Valery G. Romanovski

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We give a short proof of Urabe’s criteria for the isochronicity of periodical solutions of the equation ẍ+ g(x) = 0. We show that apart from the harmonic oscillator there exists a large family of isochronous potentials which must be all non-polynomial and not symmetric (even function of the coordinate x). PACS numbers: 46.10.+z, 95.10.Ce AMS classification(More)
We investigate some properties of the WKB series for arbitrary analytic potentials and then specifically for potentials x (N even), where more explicit formulae for the WKB terms are derived. Our main new results are: (i) We find the explicit functional form for the general WKB terms σ k, where one has only to solve a general recursion relation for the(More)
K e y w o r d s C e n t e r problem, Li~nard system, Polynomial systems, Local integrability. Consider the system 1. I N T R O D U C T I O N n dx dy x + E Y~ (x, y) = Y(x , y), (1) d---[ = y + Xs (x, y) = X (x, y), d--t -s=2 s=2 where Xs(x, y), Ys(x, y) are polynomials in x and y of degree s. Conversion to polar coordinates shows tha t near the origin(More)
Necessary conditions and distinct sufficient conditions are derived for the system _ x 1⁄4 xð1 a20x a11xy a02yÞ, _ y 1⁄4 yð qþ b20x þ b11xyþ b02yÞ to admit a first integral of the form Uðx; yÞ 1⁄4 xqyþ in a neighborhood of the origin, in which case the origin is termed a 1 : q resonant center. Necessary and sufficient conditions are obtained for odd q; q 6(More)