2015/05/26 by Glutsyuk, Alexey, Rybnikov, Leonid · 2 citations
#37C10 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1505.06975
We consider two-parametric families of non-autonomous ordinary differential equations on the two-torus with the 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 \it phase-lock areas are those level sets of the rotation number function ρ=ρ(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)=asin(mx)+bcos(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.