Controlling intermediate dynamics in a family of quadratic maps

TL;DR

This paper explores how composing quadratic maps can generate multiple independent bifurcation diagrams by controlling intermediate dynamics, leading to complex phase space structures and novel bifurcation behaviors.

Contribution

It introduces a method to produce multiple independent bifurcation diagrams through the composition of quadratic maps, revealing new bifurcation mechanisms.

Findings

Multiple independent bifurcation diagrams can be generated by map composition.

Prohibition of period doubling bifurcations leads to saddle-node bifurcations.

Distinct parameter values produce k-independent bifurcation structures.

Abstract

The intermediate dynamics of composed one-dimensional maps is used to multiply attractors in phase space and create multiple independent bifurcation diagrams which can split apart. Results are shown for the composition of k-paradigmatic quadratic maps with distinct values of parameters generating k-independent bifurcation diagrams with corresponding k orbital points. For specific conditions, the basic mechanism for creating the shifted diagrams is the prohibition of period doubling bifurcations transformed in saddle-node bifurcations.