arXiv subjects

Ivan O. Shevchenko

Publications and source records attributed to Ivan O. Shevchenko.

2 recordsLinked to original sources

arXiv2026

Stability and Bifurcations of Planar Switched Linear and Homogeneous Systems

We prove new necessary and sufficient conditions for uniform asymptotic stability under arbitrary switching of two-dimensional switched homogeneous systems with a finite number of subsystems using a worst-case switching analysis. The novelty of our approach is in its explicit nature, which allows us to then study in detail the codimension-one bifurcations of stability of the origin in switched linear systems and further conclude new local and global stability results for certain classes of nonlinear switched systems. In particular, we formulate an analogue of Lyapunov's indirect method for $\mathcal{C}^{1}$ switched nonlinear systems and derive a new method for determining the existence of a bounded basin of attraction for a class of switched nonlinear systems.

math.DS↗

arXiv2025

Computability of Brjuno-like functions

In his seminal paper from 1936, Alan Turing introduced the concept of non-computable real numbers and presented examples based on the algorithmically unsolvable Halting problem. We describe a different, analytically natural mechanism for the appearance of non-computability. Namely, we show that additive sampling of orbits of certain skew products over expanding dynamics produces Turing non-computable reals. We apply this framework to Brjuno-type functions to demonstrate that they realize bijections between computable and lower-computable numbers, generalizing previous results of M. Braverman and the second author for the Yoccoz-Brjuno function to a wide class of examples, including Wilton's functions and generalized Brjuno functions.

math.DS↗