A note on exact solutions of the logistic map.

Area of Science:

• Dynamical Systems and Chaos Theory
• Nonlinear Dynamics
• Mathematical Physics

Background:

• The logistic map is a fundamental model in chaos theory, demonstrating complex behaviors like period doubling as a parameter increases.
• Exact analytical solutions for the logistic map are known only for specific parameter values.
• Understanding the full range of logistic map dynamics is crucial for various scientific fields.

Purpose of the Study:

• To investigate the existence of general solutions for the logistic map beyond the previously known exact solution cases.
• To introduce and analyze a novel special function required for these general solutions.
• To develop and present a numerical method for calculating this new function.

Main Methods:

• Analysis of the logistic map iterations for a broader range of the control parameter.
• Derivation of general solutions involving a newly defined special function.
• Development of a numerical algorithm for computing the special function.
• Graphical representation and analysis of the special function's properties.

Main Results:

• Demonstration that general solutions exist for the logistic map for parameter values beyond those with known exact solutions.
• Introduction of a special function, not expressible in terms of standard analytical functions, as key to these general solutions.
• A proposed numerical method for calculating this special function, accompanied by illustrative graphs and property analysis.

Conclusions:

• The study expands the analytical understanding of the logistic map by providing general solutions for all parameter values.
• The newly identified special function and its numerical computation offer new tools for analyzing chaotic systems.
• This work contributes to a more complete theory of the logistic map and its applications in nonlinear dynamics.