Theory Of Rational Approximation Research Articles

Strange Nonchaotic Attractors in Memristor-Based Shimizu–Morioka Oscillator

The existence of strange nonchaotic attractors (SNAs) is examined using a memristor-based Shimizu–Morioka oscillator. In particular, we analyze the birth of SNAs by varying one of the frequency of external quasi-periodic forces. The occurrence and its mechanism are further analyzed through rational approximation theory. Following that, we also perform different nonlinear dynamical characterizations, including Poincaré surface sections, singular continuous spectrum, separation of nearby trajectories, 0–1 test, as well as finite-time Lyapunov exponents for more precise validation. Finally, the experimental evidence for the occurrence of SNA is inspected through a real-time analog circuit in the laboratory.

Padé Approximation and Hypergeometric Functions: A Missing Link with the Spectrum of Delay-Differential Equations

It is well known that rational approximation theory involves degenerate hyperge-ometric functions and, in particular, the Padé approximation of the exponential function is closely related to Kummer hypergeometric functions. Recently, in the context of the study of the exponential stability of the trivial solution of delay-differential equations, a new link between the degenerate hypergeometric function and the zeros distribution of the characteristic function associated with linear delay-differential equations was emphasized. Such a link allowed the characterization of a property of time-delay systems known as multiplicity-induced-dominancy (MID), which opened a new direction in designing low-complexity controllers for time-delay systems by using a partial pole placement idea. Thanks to their relations to hypergeometric functions, we explore in this paper links between the spectrum of delay-differential equations and Padé approximations of the exponential function. This note exploits and further comments recent results from [I. Boussaada, G. Mazanti and S-I. Niculescu. 2022, Comptes Rendus. Mathématique] and [I. Boussaada, G. Mazanti and S-I. Niculescu. 2022, Bulletin des Sciences Mathématiques].

MIMO frequency domain system identification using matrix-valued orthonormal functions

In this paper, we propose a two-stage algorithm utilizing the Cauchy integral and the matrix-valued adaptive Fourier decomposition (abbreviated as matrix AFD) to identify transfer functions of linear time-invariant (LTI) multi-input multi-output (MIMO) systems in the continuous time case. In recent work of Alpay et al. (2017), a theory of adaptive rational approximation to matrix-valued Hardy space functions on the unit disk was established. The matrix-valued function theory has great potential in applications, in views of the practice of its scalar-valued counterparts. The algorithm and application aspects of the mentioned theory of Alpay et al. (2017) have not been developed. The theory was only written for the unit disk case corresponding to the discrete time systems. The contributions of the present paper are 3-fold. First, we construct an analogous adaptive approximation theory for complex matrix-valued Hardy space functions defined on a half of the complex plane, corresponding to the Laplace transforms of signals of finite energy whose Fourier transforms are supported on a half of the frequency domain. The half plane model corresponds to signals defined in the whole axis range which is an alternative case to signals defined in a compact interval. The second fold contribution lays on maximal selection of the pair (a,P) where a is a point of the right-half plane and P is an orthogonal projection. We show that the optimal selection of P is dependent on a when a is first fixed, that is P=P(a), where P:a→P(a) has an explicit corresponding relation. Due to this relation we reduce the maximal selection of the pair (a,P) to only that of the parameter a. This result can be extended to the compact intervals case as well. The third fold is, with the precise rule from a to P(a), we develop a practical algorithm for the adaptive approximation to the transfer function. Through an example we show that the proposed algorithm is effective in both the noise-free and noisy cases.

New correlations for heat transfer in parallel-plate ducts at low Peclet number

Calculations of heat transfer in duct flows generally neglect the effect of axial heat conduction within the fluid. However, in some situations, this effect must be taken into account. The decision on whether or not to consider this phenomenon is driven by the value of the Peclet number. A low value, with 100 commonly considered the threshold, indicates that axial conduction, i.e., conduction parallel to the flow, should be accounted for in the calculations. Microfluidic devices, systems using liquid metal coolant or microchannel heat exchangers are some examples of applications where low Peclet flow conditions may be encountered. While exact analytical solutions exist for such cases, they are often impractical to use for common engineering design purposes due to their mathematical complexity. The present work proposes three new correlations for the estimation of the fully developed Nusselt number, thermal entry length and local Nusselt number in the entrance region of a laminar flow within a parallel-plate duct with constant wall temperature. These correlations allow the simple and accurate evaluation of useful engineering quantities, thus avoiding lengthy and complex calculations. They were developed using rational approximation theory and are effective on a wide range of Peclet numbers and axial coordinates. Accuracy of under 1% relative error is achieved with each of the three correlations when results are compared against exact analytical series solution values.

Approximating the [formula omitted]th root by composite rational functions

A landmark result from rational approximation theory states that x1∕p on [0,1] can be approximated by a type-(n,n) rational function with root-exponential accuracy. Motivated by the recursive optimality property of Zolotarev functions (for the square root and sign functions), we investigate approximating x1∕p by composite rational functions of the form rk(x,rk−1(x,rk−2(⋯(x,r1(x,1))))). While this class of rational functions ceases to contain the minimax (best) approximant for p≥3, we show that it achieves approximately pth-root exponential convergence with respect to the degree. Moreover, crucially, the convergence is doubly exponential with respect to the number of degrees of freedom, suggesting that composite rational functions are able to approximate x1∕p and related functions (such as |x| and the sector function) with exceptional efficiency.

Lightning Solvers for Potential Flows

We present a method for computing potential flows in planar domains. Our approach is based on a new class of techniques, known as “lightning solvers”, which exploit rational function approximation theory in order to achieve excellent convergence rates. The method is particularly suitable for flows in domains with corners where traditional numerical methods fail. We outline the mathematical basis for the method and establish the connection with potential flow theory. In particular, we apply the new solver to a range of classical problems including steady potential flows, vortex dynamics, and free-streamline flows. The solution method is extremely rapid and usually takes just a fraction of a second to converge to a high degree of accuracy. Numerical evaluations of the solutions are performed in a matter of microseconds and can be compressed further with novel algorithms.

Observation of Strange Nonchaotic Dynamics in the Frame of State-Controlled Cellular Neural Network-Based Oscillator

Abstract In this paper, the strange nonchaotic dynamics of a quasi-periodically driven state-controlled cellular neural network (SC-CNN) based on a simple chaotic circuit is investigated using hardware experiments and numerical simulations. We report here two different routes to strange nonchaotic attractors (SNAs) taken by this SC-CNN based circuit system. These routes were confirmed using rational approximation (RA) theory, finite time Lyapunov exponents, spectrum of the largest Lyapunov exponents and their variance, and phase sensitivity exponent. It is observed that the results from both computer simulations as well as laboratory experiments have spectacular resemblance.

Functional calculus forC0-groups using type and cotype

We study the functional calculus properties of generators of $C_{0}$-groups under type and cotype assumptions on the underlying Banach space. In particular, we show the following. Let $-iA$ generate a $C_{0}$-group on a Banach space $X$ with type $p\in[1,2]$ and cotype $q\in[2,\infty)$. Then $A$ has a bounded $\mathcal{H}^{\infty}$-calculus from $\mathrm{D}_{A}(\tfrac{1}{p}-\tfrac{1}{q},1)$ to $X$, i.e. $f(A):\mathrm{D}_{A}(\tfrac{1}{p}-\tfrac{1}{q},1)\to X$ is bounded for each bounded holomorphic function $f$ on a sufficiently large strip. As a corollary of our main theorem, for sectorial operators we quantify the gap between bounded imaginary powers and a bounded $\mathcal{H}^{\infty}$-calculus in terms of the type and cotype of the underlying Banach space. For cosine functions we obtain similar results as for $C_{0}$-groups. We extend our results to $R$-bounded operator-valued calculi, and we give an application to the theory of rational approximation of $C_{0}$-groups.

Fast Computation of the Matrix Exponential for a Toeplitz Matrix

The computation of the matrix exponential is a ubiquitous operation in numerical mathematics, and for a general, unstructured $n\times n$ matrix it can be computed in $\mathcal{O}(n^3)$ operations. An interesting problem arises if the input matrix is a Toeplitz matrix, for example as the result of discretizing integral equations with a time invariant kernel. In this case it is not obvious how to take advantage of the Toeplitz structure, as the exponential of a Toeplitz matrix is, in general, not a Toeplitz matrix itself. The main contribution of this work are fast algorithms for the computation of the Toeplitz matrix exponential. The algorithms have provable quadratic complexity if the spectrum is real, or sectorial, or more generally, if the imaginary parts of the rightmost eigenvalues do not vary too much. They may be efficient even outside these spectral constraints. They are based on the scaling and squaring framework, and their analysis connects classical results from rational approximation theory to matrices of low displacement rank. As an example, the developed methods are applied to Merton's jump-diffusion model for option pricing.

Dynamic bifurcation and strange nonchaos in a two-frequency parametrically driven nonlinear oscillator

A periodically forced series LCR circuit with Chua’s diode as a nonlinear element exhibits slow passage through Hopf bifurcation. This slow passage leads to a delay in the Hopf bifurcation. The delay in this bifurcation is a unique quantity and it can be predicted using various numerical analysis. We find that when an additional periodic force is added to the system, the delay in bifurcation becomes chaotic which leads to an unpredictability in bifurcation delay. Further, we study the bifurcation of the periodic delay to chaotic delay in the slow passage effect through strange nonchaotic delay. We also report the occurrence of strange nonchaotic dynamics while varying the parameter of the additional force included in the system. We observe that the system exhibits a hitherto unknown dynamical transition to a strange nonchaotic attractor. With the help of Lyapunov exponent, we explain the new transition to strange nonchaotic attractor and its mechanism is studied by making use of rational approximation theory. The birth of SNA has also been confirmed numerically, using Poincaré maps, phase sensitivity exponent, the distribution of finite-time Lyapunov exponents and singular continuous spectrum analysis.

The Gonchar-Stahl -theorem and associated directions in the theory of rational approximations of analytic functions

The Gonchar-Stahl -theorem characterizes the rate of convergence of best uniform (Chebyshev) rational approximations (with free poles) for one basic class of analytic functions. The theorem itself, modifications and generalizations of it, methods involved in its proof and other related details constitute an important subfield in the theory of rational approximations of analytic functions and complex analysis. This paper briefly outlines the essentials of the subfield. The fundamental contributions of A. A. Gonchar and H. Stahl are at the heart of the exposition. Bibliography: 70 titles.

Hermite–Padé approximation, isomonodromic deformation and hypergeometric integral

We develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations. We show that a certain duality in Hermite’s two approximation problems for functions leads to the Schlesinger transformations, i.e. transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant. Since approximants and remainders are described by block-Toeplitz determinants, one can clearly understand the determinantal structure in isomonodromic deformations. We demonstrate our method in a certain family of Hamiltonian systems of isomonodromy type including the sixth Painleve equation and Garnier systems; particularly, we present their solutions written in terms of iterated hypergeometric integrals. An algorithm for constructing the Schlesinger transformations is also discussed through vector continued fractions.

The Gonchar-Stahl $\rho^2$-theorem and associated directions in the theory of rational approximations of analytic functions

Теорема Гончара-Шталя о $\rho^2$ характеризует скорость сходимости наилучших (чебышeвских) равномерных рациональных приближений (со свободными полюсами) для одного из важнейших классов аналитических функций. Сама эта теорема, ее варианты и обобщения, методы, используемые в доказательстве, и прочие моменты составляют важную подобласть теории рациональных приближений аналитических функций и комплексного анализа. В статье вкратце очертены контуры этой подобласти. В центре изложения находится фундаментальный вклад А. А. Гончара и Г. Шталя в эту теорию. Библиография: 70 названий.

On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

For an interval E=[a,b] on the real line, let μ be either the equilibrium measure, or the normalized Lebesgue measure of E, and let Vμ denote the associated logarithmic potential. In the present paper, we construct a function f which is analytic on E and possesses four branch points of second order outside of E such that the family of the admissible compacta of f has no minimizing elements with regard to the extremal theoretic-potential problem, in the external field equals V−μ.

Method of interior variations and existence of S-compact sets

The variation of equilibrium energy is analyzed for three different functionals that naturally arise in solving a number of problems in the theory of constructive rational approximation of multivalued analytic functions. The variational approach is based on the relationship between the variation of the equilibrium energy and the equilibrium measure. In all three cases the following result is obtained: for the energy functional and the class of admissible compact sets corresponding to the problem, the arising stationary compact set is fully characterized by a certain symmetry property.

Rational approximation of analytic functions

The paper provides an overview of the author’s contribution to the theory of constructive rational approximations of analytic functions. The results presented are related to the convergence theory of Pade approximants and of more general rational interpolation processes, which significantly expand the classical theory’s framework of continuous fractions, to inverse problems in the theory of Pade approximants, to the application of multipoint Pade approximants (solutions of Cauchy-Jacobi interpolation problem) in explorations connected with the rate of Chebyshev rational approximation of analytic functions and to the asymptotic properties of Pade-Hermite approximation for systems of Markov type functions.

An inductive proof of the Feinstein-Heath Swiss cheese “Classicalisation” theorem

A theory of allocation maps has been developed by J. F. Feinstein and M. J. Heath in order to prove a theorem, using Zorn’s lemma, concerning the compact plane sets known as Swiss cheese sets. These sets are important since, as domains, they provide a good source of examples in the theory of uniform algebras and rational approximation. In this paper we take a more direct approach when proving their theorem by using transfinite induction and cardinality. An explicit reference to a theory of allocation maps is no longer required. Instead we find that the repeated application of a single operation developed from the final step of the proof by Feinstein and Heath is enough.

Swiss cheeses, rational approximation and universal plane curves

We consider the compact plane sets known as <i>Swiss cheese sets</i>, which are a useful source of examples in the theory of uniform algebras and rational approximation. We develop a theory of <i>allocation maps</i> connected to such sets and we use th

Bergman polynomials on an archipelago: Estimates, zeros and shape reconstruction

Growth estimates of complex orthogonal polynomials with respect to the area measure supported by a disjoint union of planar Jordan domains (called, in short, an archipelago) are obtained by a combination of methods of potential theory and rational approximation theory. The study of the asymptotic behavior of the roots of these polynomials reveals a surprisingly rich geometry, which reflects three characteristics: the relative position of an island in the archipelago, the analytic continuation picture of the Schwarz function of every individual boundary and the singular points of the exterior Green function. By way of explicit example, fine asymptotics are obtained for the lemniscate archipelago | z m − 1 | < r m , 0 < r < 1 , which consists of m islands. The asymptotic analysis of the Christoffel functions associated to the same orthogonal polynomials leads to a very accurate reconstruction algorithm of the shape of the archipelago, knowing only finitely many of its power moments. This work naturally complements a 1969 study by H. Widom of Szegő orthogonal polynomials on an archipelago and the more recent asymptotic analysis of Bergman orthogonal polynomials unveiled by the last two authors and their collaborators.

The Numerical Computation of Rational Structures and Asymptotic Standard Deviations in Causal Time Series Data

Problem statement: The specific properties of data series are of primary importance in several sciences. In the field of time series analysis, several researchers have considered the rational approximation theory, particularly the Pade Approximation and Orthogonal Polynomials. Approach: In this study, an approach for the statistical significance of two numerical methods, the r-s and q-d algorithm, was proposed which made possible to identify and compute certain rational structures associated with chronological data. Consideration was given to both univariate and multivariate cases. Results: Both algorithms were illustrated empirically through the use of simulated ARMA and TF models and economic data, some of which were taken from previous studies to compare results. Conclusion: This study highlighted the usefulness of several numerical methods (which were all closely related to the PA and OP) in identifying the rational structures associated with data series.