arXiv subjects

Yury Kudryashov

Publications and source records attributed to Yury Kudryashov.

12 recordsLinked to original sources

arXiv2026

Integral Curves and Flows on Banach Manifolds in Lean

We present a formalisation of the existence and uniqueness theorems of integral curves of vector fields on Banach manifolds in the Lean theorem prover. First, we formalize properties of differential equations on Banach spaces (the Picard-Lindelöf theorem, the Grönwall inequality, and corollaries), and then transfer results to abstract Banach manifolds. Built upon the differential and integral calculus and Banach manifolds libraries in Mathlib, our work aims to lay the foundation for dynamical systems and differential geometry libraries that are general, robust, and friendly to classical mathematicians.

math.DG↗

arXiv2025

Aristotle: IMO-level Automated Theorem Proving

We introduce Aristotle, an AI system that combines formal verification with informal reasoning, achieving gold-medal-equivalent performance on the 2025 International Mathematical Olympiad problems. Aristotle integrates three main components: a Lean proof search system, an informal reasoning system that generates and formalizes lemmas, and a dedicated geometry solver. Our system demonstrates state-of-the-art performance with favorable scaling properties for automated theorem proving.

cs.AI↗

arXiv2021

Circle homeomorphisms with breaks with no $C^{2-ν}$ conjugacy

The rigidity theory for circle homeomophisms with breaks was studied intensively in the last 20 years. It was proved that under mild conditions of the Diophantine type on the rotation number any two $C^{2+α}$ smooth circle homeomorphisms with a break point are $C^1$ smoothly conjugate to each other, provided that they have the same rotation number and the same size of the break. In this paper we prove that the conjugacy may not be $C^{2-ν}$ even if the maps are analytic outside of the break points. This result shows that the rigidity theory for maps with singularities is very different from the linearizable case of circle diffeomorphisms where conjugacy is arbitrarily smooth, or even analytic, for sufficiently smooth diffeomorphisms.

math.DS↗

arXiv2021

Families of vector fields with many numerical invariants

We study bifurcations in finite-parameter families of vector fields on $S^2$. Recent papers by Yu. Ilyashenko, N. Goncharuk, Yu. Kudryashov, I. Schurov, and N. Solodovnikov provide examples of (locally generic) structurally unstable families of 3-parameter vector fields: generic close 3-parameter families experience different bifurcations. In this paper, we use these results to construct new examples of few-parameter generic families of planar vector fields such that their classification has many invariants. In particular, we construct (a) 3-parameter families with infinitely many numerical invariants; (b) 4-parameter families with arbitrarily many "robust" numerical invariants; (c) 5-parameter families with functional invariants.

math.DS↗

arXiv2019

Bifurcations of the polycycle "tears of the heart": multiple numerical invariants

"Tears of the heart" is a hyperbolic polycycle formed by three separatrix connections of two saddles. It is met in generic 3-parameter families of planar vector fields. In [arXiv:1506.06797], it was discovered that generically, the bifurcation of a vector field with "tears of the heart" is structurally unstable. The authors proved that the classification of such bifurcations has a numerical invariant. In this article, we study the bifurcations of "tears of the heart" in more detail, and find out that the classification of such bifurcation may have arbitrarily many numerical invariants.

math.DS↗

arXiv2018

Spectra of quadratic vector fields on $\mathbb{C}^2$: The missing relation

Consider a quadratic vector field on $\mathbb{C}^2$ having an invariant line at infinity and isolated singularities only. We define the extended spectra of singularities to be the collection of the spectra of the linearization matrices of each of the singular points over the affine part, together with all the characteristic numbers (i.e. Camacho-Sad indices) at infinity. This collection consists of 11 complex numbers, and is invariant under affine equivalence of vector fields. In this paper we describe all polynomial relations among these numbers. There are 5 independent polynomial relations; four of them follow from the Euler-Jacobi, the Baum-Bott and the Camacho-Sad index theorems, and are well known. The fifth relation was, until now, completely unknown. We provide an explicit formula for the missing 5th relation, discuss it's meaning and prove that it cannot be formulated as an index theorem.

math.CV↗

arXiv2018

Classification of generic semigroup actions of circle diffeomorphisms

We study topological properties of semi-group actions on the circle by orientation-preserving homeomorhisms. We prove that a generic action either possesses a forward-invariant interval-domain (i.e. a finite union of disjoint circle arcs), or is two-sided minimal (and, moreover, by a small perturbation one can create a map with arbitrary rotation number in the semigroup). For the minimal case, we also study conditions required for global synchronization.

math.DS↗

arXiv2017

Genera of non-algebraic leaves of polynomial foliations of $\mathbb C^2$

In this article, we prove two results. First, we construct a dense subset in the space of polynomial foliations of degree $n$ such that each foliation from this subset has a leaf with at least $\frac{(n+1)(n+2)}2-4$ handles. Next, we prove that for a generic foliation invariant under the map $(x, y)\mapsto (x, -y)$ all leaves have infinitely many handles.

math.CV↗

arXiv2017

Cheap Complex Limit Cycles

Consider a holomorphic foliation with singularities of a 2-dimensional complex manifold. In this article we prove a new sufficient condition for this foliation to have countably many homologically independent complex limit cycles. In particular, if all leaves of a foliation are dense in the phase space, and it has a complex hyperbolic singular point, then it has infinitely many homologically independent complex limit cycles.

math.CV↗

arXiv2016

Global bifurcations in the two-sphere: a new perspective

We construct an open set of structurally unstable three parameter families whose weak and so called moderate topological classification defined below has a numerical invariant that may take an arbitrary positive value. Here and below "families" are "families of vector fields in the two-sphere". This result disproves an Arnold's conjecture of 1985. Then we construct an open set of six parameter families whose moderate topological classification has a functional invariant. This invariant is an arbitrary germ of a smooth map $(\mathbb R_+, a)\to(\mathbb R_+, b)$. More generally, for any positive integers $d$ and $d'$, we construct an open set of families whose topological classification has a germ of a smooth map $\left(\mathbb R_+^d, a\right)\to\left(\mathbb R_+^{d'}, b\right)$ as an invariant. Any smooth germ of this kind may be realized as such an invariant. These results open a new perspective of the global bifurcation theory in the two sphere. This perspective is discussed at the end of the paper.

math.DS↗

arXiv2015

Bounded limit cycles of polynomial foliations of $\mathbb CP^2$

In this article we prove in a new way that a generic polynomial vector field in $\mathbb C^2$ possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set for quadratic vector fields, and provides limit cycles that stay in a bounded domain.

math.CV↗

arXiv2013

Bony attractors in higher dimension

In this article, we extend the phenomena of a bony attractor from a rather artificial class of step skew products to the class of diffeomorphisms on the Cartesian product of the two-dimensional torus by a sphere of arbitrary dimension.

math.DS↗