Bifurcation and k-cycles of a finite-dimensional iterative map, with applications to logistic delay equations Original Research Article

This paper analyzes the local bifurcation behavior of the limit sets and k-cycles for a family of smooth iterative maps g(A,·):View the MathML sourcem → View the MathML source defined by un+1 = g(A, un, un−1,…, un−m+1), where A is a real parameter. Such maps arise from many processes, including the numerical solution of ordinary and delay differential equations. A linear stability 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 regions 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.