The asymptotic behaviour of certain difference equations with proportional delays

• Published in: Computers & mathematics with applications (1987) Vol. 28; no. 1; pp. 141 - 152
• Main Author: Iserles, A.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.08.1994
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/0898-1221(94)00102-2

This paper is concerned with the nonstationary linear difference equation y n = δy n−1 + μ(y [n/2] + y [(n−1)/2]), y 0 = 1 . We demonstrate by Fourier techniques that the sequence is asymptotically stable when |λ| < 1 and |μ| < ( 1 2 )|1 − λ| , but our main effort is devoted to the marginal case | λ| = 1. We derive the solution explicitly as a power series in μ for λ = −1 M, thereby demonstrating that it is uniformly bounded and that, for μ≠0, its attractor contains a countable subset of distinct points. Finally, we consider a somewhat more general difference equation and prove that, for specific choices of parameters therein, the attractor is a probabilistic mixture of Julia sets.