On families of differential equations on two-torus with all phase-lock areas

Abstract

We consider two-parametric families of non-autonomous ordinary differential equations on the two-torus with the coordinates $(x,t)$ of the type $\dot{x}=v(x)+A+Bf(t)$. We study its rotation number as a function of the parameters $(A,B)$. The {\it phase-lock areas} are those level sets of the rotation number function $\rho =\rho (A,B)$ that have non-empty interiors. V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi have studied the case, when $v(x)=\sin x$ in their joint paper. They have observed the quantization effect: for every smooth periodic function $f(t)$ the family of equations may have phase-lock areas only for integer rotation numbers. Another proof of this quantization statement was later obtained in a joint paper by Yu.S.Ilyashenko, D.A.Filimonov, D.A.Ryzhov. This implies the similar quantization effect for every $v(x)=a\sin (mx)+b\cos (mx)+c$ and rotation numbers that are multiples of $\frac{1}{m}$. We show that for every other analytic vector field $v(x)$ (i.e., having at least two Fourier harmonics with non-zero non-opposite degrees and nonzero coefficients) there exists an analytic periodic function $f(t)$ such that the corresponding family of equations has phase-lock areas for all the rational values of the rotation number.

arXiv comment15 pages. To appear in Nonlinearity. This version differs from the previous one by terminology (the "Arnold tongues" are changed to "phase-lock areas"), additional historical remarks and bibliography and minor English corrections