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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 ˙ 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 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) = 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. * CNRS, France (UMR 5669 (UMPA, ENS de Lyon) and UMI 2615 (Lab. J.-V.Poncelet)).
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Submitted on : Friday, December 9, 2016 - 11:56:56 PM
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Alexey Glutsyuk, Leonid Rybnikov. On families of differential equations on two-torus with all phase-lock areas. Nonlinearity, 2017, 30 (1), pp.61-72. ⟨10.1088/0951-7715/30/1/61⟩. ⟨ensl-01413585⟩



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