. Hajek, O., Dynamical Systems in the Plane, London, 1968.

  2. Mychka, E.Yu., The Space L(X) of Local Dynamical Systems and the V. V. Filippov Space A ceu(X), Differ. Uravn., 2010, vol. 46, no. 4, pp. 499–505.

    MathSciNet  Google Scholar 

  3. Filippov, V.V., Prostranstva reshenii obyknovennykh differentsial’nykh uravnenii (Solution Spaces of Ordinary Differential Equations), Moscow: Izd. Moskov. Univ., 1993.

    Google Scholar 

  4. Filippov, V.V., Asymptotic Integration of Ordinary Differential Equations with Discontinuities in the Right-Hand Side, Dokl. RAN, 1991, vol. 321, no. 3, pp. 482–485.

    Google Scholar 

  5. Gabdrakhmanov, S.R., Klebanov, B.S., and Filippov, V.V., On the Asymptotic Integration of Quasilinear Differential Equations, Mat. Zametki, 2001, vol. 70, no. 3, pp. 346–355.

    MathSciNet  Google Scholar 

  6. Filippov, A.F., On Stability and Instability with Respect to the First Approximation, Differ. Uravn., 2000, vol. 36, no. 3, pp. 475–485.

    Google Scholar 

  7. Hartman, Ph., Ordinary Differential Equations, New York, 1969. Translated under the title Obyknovennye differentsial’nye uravneniya, Moscow, 1970.

  8. Mychka, E.Yu., On the Structure of a Neighborhood of an Isolated Stationary Point of a Local Dynamical System on a Plane, Differ. Uravn., 2011, vol. 47, no. 2, pp. 195–208.

    Google Scholar 

  9. Sugaipova, L.S., Investigation of a Singular Point by an Axiomatic Method, Vestnik Moskov. Univ. Ser. 1 Mat. Mekh., 2004, no. 5, pp. 3–6.

  10. Stepanov, V.V. and Nemytskii, V.V., Kachestvennaya teoriya differentsial’nykh uravnenii (Qualitative Theory of Differential Equations), Moscow, 1949.

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    E. Yu. Mychka

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  1. E. Yu. Mychka
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Original Russian Text © E.Yu. Mychka, 2012, published in Differentsial’nye Uravneniya, 2012, Vol. 48, No. 2, pp. 183–195.

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Mychka, E.Y. On the structure of a neighborhood of an isolated equilibrium of a local planar dynamical system admiting the first approximation. Diff Equat 48, 189–201 (2012). https://doi.org/10.1134/S0012266112020036

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  • DOI: https://doi.org/10.1134/S0012266112020036

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