ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

On Polynomial Double Periodic Differential Equations


Eugenii Konstantinovich Ershov

St. Petersburg State University of Architecture and Civil Engineering,


In the paper we consider first order ordinary differential equations with right-hand side that is periodic in both variables and is a trigonometric polynom of degree n with respect to the dependent variable. We prove that every such an equation with rational rotation number p/q, where q does not exceed n, can be approximated by a structurally stable polynomial equation of degree n. It is also shown that in the case when n=1 the existence of such an approximation implies q=1.


  1. Pliss V. A. Nelocal’nye problemi teorii kolebanii [Nonlocal problems of the theory of oscillations]. Moscow-Leningrad, Nauka Publ., 1964. 368 p
  2. Pliss V. A. [On structural stability of differential equations on the torus. ] Vestnik Leningradskogo Universiteta, Ser. Mat. Mech. Astron., 1960; (3):15-23. (In Russ. )
  3. Ershov E. K. [On structurally stable polynomial diffeomorphisms of the circle. ] Differentsial’nye Uravneniya, 1988; (4):687-689. (In Russ. )
  4. Ershov E. K. [On the number of cycles of some differential equations on a two-dimensional torus. ] Differentsial’nye Uravneniya, 1991; (12):2167-2169. (In Russ. )
  5. Arnol'd V. I., Yu. S. Ilyashenko, Obyknovennye Differentsial’nye Uravneniya I. Dynamicheskie systemi [Ordinary differential equations I. Dynamical systems]. In: Sovrem. Probl. Mat., Fund. Naprav., vol. 1, VINITI, Moscow, 1985, 7-149
  6. Arnol'd V. I., [Remarks on the perturbation theory for problems of Mathieu type. ] Uspekhi Matematicheskikh Nauk, 1983; (4): 189-203. (In Russ. )

Full text (pdf)