Towards the General Theory of Global Planar Bifurcations
This is an outline of a theory to be created, as it was seen in April 2015. An addendnum to the proofs at the end of the chapter describes the recent developments.
Nobel prize winner I. Prigogine stands for peace, against the arms race, against the use of science for destruction of man and humanity. In his opinion, in the sphere of human capabilities it is essential to change the trajectory of civilization development. At the bifurcation points, unprecedented changes are possible. Instability is not a sign of weakness, but of the vitality of the system. Globalization should not mean unification, but pluralism and diversity of cultures. Science of the future needs to give a systematic explanation ofmegaera and microcosm. A sign of hope is that interest in studying nature and the desire to participate in cultural life has never been greater than today. We do not need any "post-humanity". Man, as he is today, with all his problems, joys and sorrows, is able to understand this and to keep himself in the next generations. The challenge is to find a narrow path between globalization and preservation of cultural pluralism, between violence and political solutions, between the culture of war and the culture of reason.
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.