Singularity confinement for a class of m -th order difference equations of combinatorics

2007 ◽  
Vol 366 (1867) ◽  
pp. 877-922 ◽  
Author(s):  
Mark Adler ◽  
Pierre van Moerbeke ◽  
Pol Vanhaecke
Keyword(s):  
Finite Number ◽  
Difference Equations ◽  
Random Permutations ◽  
Order Difference ◽  
Painlevé Property ◽  
Special Cases ◽  
Singularity Confinement ◽  
Time Differential ◽  
Free Parameters ◽  
The Difference

In a recent publication, it was shown that a large class of integrals over the unitary group U ( n ) satisfy nonlinear, non-autonomous difference equations over n , involving a finite number of steps; special cases are generating functions appearing in questions of the longest increasing subsequences in random permutations and words. The main result of the paper states that these difference equations have the discrete Painlevé property ; roughly speaking, this means that after a finite number of steps the solution to these difference equations may develop a pole (Laurent solution), depending on the maximal number of free parameters, and immediately after be finite again (‘ singularity confinement ’). The technique used in the proof is based on an intimate relationship between the difference equations (discrete time) and the Toeplitz lattice (continuous time differential equations); the point is that the Painlevé property for the discrete relations is inherited from the Painlevé property of the (continuous) Toeplitz lattice.

scholarly journals Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

2019 ◽  
pp. 1-12
Author(s):  
Abdualrazaq Sanbo ◽  
Elsayed M. Elsayed ◽  
Faris Alzahrani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Analytical Study ◽  
Initial Conditions ◽  
Specific Form ◽  
Positive Real ◽  
Rational Difference Equations ◽  
Special Cases ◽  
The Difference

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

scholarly journals Existence and Multiplicity Results of Homoclinic Solutions for the DNLS Equations with Unbounded Potentials

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Defang Ma ◽  
Zhan Zhou
Keyword(s):  
Difference Equations ◽  
Mountain Pass Theorem ◽  
Sufficient Conditions ◽  
Homoclinic Solutions ◽  
Fountain Theorem ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Special Cases ◽  
Discrete Nonlinear Schrödinger Equations ◽  
The Difference

A class of difference equations which include discrete nonlinear Schrödinger equations as special cases are considered. New sufficient conditions of the existence and multiplicity results of homoclinic solutions for the difference equations are obtained by making use of the mountain pass theorem and the fountain theorem, respectively. Recent results in the literature are generalized and greatly improved.

scholarly journals Comparison results and linearized oscillations for higher-order difference equations

1992 ◽  
Vol 15 (1) ◽  
pp. 129-142 ◽  
Author(s):  
G. Ladas ◽  
C. Qian
Keyword(s):  
Linear Equation ◽  
Difference Equations ◽  
Bounded Solution ◽  
Higher Order ◽  
Comparison Result ◽  
Oscillation Theorem ◽  
Order Difference ◽  
Comparison Results ◽  
The Difference

Consider the difference equationsΔmxn+(−1)m+1pnf(xn−k)=0,   n=0,1,…        (1)andΔmyn+(−1)m+1qng(yn−ℓ)=0,   n=0,1,….       (2)We establish a comparison result according to which, whenmis odd, every solution of Eq.(1) oscillates provided that every solution of Eq.(2) oscillates and, whenmis even, every bounded solution of Eq.(1) oscillates provided that every bounded solution of Eq.(2) oscillates. We also establish a linearized oscillation theorem according to which, whenmis odd, every solution of Eq.(1) oscillates if and only if every solution of an associated linear equationΔmzn+(−1)m+1pzn−k=0,   n=0,1,…         (*)oscillates and, when m is even, every bounded solution of Eq.(1) oscillates if and only if every bounded solution of (*) oscillates.

From chaos to order. Difference equations in one ecological problem

2016 ◽  
Vol 31 (5) ◽  
Author(s):  
Georgy K. Kamenev ◽  
Oleg P. Lyulyakin ◽  
Dmitry A. Sarancha ◽  
Nikolai A. Lysenko ◽  
Valery O. Polyanovsky
Keyword(s):  
Population Dynamics ◽  
Numerical Analysis ◽  
Difference Equations ◽  
Computational Experiments ◽  
Ecological Problem ◽  
Order Difference ◽  
Resonance Parameters ◽  
The Difference ◽  
Natural Series ◽  
Methods Of Approximation

AbstractIn this paper we consider properties of the difference equations (discrete mappings) obtained in the study of the population dynamics of lemmings. A bifurcation scenario is proposed for obtained equations. Certain stability zones appear under this scenario with periods varying in order of natural series and also zones with more complicated modes. The study of transitional zones (‘ordering of the chaos’) is performed with the use of analytic calculations and computational experiments. Numerical analysis of mappings uses the methods of approximation of implicitly specified sets allowing us to construct and visualize sets of ‘resonance’ parameters including the front of the so-called singularity of ‘blue sky’.

scholarly journals Growth of Meromorphic Solutions of Some -Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Guowei Zhang
Keyword(s):  
Difference Equation ◽  
Fixed Points ◽  
Difference Equations ◽  
Meromorphic Solutions ◽  
Order Difference ◽  
Complex Difference ◽  
Complex Difference Equations ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Difference

We estimate the growth of the meromorphic solutions of some complex -difference equations and investigate the convergence exponents of fixed points and zeros of the transcendental solutions of the second order -difference equation. We also obtain a theorem about the -difference equation mixing with difference.

scholarly journals Growth of meromorphic solutions of some difference equations

2010 ◽  
Vol 4 (2) ◽  
pp. 309-321 ◽  
Author(s):  
Xiu-Min Zheng ◽  
Zong-Xuan Chen ◽  
Tu Jin
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Higher Order ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Order Difference ◽  
Difference Analogue ◽  
Algebraic Difference ◽  
The Difference

We investigate higher order difference equations and obtain some results on the growth of transcendental meromorphic solutions, which are complementary to the previous results. Examples are also given to show the sharpness of these results. We also investigate the growth of transcendental entire solutions of a homogeneous algebraic difference equation by using the difference analogue of Wiman-Valiron Theory.

scholarly journals Distribution of iterates of first order difference equations

1980 ◽  
Vol 22 (1) ◽  
pp. 133-143 ◽  
Author(s):  
James B. McGuire ◽  
Colin J. Thompson
Keyword(s):  
Invariant Measure ◽  
Difference Equation ◽  
Lebesgue Measure ◽  
Difference Equations ◽  
Absolutely Continuous ◽  
Order Difference ◽  
First Order ◽  
Order Difference Equation ◽  
The Difference ◽  
Biological Context

An invariant measure which is absolutely continuous with respect to Lebesgue measure is constructed for a particular first order difference equation that has an extensive biological pedigree. In a biological context this invariant measure gives the density of the population whose growth is governed by the difference equation. Further asymptotically universal results are obtained for a class of difference equations.

scholarly journals Behaviors of the Solutions via Their Closed-Form Formulas for Two Rational Third-Order Difference Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Raafat Abo-Zeid ◽  
Abdul Qadeer Khan
Keyword(s):  
Closed Form ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Third Order ◽  
Order Difference ◽  
The Difference

In this work, we derive the solution formulas and study their behaviors for the difference equations x n + 1 = α x n x n − 3 / − β x n − 3 + γ x n − 2 , n ∈ ℕ 0 and x n + 1 = α x n x n − 3 / β x n − 3 − γ x n − 2 , n ∈ ℕ 0 with real initials and positive parameters. We show that there exist periodic solutions for the second equation under certain conditions when β 2 < 4 α γ . Finally, we give some illustrative examples.

scholarly journals Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

AbstractIn this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.

scholarly journals Oscillation of nonlinear third-order difference equations with mixed neutral terms

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jehad Alzabut ◽  
Martin Bohner ◽  
Said R. Grace
Keyword(s):  
Difference Equations ◽  
Riccati Transformation ◽  
Third Order ◽  
Order Difference ◽  
New Approach ◽  
Comparison Technique

AbstractIn this paper, new oscillation results for nonlinear third-order difference equations with mixed neutral terms are established. Unlike previously used techniques, which often were based on Riccati transformation and involve limsup or liminf conditions for the oscillation, the main results are obtained by means of a new approach, which is based on a comparison technique. Our new results extend, simplify, and improve existing results in the literature. Two examples with specific values of parameters are offered.

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