Non-periodic Functions Research Articles

Finite-time synchronization of Lorenz chaotic systems: theory and circuits

This paper addresses the problem of finite-time master–slave synchronization of Lorenz chaotic systems from a control theoretic point of view. We propose a family of feedback couplings which accomplish the synchronization of Lorenz chaotic systems based on Lyapunov stability theory. These feedback couplings are based on non-periodic functions. A finite horizon can be arbitrarily established by ensuring that chaos synchronization is achieved at established time. An advantage is that some of the proposed feedback couplings are simple and easy to implement. Both mathematical investigations and numerical simulations followed by a Pspice experiment are presented to show the feasibility of the proposed method.

Approximation by algebraic polynomials in metric spaces $L_{\psi}$

In the space $L_{\psi}[-1;1]$ of non-periodic functions with metric $\rho(f,0)_{\psi} = \int\limits_{-1}^1 \psi(|f(x)|)dx$, where $\psi$ is a function of the type of modulus of continuity, we study Jackson inequality for modulus of continuity of $k$-th order in the case of approximation by algebraic polynomials. It is proved that the direct Jackson theorem is true if and only if the lower dilation index of the function $\psi$ is not equal to zero.

FRACTIONAL DUFFING'S EQUATION AND GEOMETRICAL RESONANCE

We investigate the Fractional Duffing equation in the presence of nonharmonic external perturbations. We have applied the concept of Geometrical Resonance to this equation. We have obtained the conditions that should be satisfied by the external driving forces in order to produce high-amplitude periodic oscillations avoiding chaos. We also show that, for Duffing's equation with fractional damping, the perturbations that satisfy the Geometrical Resonance conditions are nonperiodic functions.

Classical Oscillator Model on Canonical, Lie-Algebraic Deformation and Heisenberg-Weyl Algebra Deformation Nonrelativistic Space

The oscillator model on nonrelativistic Canonical (soft), Lie-Algebraic deformation noncommutative space and deformed Heisenberg-Weyl Algebra noncomutative phase space are analyzed. For canonical deformation the additional dynamical effects are absent. For two kinds of Lie-Algebraic deformation space the additional effects are generated. Remarkably angular frequency ω ρ is stepfunction, non-periodic function, and contains imaginary frequency. The corresponding particle trajectory varies greatly along with the deformation and time parameter $\hat{k}$ , t. For deformed Heisenberg-Weyl Algebra noncommutative phase space the additional corrections are generated. K ρ ,ω ρ and particle trajectory have constant second-order correction with deformation parameters Θ and Π, but the particle still keep trajectory of classical oscillator.

On the best polynomial approximation in the space L 2 and widths of some classes of functions

We consider the problem of the best polynomial approximation of 2 -periodic functions in the space L2 in the case where the error of approximation En1 .f / is estimated via the k th-order modulus of continuityk.f / in which the Steklov operator Shf is used instead of the operator of transla- tion Thf.x/D f.xCh/: For the classes of functions defined by using the indicated characteristic of smoothness, we determine the exact values of various n-widths. 1. In the solution of some problems of the approximation theory of functions of real variable, it is cus- tomary to use various modifications of the classic definition of the modulus of continuity (see, e.g., (1-11)). In numerous cases, this is explained by the specific conditions of the analyzed problems and enables one to get the results illustrating the profound nature of these problems. Thus, for the approximation of nonperiodic functions by algebraic polynomials, Potapov and his colleagues proposed various modifications of the classic definition of the modulus of continuity based on the use of different averaging operators instead of the operator of translation T h f.x/Df.xCh/ (see, e.g., (3, 4)). In the case of approximation of 2 -periodic functions, Abilov and Abilova used the Steklov function S h .f / instead of the operator of translation Thf (5). A similar approach was applied by Kokilashvili and Yildirir in (8) and by the authors in (10). In the present paper, we continue our investigations in this direction. Let L2 L2.a0;2c/ be the space of 2 -periodic Lebesgue measurable functions with the norm

Inequalities for derivatives of functions on an axis with nonsymmetrically bounded higher derivatives

For nonperiodic functions $ x\in L_{\infty}^r(R) $ defined on the entire real axis, we prove analogs of the Babenko inequality. The obtained inequalities estimate the norms of derivatives $ \left\| {x_{\pm}^{(k) }} \right\|{L_{q[a,b] }} $ on an arbitrary interval [a, b] ⊂ R such that x (k)(a) = x (k)(b) = 0 via local L p -norms of the functions x and uniform nonsymmetric norms of the higher derivatives x(r) of these functions.

An elementary method of calculating an explicit form of Young measures in some special cases

We present an elementary method of explicit calculation of Young measures for a certain class of functions. This class contains, in particular, functions of a highly oscillatory nature which appear in optimization problems and homogenization theory. In engineering such situation occurs, for instance, in nonlinear elasticity (solid–solid phase transition in certain elastic crystals). Young measures associated with oscillating minimizing sequences gather information about their oscillatory nature and therefore about underlying microstructure. The method presented in this article makes no use of functional analytic tools. There is no need to use a generalized version of the Riemann–Lebesgue lemma and to calculate weak* limits of functions. The main tool is the change of variable theorem. The method applies both to sequences of periodic and nonperiodic functions.

High-smoothness continuations for Fourier approximations of nonperiodic functions

High-smoothness continuations for Fourier approximations of nonperiodic functions

High-order numerical methods for nonlinear acoustics: A Fourier Continuation approach

Dispersion errors, which result from the use of low-order numerical methods in wave-propagation and transport problems, can have a devastating impact on the accuracy of acoustic simulations. These errors are especially problematic in settings containing nonlinear acoustic waves that propagate many times their fundamental wavelength. In these cases, the use of high-order numerical schemes is vital for the accurate and efficient evaluation of the acoustic field. We present a class of high-order time-domain solvers for the treatment of nonlinear acoustic propagation problems. These solvers are based on the Fourier Continuation method, which produces rapidly-converging Fourier series expansions of non-periodic functions (thereby avoiding the Gibbs phenomenon), and are capable of accurately and efficiently simulating nonlinear acoustic fields in large, complex domains.

Inequalities for norms of derivatives of non-periodic functions with non-symmetric constraints on higher derivatives

For non-periodic functions $x \in L^r_{\infty}(\mathbb{R})$ defined on the whole real line we established the analogs of certain inequality of V.F. Babenko.

Estimates for functionals with a known finite set of moments in terms of deviations of operators constructed with the use of the Steklov averages and finite differences

A new technique for deriving estimates indicated in the title is presented. This technique allows one, in particular, to establish Jackson type inequalities for moduli of continuity of arbitrary order with relatively small constants. The results obtained hold in different spaces of periodic and nonperiodic functions. Bibliography: 13 titles.

Approximation error in regularized SVD-based Fourier continuations

We present an analysis of the convergence of recently developed Fourier continuation techniques that incorporates the required truncation of the Singular Value Decomposition (SVD). Through the analysis, the convergence of SVD-based continuations are related to the convergence of any Fourier approximation of similar form, demonstrating the efficiency and accuracy of the numerical method. The analysis determines that the Fourier continuation approximation error can be bounded by a key value that depends only on the parameters of the Fourier continuation and on the points over which it is applied. For any given distribution of points, a finite number of calculations can be performed to obtain this important value. Our numerical computations on evenly spaced points show that as the number of points increases, this quantity converges to a fixed value, allowing for broad conclusions on the convergence of Fourier continuations calculated with truncated SVDs. We conclude that Fourier continuations can obtain super-algebraic or even exponential convergence on evenly spaced points for non-periodic functions until the convergence is limited by a parameter normally chosen near the machine precision accuracy threshold.

Sobolev smoothing of SVD-based Fourier continuations

A method for calculating Sobolev smoothed Fourier continuations is presented. The method is based on the recently introduced singular value decomposition based Fourier continuation approach. This approach allows for highly accurate Fourier series approximations of non-periodic functions. These super-algebraically convergent approximations can be highly oscillatory in an extended region, contaminating the Fourier coefficients. It is shown that through solving a subsequent least squares problem, a Fourier continuation can be produced which has been dramatically smoothed in that the Fourier coefficients exhibit a prescribed rate of decay as the wave number increases. While the smoothing procedure has no significant negative effect on the accuracy of the Fourier series approximation, in some situations the smoothed continuations can actually yield increased accuracy in the approximation of the function and its derivatives.

Deterministic homogenization on problems of minimal hypersurface type

We study here the deterministic homogenization for a particular family of convex integral functionals, namely problems of minimal hypersurfaces. By the sigma-convergence method relooked especially in appropriate representation of non-periodic functions and in the extension of sigma-convergence on the L1-space, we obtain a general homogenization result, and thanks to the particularity of minimal hypersurfaces, we derive a macroscopic integral functional. We then provide a number of physical applications (including the periodic case) of this result.

Turyn‐Type Sequences: Classification, Enumeration, and Construction

AbstractTuryn‐type sequences, , are quadruples of ‐sequences , with lengths , respectively, where the sum of the nonperiodic autocorrelation functions of and twice that of is a δ‐function (i.e., vanishes everywhere except at 0). Turyn‐type sequences are known to exist for all even n not larger than 36. We introduce a definition of equivalence to construct a canonical form for in general. By using this canonical form, we enumerate the equivalence classes of for . We also construct the first example of Turyn‐type sequences .

A New Formulation of the Fast Fractional Fourier Transform

By using a spectral approach, we derive a Gaussian-like quadrature of the continuous fractional Fourier transform. The quadrature is obtained from a bilinear form of eigenvectors of the matrix associated to the recurrence equation of the Hermite polynomials. These eigenvectors are discrete approximations of the Hermite functions, which are eigenfunctions of the fractional Fourier transform operator. This new discrete transform is unitary and has a group structure. By using some asymptotic formulas, we rewrite the quadrature in terms of the fast Fourier transform (FFT), yielding a fast discretization of the fractional Fourier transform and its inverse in closed form. We extend the range of the fractional Fourier transform by considering arbitrary complex values inside the unit circle and not only at the boundary. We find that this fast quadrature evaluated at $z=i$ becomes a more accurate version of the FFT and can be used for nonperiodic functions.

The method of odd continuation for Fourier approximations of nonperiodic functions

The method of odd continuation for Fourier approximations of nonperiodic functions

Estimates for functionals with a known moment sequence in terms of deviations of Steklov type means

Some estimates for functionals indicated in the title are established. As implications, Jackson type inequalities with constants smaller than the previously known ones are obtained. The results hold in various spaces of both periodic and nonperiodic functions. Bibliography: 9 titles.

The rate of decrease of constants in Jackson type inequalities in dependence of the order of the continuity modulus

Jackson type inequalities for moduli of continuity of arbitrary order are established with the use of linear approximation methods. The constants are smaller than known previously. The results hold in different spaces of periodic and nonperiodic functions. Bibliography: 19 titles.

On the convergence of expansions in polyharmonic eigenfunctions

We consider expansions of smooth, nonperiodic functions defined on compact intervals in eigenfunctions of polyharmonic operators equipped with homogeneous Neumann boundary conditions. Having determined asymptotic expressions for both the eigenvalues and eigenfunctions of these operators, we demonstrate how these results can be used in the efficient computation of expansions. Next, we consider the convergence. We establish the key advantage of such expansions over classical Fourier series–namely, both faster and higher-order convergence–and provide a full asymptotic expansion for the error incurred by the truncated expansion. Finally, we derive conditions that completely determine the convergence rate.