arXiv subjects

Mikhail Anikushin

Publications and source records attributed to Mikhail Anikushin.

16 recordsLinked to original sources

arXiv2026

Frequency conditions for the global stability of nonlinear delay equations with several equilibria

In our adjacent work, we developed a spectral comparison principle for compound cocycles generated by delay equations. It allows to derive frequency inequalities for the uniform exponential stability of such cocycles by means of their comparison with stationary problems. Such inequalities are hard to verify purely analytically, and in this work we develop approximation schemes to verify some of the arising frequency inequalities. Besides some general theoretical results, in applications we stick to the case of scalar equations. By means of the Suarez-Schopf delayed oscillator and the Mackey-Glass equations, we demonstrate applications of the theory to reveal regions in the space of parameters where the absence of closed invariant contours can be guaranteed. Since the frequency inequalities are robust, so close systems also satisfy them, we expect the method to actually imply the global stability, as in known finite-dimensional results utilizing variants of the closing lemma, which is still awaiting developments in infinite dimensions.

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arXiv2026

Spectral comparison of compound cocycles generated by delay equations in Hilbert spaces

We study linear cocycles generated by nonautonomous delay equations in a suitable Hilbert space and their extensions, called compound cocycles, to exterior powers. Using a recent version of the frequency theorem, we develop analytical techniques for comparing spectral properties, such as uniform exponential dichotomies, between such cocycles and semigroups generated by stationary equations. These methods are based on properties related to regularity and structure in PDEs associated with delay equations. In particular, the developed machinery leads to effective robust criteria that guarantee the absence of closed invariant contours on global attractors arising in nonlinear problems and are expected to ensure global stability.

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arXiv2025

Variational description of uniform Lyapunov exponents via adapted metrics on exterior products

In this work, we present a comprehensive study of the relationship among uniform Lyapunov exponents, the Liouville trace formula, and adapted metrics for cocycles in Hilbert spaces. First, we prove that uniform Lyapunov exponents can be approximated by constructing adapted metrics on exterior products. Next, we develop a general computational theory in an abstract setting, establish a generalized Liouville trace formula, and pose and discuss the symmetrization problem related to computations. Third, we discuss ergodic properties and upper semicontinuity in the context of subadditive families over a noncompact base. Furthermore, we use adapted metrics and the trace formula to obtain, for the first time, effective dimension estimates for a general class of delay equations. In particular, we illustrate this approach by deriving upper estimates for the Lyapunov dimension of global attractors in the Mackey--Glass equations and the periodically forced Suarez--Schopf delayed oscillator. As the delay value tends to infinity, the estimates appear to be asymptotically sharp.

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arXiv2025

Nonlinear constrained optimization of Schur test functions

We apply the iterative nonlinear programming method, previously proposed in our earlier work, to optimize Schur test functions and thereby provide refined upper bounds for the norms of integral operators. As an illustration, we derive such bounds for transfer operators associated with twofold additive compound operators that arise in the study of delay equations. This is related to the verification of frequency inequalities that guarantee the global stability of nonlinear delay equations through the generalized Bendixson criterion.

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arXiv2025

Robust upper estimates for topological entropy via nonlinear constrained optimization over adapted metrics

We present an analytical-numerical method providing robust upper estimates for the topological entropy or, more generally, uniform volume growth exponents of differentiable mappings. By introducing varying metrics, we simplify the analysis at the cost of generally rougher bounds, but keeping the prospect of choosing more relatable metrics to refine the estimates. With any covering of an invariant set by a finite number of cubes, we associate a graph describing overlaps (edges) of the cubes (vertices) under iterates of the mapping. Weighing vertices according to a given metric, we reduce the problem to finding simple cycles with maximal relative weights. Then we develop an algorithm concerned with iterative resolving nonlinear programming problems for optimization of maximal relative weights in general smooth families of metrics which may involve interpolation or neural networks models. We describe applications of the algorithm to compute the largest uniform Lyapunov exponent and uniform Lyapunov dimension for the Hénon and Rabinovich systems justifying the Eden conjecture at stationary and periodic points respectively.

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arXiv2025

Inertial manifolds via spatial averaging: a control-theoretic perspective

We develop a functional-analytical machinery for studying the quadratic regulator problem arising from spectra perturbations of infinite-dimensional dynamical systems. In particular, we are interested in applications to inertial manifolds theory. For certain nonautonomous Hamiltonian systems associated with such problems, we show the existence and uniform nonoscillation of stable Lagrangian bundles. This is done within the context of the classical frequency condition for stationary problems, as well as for nonstationary problems arising under the conditions of the Spatial Averaging Principle of J. Mallet-Paret and G.R. Sell.

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arXiv2023

Frequency theorem and inertial manifolds for neutral delay equations

We study the quadratic regulator problem for linear control systems in Hilbert spaces, where the cost functional is in some sense unbounded. Our motivation comes from delay equations with the feedback part containing discrete delays or, in other words, measurements given by $δ$-functionals, which are unbounded in $L_{2}$. Working in an abstract context in which such (and many others, including parabolic boundary control problems) equations can be treated, we obtain a version of the Frequency Theorem (following the works of V.A. Yakubovich and A.L. Likhtarnikov), which guarantees the existence of a unique optimal process and shows that the optimal cost is given by a quadratic Lyapunov-like functional. In our adjacent works it is shown that such functionals can be used to construct inertial manifolds and allow to treat and extend many works in the field in a unified manner. Here we concentrate on applications to delay equations and especially mention the works of R.A. Smith on developments of convergence theorems and the Poincaré-Bendixson theory; Yu. A. Ryabov, R.D. Driver and C. Chicone on inertial manifolds for equations with small delays and their recent generalization for equations of neutral type given by S. Chen and J. Shen.

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arXiv2022

Nonlinear semigroups for delay equations in Hilbert spaces, inertial manifolds and dimension estimates

We study the well-posedness of nonautonomous nonlinear delay equations in $\mathbb{R}^{n}$ as evolutionary equations in a proper Hilbert space. We present a construction of solving operators (nonautonomous case) or nonlinear semigroups (autonomous case) for a large class of such equations. The main idea can be easily extended for certain PDEs with delay. Our approach has lesser limitations and much more elementary than some previously known constructions of such semigroups and solving operators based on the theory of accretive operators. In the autonomous case we also study differentiability properties of these semigroups in order to apply various dimension estimates using the Hilbert space geometry. However, obtaining effective dimension estimates for delay equations is a nontrivial problem and we explain it by means of a scalar delay equation. We also discuss our adjacent results concerned with inertial manifolds and their construction for delay equations.

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arXiv2022

Hidden and unstable periodic orbits as a result of homoclinic bifurcations in the Suarez-Schopf delayed oscillator and the irregularity of ENSO

We revisit the classical Suarez-Schopf delayed oscillator. Special attention is paid to the region of linear stability in the space of parameters. By means of the theory of inertial manifolds developed in our adjacent papers, we provide analytical-numerical evidence for the existence of two-dimensional inertial manifolds in the model. This allows to suggest a complete qualitative description of the dynamics in the region of linear stability. We show that there are two subregions corresponding to the existence of hidden or self-excited periodic orbits. These subregions must be separated by a curve on which homoclinic "figure eights", bifurcating into a single one or a pair of unstable periodic orbits, should exist. We relate the observed hidden oscillations and homoclinics to the irregularity theories of ENSO and provide numerical evidence that chaotic behavior may appear if a small periodic forcing is applied to the model. We also use parameters from the Suarez-Schopf model to discover hidden and self-excited asynchronous periodic regimes in a ring array of coupled lossless transmission lines studied by J. Wu and H. Xia.

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arXiv2022

Inertial manifolds and foliations for asymptotically compact cocycles in Banach spaces

We study asymptotically compact nonautonomous dynamical systems given by abstract cocycles in Banach spaces. Our main assumptions are given by a squeezing property in a quadratic cone field (given by a family of indefinite quadratic Lyapunov-like functionals) and asymptotic compactness. Under such conditions it is possible to reconstruct foliations as in the theory of normally hyperbolic manifolds. Our approach allows to unify many "practical" theories of local and nonlocal invariant manifolds, such as stable/unstable/center manifolds and inertial manifolds, and, moreover, it often leads to optimal conditions for their existence. We give applications for semilinear parabolic equations and neutral delay equations, where the Frequency Theorem is used to construct constant cone fields.

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arXiv2021

Frequency theorem for parabolic equations and its relation to inertial manifolds theory

We obtain a version of the Frequency Theorem (a theorem on solvability of certain operator inequalities), which allows to construct quadratic Lyapunov functionals for semilinear parabolic equations. We show that the well-known Spectral Gap Condition, which was used in the theory of inertial manifolds by C. Foias, R. Temam and G. R. Sell, is a particular case of some frequency inequality, which arises within the Frequency Theorem. In particular, this allows to construct inertial manifolds for semilinear parabolic equations (including also some non-autonomous problems) in the context of a more general geometric theory developed in our adjacent works. This theory is based on quadratic Lyapunov functionals and generalizes the frequency-domain approach used by R. A. Smith. We also discuss the optimality of frequency inequalities and its relationship with known old and recent results in the field.

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arXiv2020

The Poincaré-Bendixson theory for certain compact semi-flows in Banach spaces

We study semiflows satisfying a certain squeezing condition with respect to a quadratic functional in some Banach space. Under certain compactness assumptions from our previous results it follows that there exists an invariant manifold, which is under more restrictive conditions is an inertial manifold. In the case of a two-dimensional manifold we obtain an analog of the Poincaré-Bendixson theorem on the trichotomy of $ω$-limit sets. Moreover, we obtain conditions for the existence of an orbitally stable periodic orbit. Our approach unifies a series of papers by R.~A.~Smith, establishes their connection with the theory of inertial manifolds and opens a new perspective of applications. To verify the squeezing property in applications we use recently developed versions of the frequency theorem, which guarantee the existence of the required quadratic functional if some frequency-domain condition is satisfied. We present applications of our results for nonlinear delay equations in $\mathbb{R}^{n}$ and semilinear parabolic equations and discuss perspectives of applications to parabolic problems with delay and boundary controls.

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arXiv2020

A non-local reduction principle for cocycles in Hilbert spaces

We study cocycles (non-autonomous dynamical systems) satisfying a certain squeezing condition with respect to the quadratic form of a bounded self-adjoint operator acting in a Hilbert space. We prove that (under additional assumptions) the orthogonal projector maps the fibres of some invariant set, containing bounded trajectories, in a one-to-one manner onto the negative subspace of the operator. This allows to reduce interesting dynamics onto this invariant set, which in some cases can be considered as a kind of inertial manifold for the cocycle. We consider applications of the reduction principle for periodic cocycles. For such cocycles we give an extension of the Massera second theorem, obtain the conditions for the existence of a Lyapunov stable periodic trajectory and prove convergence-type results, which we apply to study nonlinear periodic in time delayed-feedback equations posed in a proper Hilbert space and parabolic problems with a nonlinear periodic in time boundary control. The required operator is obtained as a solution to certain operator inequalities with the use of the Yakubovich-Likhtarnikov frequency theorem for $C_{0}$-semigroups and its properties are established from the Lyapunov inequality and dichotomy of the linear part of the problem.

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arXiv2018

On the Dimensional-like Characteristics Arising From Linear Inhomogeneous Approximations

As it follows from the theory of almost periodic functions the set of integer solutions $q$ to the Kronecker system $|ω_{j} q - θ_{j}| < \varepsilon \pmod 1$, $j=1,\ldots,m$, where $1,ω_{1},\ldots,ω_{m}$ are linearly independent over $\mathbb{Q}$, is relatively dense in $\mathbb{R}$. The latter means that there is $L(\varepsilon)>0$ such that any segment of length $L(\varepsilon)$ contains at least one integer solution to the Kronecker system. We give some lower and upper non-effective (asymptotic) estimates for $L(\varepsilon)$ and, in particular, show that $L(\varepsilon) = \left(\frac{1}{\varepsilon}\right)^{m+o(1)}$ as $\varepsilon \to 0$ for many cases, including algebraic numbers as well as badly approximable numbers. We use methods of dimension theory and Diophantine approximations of $m$-tuples satisfying the Diophantine condition.

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arXiv2017

Badly Approximable Numbers and the Growth Rate of the Inclusion Length of an Almost Periodic Function

We study the growth rate of the inclusion length of an almost periodic function. For a given a. p. function such growth rate depends on the algebraic structure of Fourier exponents, i. e. on how good they can be approximated by rational numbers. In additional, as appears from the definition, the inclusion length carries some information about the translation numbers (almost periods). Our result is a lower bound of the growth rate of the inclusion interval of a quasiperiodic function (theorem 3). Here we use methods from dimension theory. We do not assume anything about exponents, but rationally independence. This suggest an idea that this lower bound can be reached (in asymptotic sense) for some "bad" exponents. Koichiro Naito in his papers on estimates of the fractal dimension of almost periodic attractors proved an upper bound of the inclusion length for some class of a.p. functions, using simultaneous Diophantine approximations. For the special case of badly approximable exponents we can see that the both estimates (if we consider them as asymptotic estimates) are coincide (see theorem 4). We hope that ideas and results presented in this paper can be useful not only to understand the nature of badly approximable numbers and almost periods, but also for more detailed understanding of the structure of almost periodic attractors.

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arXiv2017

Dimension Theory Approach to the Complexity of Almost Periodic Trajectories

We introduce and study a dimensional-like characteristic of an uniformly almost periodic function, which we call the Diophantine dimension. By definition, it is the exponent in the asymptotic behavior of the inclusio length. Diophantine dimension is connected with recurrent and ergodic properties of an almost periodic function. We get some estimates of the Diophantine dimension for certain quasiperiodic functions and present methods to investigate such a characteristic for almost periodic trajectories of evolution equations. Also we discuss the link between the presented approach and the so called effective versions of the Kronecker theorem.

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