Analysis of Local Bifurcations in One-Dimensional Systems

Abstract

This article delves deeper into the study of local bifurcation in maps, although only in the context of a single dimension. The various local bifurcation types common to one-dimensional maps each have their own specific conditions that must be fulfilled. An object divides into two when it bifurcates. After a bifurcation, a family of one-parameter functions retains its stable or cyclic point structure. A bifurcation in the iterative process happens when a parameter is altered, causing a qualitative change in behavior. The phenomena of local splitting can be induced by adjusting just one parameter. Both the norm form of the transcritical split and the stability of the saddle node are highlighted. As with supercritical and subcritical bifurcations, the stability of the pitchfork bifurcation is assessed, as is its norm form.

Jagannath University Journal of Science, Volume 10, Number II, Dec. 2023, pp. 108-123