A nonlinear difference equation with two parameters

Abstract

The paper is mainly concerned with the difference equation where 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 given value of m, which makes it effectively a one-parameter problem, and it turns out that the changes 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 reversed for larger values of k where stable 4-cycle solutions precede stable 2-cycle solutions. Similar anomalies arise for the 3-cycle solutions.

A non-linear difference equation with two parameters. II

“…When \L X \ and \L 2 \ were both less than 0.8, /? 2 was obtained from a quadratic equation 2 and we can use equation (3.14) to determine the corresponding values for /? 1# Each pair was then substituted in equation (3.11) and the pair which gave better agreement was taken as the appropriate pair in the subsequent calculations.…”

“…However it can be proved that no C3 solutions are possible in this instance, so the process must come to a stop at some intermediate stage. Perhaps the simplest way of showing that no C3 solutions can occur is to use the result (from Paper I) that, for a given value of m, C3 solutions exist for K l < k < K 2 , where m > j3 and However, the behaviour of the solution depends on the value of k. To illustrate this we can start with k = 0.5, which gives F(y n ) = 1/{1 + (y n -m) 2 }. In this case there is a single equilibrium solution for each m and the equilibrium solution is stable.…”

“…(More precise values for m 1 and other critical values are given later.) The next critical value, m 2 , is roughly 2.31 and for m 1 < m < m 2 , there is a single C4 solution for each k, with two sub-intervals for κ in which the solutions are stable. One of these has k 5 as its lower limit while the other has k 6 as its upper limit.…”

“…One of these has k 5 as its lower limit while the other has k 6 as its upper limit. For m > m 2 , there are three C4 solutions for some values of k, and it is possible to have two more sub-intervals for κ where stable C4 solutions occur. In Paper I it was noted that for m = 3, superstable C4 solutions could be identified for four different values of k, and in fact each of these superstable solutions occurs in a separate "window of stability" whose extent is determined in the course of this paper.…”

“…This suggested that it would be worth examining points where L x = 0 and L 2 = 0 and finding what happens to the C4 solutions at these points. From equations (3.17) For a finite C4 solution, fi 2 must be finite and this implies that L l cannot be zero without having N t zero also (from equation (3.16) 2 , it follows that G and its first order partial derivatives are zero at points where L x , L 2 , N l and A^ are all zero. However, although this is a sufficient condition for a double root it is not a necessary condition.…”