Nonlinear Recurrence Relations Research Articles

Discrete classical orthogonal polynomials

We find necessary and sufficient conditions for the difference equation of hypergeometric type to have polynomial solutions , which are orthogonal, that is Traditionallydμ(x) is a positive measure but here we allow it to be a signed measure. We then show that the usual restrictions on parameters in discrete classical orthogonal polynomials can be relaxed. We also derive functional Rodrigues' formula for discrete classical orthogonal polynomials. Finally, we give a nonlinear recurrence relation characterizing discrete classical orthogonal polynomials. This is the first nonlinear characterization of discrete classical orthogonal polynomials whereas several nonlinear characterizations are known for classical orthogonal polynomials.

Functional logistic mapping

The non-linear recurrence scheme g n + 1 ( x) = 4λ(1 − g n ( x)) · g n ( x) is studied where λ is the growth parameter 0 ⩽ λ ⩽ 1 and g n ( x) is a function of −1 ⩽ x ⩽1 instead of a simple number as is the case in the classical logistic mapping scheme. This kind of a new functional logistic mapping is solved by the ansatz g n ( x) = 0.5(1 − S 2 n ( x)). It is shown that the generated novel polynomials, S 2 n ( x) with n = 0, 1, 2, 3, …, possess the non-linear recurrence relation S 2 n + 1 ( x) = 2λ S 2 n 2( x) − (2λ − 1) and represent some kind of a set of generalized Chebyshev polynomials. Their properties are discussed thoroughly for various λ-values, i.e. in the ‘smooth’, the bifurcated and the chaotic regimes.

Generalized KdV equation for fluid dynamics and quantum algebras

We generalize the non-linear one-dimensional equation of a fluid layer for any depth and length as an infinite order differential equation for the steady waves. This equation can be written as a q-differential one, with its general solution written as a power series expansion with coefficients satisfying a nonlinear recurrence relation. In the limit of long and shallow water (shallow channels) we reobtain the well known Korteweg-de-Vries equation together with its single-soliton solution.

On recurrence relations for order statistics

The main purpose of this paper is to provide a unified approach to the treatment of linear recurrence relations for single or pairs of order statistics. Suppose such a relation has been proved in the simplest case when X 1, …, X n are independent variates having an arbitrary absolutely continuous distribution. It is pointed out that the same relation continues to hold when the X's are exchangeable, whether continuous or not. As has recently become well known, further generalizations are possible when the X's have any joint distribution. Attention is also drawn to a useful nonlinear recurrence relation due to Boncelet (1987).

Nonexistence criteria for positive solutions of a nonlinear recurrence relation

In this paper, we derive several nonexistence criteria for eventually positive solutions of a class of nonlinear recurrence relation. These criteria are then used to obtain oscillation criteria for an associated nonlinear neutral difference equation with delay.

Parallelizing iterative loops with conditional branching

This paper considers automatic restructuring of loops with branching for parallel processing, especially a class of loops termed conditional cyclic loops. A cyclic loop possesses a dependence cycle caused by branching across loop iterations, which makes it difficult to parallelize. In general, parallel execution of a cyclic loop provides little benefit due to the need of solving a full-order nonlinear Boolean recurrence relation. However, the Boolean recurrence in practice is often of simpler forms. With the simpler forms, the number of possible predicate values of branching is reduced drastically compared to a general cyclic loop, These simple forms of cyclic loops found in practice can be parallelized for O(p/ log p) speedup with p processors. >

Nonlinear recurrence relations and induced orthogonal polynomials

We consider a class of polynomials Q n ( x) defined by Q n(x) = (x + b n) P n−1 (x) + d nP n (x), n = 0, 1, 2,…,d o ≠ 1 , where the P n ( x) are polynomials orthogonal with respect to some real linear functional. We ask what conditions the sequences b n , d n must satisfy so that the polynomials Q n ( x) also form an orthogonal set. It turns out that the quantities b n , d n must satisfy second order nonlinear recurrences, but that these recurrences reduce to first order recurrences. The values b 0, b 1, d 0, d 1 are arbitrary. We determine the weight function for the polynomials and discuss a number of special cases. We show that a specialization of our results leads to some polynomials discussed by Koornwinder.

Response of a Periodically Driven Impact Oscillator

The response of a single-degree-of-freedom oscillator that can impact a surface with prescribed harmonic motion is investigated through experimental and numerical means. The test apparatus consisted of a stainless steel cantilever beam that could collide with a voice coil shaker which was driven at a specified frequency and amplitude. By collocating the contact and measurement points at a particular location near the free end of the beam, the response was dominated by the beam's fundamental mode. Response time records, frequency spectra and state space trajectories that were measured are compared to those predicted by a non-linear recurrence relation for the oscillator's state from one impact to the next. As borne out by non-dimensionalization of the model, the amplitude of response at a given frequency of excitation is proportional to the amplitude of the surface's motion, notwithstanding the non-linearity of the impact process. On the other hand, the qualitative character of the oscillator's response depends strongly on the frequency of the surface's motion. As the excitation frequency is increased gradually over a range equalling several times the oscillator's natural frequency, the response exhibits a recurring pattern of resonance, period-doubling bifurcation, and irregular non-periodic motion.

Integral equations and exact solutions for the fourth Painlevé equation

We consider a special case of the fourth Painlevé equation given by d 2 ƞ / dξ 2 = 3 ƞ 5 + 2ξ ƞ 3 + (1/4ξ 2 - v - 1/2 ) ƞ , (1) with v a parameter, and seek solutions ƞ (ξ; v ) satisfying the boundary condition ƞ (∞)=0. (2) Equation (1) arises as a symmetry reduction of the derivative nonlinear Schrödinger (DNLS) equation, which is a completely integrable soliton equation solvable by inverse scattering techniques. Solutions of equation (1), satisfying (2), are expressed in terms of the solutions of linear integral equations obtained from the inverse scattering formalism for the DNLS equation. We obtain exact ‘bound state’ solutions of equation (1) for v = n , a positive integer, using the integral equation representation, which decay exponentially as ξ→ ± ∞ and are the first example of such solutions for the Painlevé equations. Additionally, using Bäcklund transformations for the fourth Painlevé equation, we derive a nonlinear recurrence relation (commonly referred to as a Bäcklund transformation in the context of soliton equations) for equation (1) relating ƞ (ξ; v ) and ƞ (ξ; v + 1).

Localization of electrons and electromagnetic waves in a deterministic aperiodic system.

Electron localization and optical transmission in one-dimensional systems with two components distributed according to the Rudin-Shapiro sequence are investigated. The nature of the eigenstates in the diagonal tight-binding model is studied by making use of nonlinear recurrence relations satisfied by the associated transfer matrices and their traces. It is shown that the wave functions display a wide range of features going from weak to exponential localization. Numerical computations lead to the conjecture that the localization property is generic. Nevertheless, a countable dense set of critical values of the on-site potential amplitude is found for which a class of extended states exists. Accordingly, the numerical investigation of the time evolution of the electronic wave packets displays a subdiffusive behavior when the potential amplitude becomes critical. Finally, a study of the optical properties of the Rudin-Shapiro dielectric multilayer shows strong similarities with the behavior of the disordered multilayers regarding the development of gaps in the transmission spectra.

On Bernstein-Szegö orthogonal polynomials on several intervals. II. Orthogonal polynomials with periodic recurrence coefficients

Geronimus has shown that a sequence of orthogonal polynomials ( p n ) with periodic recurrence coefficients for n ⩾ n 0 is orthogonal on a set of disjoint intervals e l = U j = 1 l = [ a 2 j − 1 , a 2 j ] with respect to a distribution of the form dψ(x)= − ∏ j=1 2l (x−a j |p v(x)|dx+dμ(x) where ρ v ,( x) = Π j = 1 v ( x − W j ) with sgn ν v ( x) = (−1) l + 1 − j on ( a 2 j − 1 , a 2 j ) for j = 1, …, l, v ⩾ l − 1, and where μ is a certain point measure with supp( μ) ⊂{ w 1, …, w v }. In this paper we show (in fact a more general result is presented) that a sequence of polynomials ( p n ) orthogonal with respect to dψ has recurrence coefficients of period N, N ⩾ l, for n ⩾ n 0, if and only if there exists a so-called Chebyshev polynomial T N of degree N on E l , where a polynomial T N is called a Chebyshev polynomial on E l if ¦ T N| attains its maximum value on E l at N + l points from E l . Furthermore it is demonstrated how to get in a simple way a (nonlinear) recurrence relation for the recurrence coefficients of the orthogonal polynomials. Results on Chebyshev polynomials on several intervals are also given.

Finite element analysis of viscous fluid flow

Abstract A finite element model for the analysis of two dimensional viscous flows is formulated using the virtual work method. The model is in part based on a finite element shell model, using the same reduced integration of quadratic interpolations for all variables[1]. Differences from preceding formulations are that integration by parts is applied to the continuity equation, yielding different loading terms which are more easily defined in some problems, and a new approach is used for the convective inertia terms, giving a clearer interpretation of their effects which are distributed to both sides of the nonlinear recurrence relation. In the case of compressible flow, for which comparatively few formulations have been proposed to date, the thermal energy equation is used to form a two stage solution and here this seems the most natural and economical approach.

A note on the iterative solution of recurrence relations

A class of Gauss-Seidel iteration schemes suitable for the stable generation of non-dominant solutions of certain third order linear recurrence relations is developed. The algorithms derived have two main advantages over existing algorithms which generally re-formulate the problem as the solution of a system of algebraic equations. Firstly, unlike existing algorithms, the algorithms developed in this paper automatically determine the size of the system to be solved in all cases and secondly they may be extended directly to an important class of nonlinear recurrence relations.

Sensitivity problems related to certain bifurcations in non-linear recurrence relations

This paper is concerned with certain qualitative aspects of the sensitivity problem in relation to small variations of a parameter of a system, the behaviour of which can be described by an autonomous recurrence relation: V n+1 = F(V n , λ) (1) V being a vector, λ the parameter. The problem consists in the determination of the bifurcation values λ 0 of λ, i.e. values such that the qualitative behaviour of a solution of (1) should be different for λ = λ 0 ± ε where ε is a small quantity. Bifurcations that correspond to a critical case in the Liapunov sense, and the crossing through this critical case, are considered. Examples of bifurcations, not connected with the presence of a critical case, and which correspond to a large deformation of the stability domain boundary of an equilibrium point, a fixed point of (1), under the effect of a parameter variation, are given where V is a two dimensional vector.

The stability of harrod's growth model of an economy

Nevile [2] has shown that if Rt is a certain measure of the rate of growth of the national income in Harrod's growth model of an economy, then Rt satisfies the non-linear recurrence relation , where 0 < k < and −1 < c < 1. The definition of Rt ([2] p. 369) is such that Rt > 0 for all t. Nevile has pointed out features of the model that indicate that it may be unstable. In this paper I propose to show that the model is, in general. unstable, but that proper choice of the initial values R0R1 apparently leads to stability. In order to do this, we require the conditions (if any) under which Rt converges.

Nonlinear Recurrence Relations for Classical Orthogonal Polynomials

Nonlinear Recurrence Relations for Classical Orthogonal Polynomials

Non-Linear Recurrence Relations for Certain Classical Functions

Non-Linear Recurrence Relations for Certain Classical Functions

Non-Linear Recurrence Relations for Certain Classical Functions

Non-Linear Recurrence Relations for Certain Classical Functions

A Markovian Decision Process

Abstract : The purpose of this paper is to discuss the asymptotic behavior of the sequence (f sub n(i)) generated by a nonlinear recurrence relation. This problem arises in connection with an equipment replacement problem, cf. S. Dreyfus, A Note on an Industrial Replacement Process.