Antimonotonicity: Concurrent creation and annihilation of periodic orbits

Abstract

The creation or annihilation of periodic orbits is the most fundamental bifurcation process in one-parameter families of maps. We call a parameter value at which a periodic orbit is created as the parameter is increased an orbitcreation value; and similarly, an orbit-annihilation value is a parameter value at which a periodic orbit is annihilated as the parameter is increased. Certain common bifurcation values of the parameter, called homoclintic-tangency values and defined precisely later, have long been known to occur only at accumulation points of orbit-creation or orbit-annihilation values. In the logistic family of one-dimensional maps