Rigorous numerical study of the density of periodic windows for the logistic map

Area of Science:

• Dynamical Systems and Chaos Theory
• Computational Mathematics

Background:

• The logistic map is a fundamental model in chaos theory, exhibiting complex dynamics including periodic windows.
• Understanding the distribution and characteristics of these periodic windows is crucial for characterizing the map's behavior.

Purpose of the Study:

• To numerically investigate periodic windows within the logistic map.
• To develop and apply efficient methods for locating and analyzing periodic windows.
• To explain the existence of parameter space regions devoid of low-period windows.

Main Methods:

• Utilizing interval arithmetic for accurate computation of periodic window endpoints.
• Developing an efficient algorithm to find the smallest period window between two existing windows.
• Applying the method to identify windows near specific parameter values and minimize maximum gaps.

Main Results:

• Accurate rigorous bounds for periodic window endpoints were computed.
• An efficient method for finding minimal period windows was successfully developed and applied.
• Periodic windows extremely close to selected parameter points were identified.
• A set of periodic windows was found to minimize the maximum gap to 4×10-9.
• The phenomenon of regions free from low-period windows was explained.

Conclusions:

• The study provides precise computational tools for analyzing periodic windows in the logistic map.
• The developed method offers efficient identification of periodic windows, advancing the understanding of chaotic systems.
• The findings contribute to explaining the complex structure of the logistic map's parameter space.