arXiv subjects

Nikita Solodovnikov

Publications and source records attributed to Nikita Solodovnikov.

3 recordsLinked to original sources

arXiv2019

New structurally unstable families of planar vector fields

We study global bifurcations in generic 3-parameter families of vector fields on $S^2$. In the recent article [arXiv:1506.06797], Ilyashenko, Kudryashov, and Schurov show that 3-parameter unfoldings of vector fields with the polycycle "tears of the heart" are structurally unstable. We consider 3-parameter unfoldings of vector fields with separatrix graphs "ears" and "glasses", and prove that these families are structurally unstable as well. We also study in more details the classical bifurcation of a saddle loop, and use it as a building block in our main example.

math.DS↗

arXiv2019

Global bifurcations in generic one-parameter families with a parabolic cycle on $S^2$

We classify global bifurcations in generic one-parameter local families of \vfs on $S^2$ with a parabolic cycle. The classification is quite different from the classical results presented in monographs on the bifurcation theory. As a by product we prove that generic families described above are structurally stable.

math.DS↗

arXiv2016

Duck factory on the two-torus: multiple canard cycles without geometric constraints

Slow-fast systems on the two-torus are studied. As it was shown before, canard cycles are generic in such systems, which is in drastic contrast with the planar case. It is known that if the rotation number of the Poincare map is integer and the slow curve is connected, the number of canard limit cycles is bounded from above by the number of fold points of the slow curve. In the present paper it is proved that there are no such geometric constraints for non-integer rotation numbers: it is possible to construct generic system with as simple as possible slow curve and arbitrary many limit cycles.

math.DS↗