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Strong isochronism of a center and a focus for systems with homogeneous nonlinearities

  • Ordinary Differential Equations
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Abstract

We suggest a new approach to studying the isochronism of the system

$$ {{dx} \mathord{\left/ {\vphantom {{dx} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = - y + p_n (x,y),{{dy} \mathord{\left/ {\vphantom {{dy} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = x + q_n (x,y), $$

where p n and q n are homogeneous polynomials of degree n. This approach is based on the normal form

$$ {{dX} \mathord{\left/ {\vphantom {{dX} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = - Y + XS(X,Y),{{dY} \mathord{\left/ {\vphantom {{dY} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = X + YS(X,Y) $$

and its analog in polar coordinates. We prove a theorem on sufficient conditions for the strong isochronism of a center and a focus for the reduced system and obtain examples of centers with strong isochronism of degrees n = 4, 5. The present paper is the first to give examples of foci with strong isochronism for the system in question.

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Authors and Affiliations

  1. Belarus State University, Minsk, Belarus

    A. E. Rudenok

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  1. A. E. Rudenok
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Original Russian Text © A.E. Rudenok, 2009, published in Differentsial’nye Uravneniya, 2009, Vol. 45, No. 2, pp. 154–161.

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Rudenok, A.E. Strong isochronism of a center and a focus for systems with homogeneous nonlinearities. Diff Equat 45, 159–167 (2009). https://doi.org/10.1134/S0012266109020025

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  • DOI: https://doi.org/10.1134/S0012266109020025

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