Investigation of bifurcations in the behavior of the process equation

TL;DR

This paper explores the complex behaviors of a multistable process equation with nonlinear feedback, analyzing bifurcations, chaos, and periodic windows through phase portraits and mathematical investigation.

Contribution

It provides a detailed analysis of bifurcations and unstable windows in the process equation, highlighting new insights into its behavior and parameter effects.

Findings

Identification of period doubling route to chaos

Characterization of unstable and periodic windows

Mathematical analysis of bifurcation patterns

Abstract

This paper investigates the different behaviors of the process equation and parameters of their occurrences. The process equation is a multistable one dimensional map with nonlinear feedback and can show various behaviors such as period doubling route to chaos, bios, unstable windows and periodic windows. In this note, we focus on different behaviors of the process equation by a deep look at phase portraits and cobweb plots. Therefore, period doubling route to chaos and unifurcations of the equation are investigated, and also the parameter of its entrance to biotic pattern is discussed. The control parameter for the process equation is g (the coupling constant). In higher g, the system shows some unstable and periodic windows among the biotic behaviors. Different patterns of these windows and the reasons of their happenings are investigated mathematically. In addition, some other types of unstable and periodic windows are discussed and Q-curves determine the difference of these windows with the previous ones.