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Sergey Tikhomirov

Publications and source records attributed to Sergey Tikhomirov.

At least 19 recordsLinked to original sources
arXiv2025

Generalized hyperbolicity for diffeomorphisms of Banach spaces

We introduce generalized hyperbolicity for nonlinear dynamics in Banach spaces. The definition allows the stable/unstable splitting to be discontinuous and requires only inclusions, rather than equalities, in the invariance conditions for both subspaces. On smooth compact manifolds, generalized hyperbolicity is equivalent to Axiom~A and the strong transversality condition, providing a finite-dimensional calibration of the proposed Banach-space theory. For $C^1$-diffeomorphisms of the whole Banach space such that $Df$ and $D(f^{-1})$ are globally bounded and $Df$ is uniformly continuous, we establish the principal dynamical consequences of generalized hyperbolicity: Lipschitz shadowing, density of periodic points in the chain-recurrent set, and robustness under perturbations small in the uniform $C^1$ distance. The shadowing result requires no continuity of the splitting, and shadowing trajectories need not be unique. Under the additional assumption that the splitting is uniformly continuous, we prove semi-structural stability. If, in addition, the subspace $E^s_x\oplus (Df(x))^{-1}E^u_{f(x)}$ is independent of $x$, we obtain structural stability.

math.DS↗
arXiv2024

Properties of the phase boundary in the parabolic problem with hysteresis

We study solutions of parabolic equations with a discontinuous hysteresis operator, described by a free interface boundary. It is established that for spatially transverse initial data from the space $W^{2-2/q}_q$ with $q > 3$, there exists a solution in the space $W^{2,1}_q$, where the interface boundary exhibits Holder continuity with an exponent $1/2$. Furthermore for initial data from the space $W^2_\infty$, it is proven that the interface boundary satisfies the Lipschitz condition. It is shown that for non-transversal initial data, solutions with an interface boundary do not exist.

math.AP↗
arXiv2024

Propagating terrace in a two-tubes model of gravitational fingering

We study a semi-discrete model for the two-dimensional incompressible porous medium (IPM) equation describing gravitational fingering phenomenon. The model consists of a system of advection-reaction-diffusion equations on concentration, velocity and pressure, describing motion of miscible liquids under the Darcy's law in two vertical tubes (real lines) and interflow between them. Our analysis reveals the structure of gravitational fingers in this simple setting - the mixing zone consists of space-time regions of constant intermediate concentrations and the profile of propagation is characterized by two consecutive traveling waves which we call a terrace. We prove the existence of such a propagating terrace for the parameters corresponding to small distances between the tubes. This solution shows the possible mechanism of slowing down the fingers' growth due to convection in the transversal direction. The main tool in the proof is a reduction to pressure-free transverse flow equilibrium (TFE) model using geometrical singular perturbation theory and the persistence of stable and unstable manifolds under small perturbations.

math.AP↗
arXiv2024

Probabilistic shadowing in linear skew products

We investigate the probability of shadowing of a random finite pseudotrajectory by an exact trajectory for linear skew products. We describe general conditions under which a random pseudotrajectory can be shadowed with polynomial (with respect to its length) precision with high probability. Examples satisfying that general condition are continuous linear skew products over Bernoulli shift, doubling map on a circle, and any Anosov linear map on a torus. The main tool used in the proof is Cramer's large deviation theorem.

math.DS↗
arXiv2024

Velocity of viscous fingers in miscible displacement: Intermediate concentration

We investigate one-phase flow in porous medium corresponding to a miscible displacement process in which the viscosity of the injected fluid is smaller than the viscosity in the reservoir fluid, which frequently leads to the formation of a mixing zone characterized by thin fingers. The mixing zone grows in time due to the difference in speed between its leading and trailing edges. The transverse flow equilibrium (TFE) model provides estimates of these speeds. We propose an enhancement for the TFE estimates, and provide its theoretical justification. It is based on the assumption that an intermediate concentration exists near the tip of the finger, which allows to reduce the integration interval in the speed estimate. Numerical simulations were conducted that corroborate the new estimates within the computational fluid dynamics model. The refined estimates offer greater accuracy than those provided by the original TFE model.

physics.flu-dyn↗
arXiv2020

New moduli components of rank 2 bundles on projective space

We present a new family of monads whose cohomology is a stable rank two vector bundle on $\mathbb{P}^3$. We also study the irreducibility and smoothness together with a geometrical description of some of these families. These facts are used to construct a new infinite series of rational moduli components of stable rank two vector bundles with trivial determinant and growing second Chern class. We also prove that the moduli space of stable rank two vector bundles with trivial determinant and second Chern class equal to 5 has exactly three irreducible rational components.

math.AG↗
arXiv2018

Series of rational moduli components of stable rank 2 vector bundles on $\mathbb{P}^3$

We study the problem of rationality of an infinite series of components, the so-called Ein components, of the Gieseker-Maruyama moduli space $M(e,n)$ of rank 2 stable vector bundles with the first Chern class $e=0$ or -1 and all possible values of the second Chern class $n$ on the projective 3-space. The generalized null correlation bundles constituting open dense subsets of these components are defined as cohomology bundles of monads whose members are direct sums of line bundles of degrees depending on nonnegative integers $a,b,c$, where $b\ge a$ and $c>a+b$. We show that, in the wide range when $c>2a+b-e,\ b>a,\ (e,a)\ne(0,0)$, the Ein components are rational, and in the remaining cases they are at least stably rational. As a consequence, the union of the spaces $M(e,n)$ over all $n\ge1$ contains an infinite series of rational components for both $e=0$ and $e=-1$. Explicit constructions of rationality of Ein components under the above conditions on $e,a,b,c$ and, respectively, of their stable rationality in the remaining cases, are given. In the case of rationality, we construct universal families of generalized null correlation bundles over certain open subsets of Ein components showing that these subsets are fine moduli spaces. As a by-product of our construction, for $c_1=0$ and $n$ even, they provide, perhaps the first known, examples of fine moduli spaces not satisfying the condition "$n$ is odd", which is a usual sufficient condition for fineness.

math.AG↗
arXiv2018

Construction of stable rank 2 vector bundles on $\mathbb{P}^3$ via symplectic bundles

In this article we study the Gieseker-Maruyama moduli spaces $\mathcal{B}(e,n)$ of stable rank 2 algebraic vector bundles with Chern classes $c_1=e\in\{-1,0\},\ c_2=n\ge1$ on the projective space $\mathbb{P}^3$. We construct two new infinite series $Σ_0$ and $Σ_1$ of irreducible components of the spaces $\mathcal{B}(e,n)$, for $e=0$ and $e=-1$, respectively. General bundles of these components are obtained as cohomology sheaves of monads, the middle term of which is a rank 4 symplectic instanton bundle in case $e=0$, respectively, twisted symplectic bundle in case $e=-1$. We show that the series $Σ_0$ contains components for all big enough values of $n$ (more precisely, at least for $n\ge146$). $Σ_0$ yields the next example, after the series of instanton components, of an infinite series of components of $\mathcal{B}(0,n)$ satisfying this property.

math.AG↗
arXiv2017

Spatially discrete reaction-diffusion equations with discontinuous hysteresis

We address the question: Why may reaction-diffusion equations with hysteretic nonlinearities become ill-posed and how to amend this? To do so, we discretize the spatial variable and obtain a lattice dynamical system with a hysteretic nonlinearity. We analyze a new mechanism that leads to appearance of a spatio-temporal pattern called {\it rattling}: the solution exhibits a propagation phenomenon different from the classical traveling wave, while the hysteretic nonlinearity, loosely speaking, takes a different value at every second spatial point, independently of the grid size. Such a dynamics indicates how one should redefine hysteresis to make the continuous problem well-posed and how the solution will then behave. In the present paper, we develop main tools for the analysis of the spatially discrete model and apply them to a prototype case. In particular, we prove that the propagation velocity is of order $a t^{-1/2}$ as $t\to\infty$ and explicitly find the rate $a$.

math.AP↗
arXiv2017

Depinning asymptotics in ergodic media

We study speeds of fronts in bistable, spatially inhomogeneous media at parameter regimes where speeds approach zero. We provide a set of conceptual assumptions under which we can prove power-law asymptotics for the speed, with exponent depending a local dimension of the ergodic measure near extremal values. We also show that our conceptual assumptions are satisfied in a context of weak inhomogeneity of the medium and almost balanced kinetics, and compare asymptotics with numerical simulations.

nlin.PS↗
arXiv2016

On absolute nonshadowability of transitive maps

We study shadowing property for random infinite pseudotrajectories of a continuous map $f$ of a compact metric space. For the cases of transitive maps and transitive attractors we prove a dichotomy: either $f$ satisfies shadowing property or random pseudotrajectory is shadowable with probability 0.

math.DS↗
arXiv2016

Rattling in spatially discrete diffusion equations with hysteresis

The paper treats a reaction-diffusion equation with hysteretic nonlinearity on a one-dimensional lattice. It arises as a result of the spatial discretization of the corresponding continuous model with so-called nontransverse initial data and exhibits a propagating microstructure --- which we call {\em rattling} --- in the hysteretic component of the solution. We analyze this microstructure and determine the speed of its propagation depending on the parameters of hysteresis and the nontransversality coefficient in the initial data.

math.AP↗
arXiv2015

Recent Advances in Reaction-Diffusion Equations with Non-Ideal Relays

We survey recent results on reaction-diffusion equations with discontinuous hysteretic nonlinearities. We connect these equations with free boundary problems and introduce a related notion of spatial transversality for initial data and solutions. We assert that the equation with transverse initial data possesses a unique solution, which remains transverse for some time, and also describe its regularity. At a moment when the solution becomes nontransverse, we discretize the spatial variable and analyze the resulting lattice dynamical system with hysteresis. In particular, we discuss a new pattern formation mechanism --- {\it rattling}, which indicates how one should reset the continuous model to make it well posed.

math.AP↗
arXiv2014

Shadowing in linear skew products

We consider a linear skew product with the full shift in the base and nonzero Lyapunov exponent in the fiber. We provide a sharp estimate for the precision of shadowing for a typical pseudotrajectory of finite length. This result indicates that the high-dimensional analog of Hammel-Yorke-Grebogi's conjecture concerning the interval of shadowability for a typical pseudotrajectory is not correct. The main technique is reduction of the shadowing problem to the ruin problem for a simple random walk.

math.DS↗
arXiv2013

Holder Shadowing on Finite Intervals

For any $θ, ω> 1/2$ we prove that, if any $d$-pseudotrajectory of length $\sim 1/d^ω$ of a diffeomorphism $f\in C^2$ can be $d^θ$-shadowed by an exact trajectory, then $f$ is structurally stable. Previously it was conjectured by Hammel-Grebogi-Yorke that for $θ= ω= 1/2$ this property holds for a wide class of non-uniformly hyperbolic diffeomorphisms. In the proof we introduce the notion of sublinear growth property for inhomogenious linear equations and prove that it implies exponential dichotomy.

math.DS↗
arXiv2013

Systems of reaction-diffusion equations with spatially distributed hysteresis

We study systems of reaction-diffusion equations with discontinuous spatially distributed hysteresis in the right-hand side. The input of hysteresis is given by a vector-valued function of space and time. Such systems describe hysteretic interaction of non-diffusive (bacteria, cells, etc.) and diffusive (nutrient, proteins, etc.) substances leading to formation of spatial patterns. We provide sufficient conditions under which the problem is well posed in spite of the discontinuity of hysteresis. These conditions are formulated in terms of geometry of manifolds defining hysteresis thresholds and the graph of initial data.

math.AP↗