A Continuous Planar Map With Many Periodic Points

摘要

The existence and structure of periodic orbits of the discrete delayed logistic equation xn=;1=rxn(1−xn−1) is studied. In the phase plane (x,y)=(xn−1,xn), the corresponding continuous planar map is Fr(x,y)=(y, ry(1−x)). The Birkhoff-Smale theorem for infinitely many periodic points in the neighborhood of a homoclinic point of a differentiable map is appropriately modified for the example; the proof resembles the geometric proof of Birkhoff in two dimensions. Related numerical work is included, and implications of the results in terms of a model for population growth are discussed.