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

TL;DR

This paper investigates families of differential equations on the two-torus, revealing that for certain vector fields, phase-lock areas can occur at all rational rotation numbers, contrasting with previously known quantization effects.

Contribution

It demonstrates that for generic analytic vector fields with multiple Fourier components, phase-lock areas can exist for all rational rotation numbers, extending previous quantization results.

Findings

Phase-lock areas can occur at all rational rotation numbers for certain vector fields.

Quantization of rotation numbers is not universal; it depends on the structure of the vector field.

Previous quantization results are specific to simpler vector fields like sine functions.

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 $\frac 1m$. 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.