scholarly journals Critical values for a nonlinear difference equation

1987 ◽  
Vol 28 (3) ◽  
pp. 340-361
Author(s):  
A. Brown
Keyword(s):  
Difference Equation ◽  
Periodic Solutions ◽  
Finite Length ◽  
Critical Values ◽  
Nonlinear Difference Equation ◽  
Equilibrium Solutions ◽  
Critical Curves ◽  
Parameter Equation ◽  
The Difference ◽  
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AbstractThe paper discusses equilibrium solutions and solutions with period two and period three for the difference equationwhere Q and A are real, positive parameters. The equation was used by Bier and Bountis [1] as an example of a difference equation whose iteration diagram can show bubbles of finite length rather than the successive bifurcations usually expected. The paper examines in more detail what kind of solution can occur for given values of Q and A and establishes a series of critical curves which demarcate the regions in the (Q, A) plane where solutions of period two or period three occur and the subregions where these periodic solutions are stable. This makes it easy to see how Q and A can be combined into a one-parameter equation which gives a bubble, or a series of bubbles, in the iteration diagram.

scholarly journals solutions of period three for a non-linear difference equation

1984 ◽  
Vol 25 (4) ◽  
pp. 451-462 ◽  
Author(s):  
A. Brown
Keyword(s):  
Mathematical Model ◽  
Difference Equation ◽  
Exact Solutions ◽  
Linear Difference Equation ◽  
Critical Values ◽  
Simple Mathematical Model ◽  
Frequency Dependent Selection ◽  
Stable Solutions ◽  
The Difference ◽  
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AbstractThe paper uses the factorisation method to discuss solutions of period three for the difference equationwhich has been proposed as a simple mathematical model for the effect of frequency dependent selection in genetics. Numerical values are obtained for the critical values of a at which solutions of period three first appear. In addition, the interval in which stable solutions are possible has been determined. Exact solutions are given for the case a = 4 and these have been used to check the results.

scholarly journals A nonlinear difference equation with two parameters

1985 ◽  
Vol 26 (4) ◽  
pp. 430-451 ◽  
Author(s):  
A. Brown
Keyword(s):  
Difference Equation ◽  
General Problem ◽  
Iteration Process ◽  
Nonlinear Difference Equation ◽  
Cubic Equation ◽  
The Difference ◽  
The Stability ◽  
Image Position ◽  
Two Parameters ◽  
Parameter Problem

AbstractThe paper is mainly concerned with the difference equationwhere k and m are parameters, with k > 0. This equation arises from a method proposed for solving a cubic equation by iteration and represents a standardised form of the general problem. In using the above equation it is essential to know when the iteration process converges and this is discussed by means of the usual stability criterion. Critical values are obtained for the occurrence of solutions with period two and period three and the stability of these solutions is also examined. This was done by considering the changes as k increases, for a give value of m, which makes it effectively a one-parameter problem, and it turns out that the change with k can differ strongly from the usual behaviour for a one-parameter difference equation. For m = 2, for example it appears that the usual picture of stable 2-cycle solutions giving way to stable 4-cycle solutions is valid for smaller values of k but the situation is recersed for larger values of k where stable 4-cycle solutions precede stable 2-cycle solutions. Similar anomalies arise for the 3-cycle solutions.

A Singular Perturbation Problem and A Neutral Differential-Difference Equation

1974 ◽  
Vol 17 (1) ◽  
pp. 77-83
Author(s):  
Edward Moore
Keyword(s):  
Difference Equation ◽  
Taylor Series Expansion ◽  
Linear Difference Equation ◽  
Constant Matrix ◽  
Perturbation Problem ◽  
Constant Coefficients ◽  
Explicit Formulae ◽  
Differential Difference Equation ◽  
The Difference ◽  
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Vasil’eva, [2], demonstrates a close connection between the explicit formulae for solutions to the linear difference equation with constant coefficients(1.1)where z is an n-vector, A an n×n constant matrix, τ>0, and a corresponding differential equation with constant coefficients(1.2)(1.2) is obtained from (1.1) by replacing the difference z(t—τ) by the first two terms of its Taylor Series expansion, combined with a suitable rearrangement of the terms.

scholarly journals The evaluation of certain continued fractions

1937 ◽  
Vol 30 ◽  
pp. vi-x
Author(s):  
C. G. Darwin
Keyword(s):  
Difference Equation ◽  
Hypergeometric Function ◽  
Difference Equations ◽  
Continued Fractions ◽  
Continued Fraction ◽  
The Difference ◽  
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1. If the approximate numerical value of e is expressed as a continued fraction the result isand it was in finding the proof that the sequence extends correctly to infinity that the following work was done. First the continued fraction may be simplified by setting down the difference equations for numerator and denominator as usual, and eliminating two out of every successive three equations. A difference equation is thus formed between the first, fourth, seventh, tenth … convergents , and this equation will generate another continued fraction. After a little rearrangement of the first two members it appears that (1) implies2. We therefore consider the continued fractionwhich includes (2), and also certain continued fractions which were discussed by Prof. Turnbull. He evaluated them without solving the difference equations, and it is the purpose here to show how the difference equations may be solved completely both in his cases and in the different problem of (2). It will appear that the work is connected with certain types of hypergeometric function, but I shall not go into this deeply.

scholarly journals On a Difference Equation due to Stirling

1917 ◽  
Vol 36 ◽  
pp. 40-60
Author(s):  
Eleanor Pairman
Keyword(s):  
Difference Equation ◽  
Factorial Series ◽  
The Difference ◽  
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In 1730 there was published Stirling's Methodus Differentialis, and in it (Prop. VIII., p. 44) he considers the Difference Equationand shows that it is satisfied by an inverse factorial series

Periodic solutions of forced Liénard equations with jumping nonlinearities under nonuniform conditions

1988 ◽  
Vol 110 (3-4) ◽  
pp. 183-198 ◽  
Author(s):  
R. Iannacci ◽  
M.N. Nkashama ◽  
P. Omari ◽  
F. Zanolin
Keyword(s):  
Differential Equation ◽  
Asymptotic Behaviour ◽  
Periodic Solutions ◽  
Homogeneous Problem ◽  
A Priori ◽  
Critical Values ◽  
Liénard Equations ◽  
Jumping Nonlinearities ◽  
Existence Of Periodic Solutions ◽  
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SynopsisThis paper is devoted to the existence of periodic solutions for the scalar forced Lienard differential equationThe key assumptions relate the asymptotic behaviour as x →± ∞of g(t; x)/x to the “critical values” of the positively 1-homogeneous problemNo condition on f, except continuity, is assumed. Our approach is based on Leray–Schauder degree techniques and a priori estimates.

scholarly journals On the Relation between Pincherle's Polynomials and the Hypergeometric Function

1920 ◽  
Vol 39 ◽  
pp. 58-62 ◽  
Author(s):  
Bevan B. Baker
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Hypergeometric Function ◽  
The Difference ◽  
Image Position

1. The Pincherle polynomials are defined as the coefficients in the expansion of {1 − 3 tx + t3}−½ in ascending powers of t. If the coefficient of tn be denoted by Pn(x), then the polynomials satisfy the difference equationand Pn(x) satisfies the differential equation

On discrete harmonic functions

1949 ◽  
Vol 45 (2) ◽  
pp. 194-206 ◽  
Author(s):  
H. A. Heilbronn
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Difference Equation ◽  
Harmonic Functions ◽  
The Difference ◽  
Discrete Harmonic Functions ◽  
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A function f(x1, x2) of two real variables x1, x2 which are restricted to rational integers will be called discrete harmonic (d.h.) if it satisfies the difference equationThis equation can be considered as the direct analogue either of the differential equationor of the integral equationin the notation normally employed to harmonic functions.

scholarly journals On a nonlinear difference equation with variable delay

2013 ◽  
Vol 46 (1) ◽  
Author(s):  
Dinh Cong Huong ◽  
Nguyen Van Mau
Keyword(s):  
Difference Equation ◽  
Periodic Solutions ◽  
Nonlinear Difference Equation ◽  
Variable Delay ◽  
Positive Periodic Solutions ◽  
Boundedness Of Solutions ◽  
The Stability

AbstractIn this paper, some results on the equi-boundedness of solutions, the stability of the zero and the existence of positive periodic solutions of nonlinear difference equation with variable delay

scholarly journals On the Dynamics of the Recursive Sequence

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Mehmet Gümüş ◽  
Özkan Öcalan ◽  
Nilüfer B. Felah
Keyword(s):  
Periodic Solution ◽  
Difference Equation ◽  
Periodic Solutions ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Recursive Sequence ◽  
Boundedness Character ◽  
The Difference

We investigate the boundedness character, the oscillatory, and the periodic character of positive solutions of the difference equation , where , , and the initial conditions are arbitrary positive numbers. We investigate the boundedness character for . Also, we investigate the existence of a prime two periodic solution for is odd. Moreover, when is even, we prove that there are no prime two periodic solutions of the equation above.

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