BIFURCATION AND K-CYCLES OF A FINITE-DIMENSIONAL ITERATIVE MAP, WITH APPLICATIONS TO LOGISTIC DELAY EQUATIONS

This paper analyzes the local bifurcation behavior of the limit sets a nd k-cycles for a family of smooth iterative maps g(A, .):R-m --> R de fined by u(n+1) = g(A, u(n), u(n-1),..., u(n-m+1)), where A is a real parameter. Such maps arise from many processes, including the numerica l solution of ordinary and delay differential equations. A linear stab ility theory establishes the local existence of a domain of attraction in which the map g(A,.) tends to a unique stable k-cycle. Included is a numerical algorithm for finding k-periodic points, stability region s and bifurcation points of the map g(A,.). Computational experiments with bifurcation diagrams for various iterative maps, including those from logistic delay equations, are presented along with the tabulated experimental and theoretical results. (C) 1997 Elsevier Science B.V.