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

Nonlinearity. 2017. Vol. 30. No. 1. P. 61–72.

Glutsyuk A., Rybnikov L. G.

We consider two-parametric families of non-autonomous ordinary differential equations on the two-torus with coordinates (x, t) of the type x'=v(x)+A+Bf(t). We study its rotation number as a function of the parameters (A, B). The phase-lock areas are those level sets of the rotation number function that have non-empty interiors. Buchstaber, Karpov and Tertychnyi studied the case when v(x)=sin x in their joint paper. They 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 Ilyashenko, Filimonov and Ryzhov. This implies a similar quantization effect for every v(x)=a sin(mx)+b cos(mx)+c and rotation numbers that are multiples of 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.

Keywords: rotation number, Lie algebra, vector field, differential equations on torus, phase-lock area

Publication based on the results of: Теория представлений и математическая физика (2017)