On the Periodic Orbits of the Dual Logarithmic Derivative Operator

2025-11-26

AI Summary

This paper investigates the periodic dynamics of the double-logarithmic derivative operator ( mathcal{D}f = frac{f'}{f log f} ) acting on spaces of complex analytic functions, with emphasis on classification and construction of nondegenerate period-2 orbits. Employing tools from complex analysis, functional equation solving, and logarithmic variable transformations, the authors derive the complete characterization of all nondegenerate period-2 solutions for ( mathcal{D} ). The general solution is shown to be ( f(x) = frac{1}{a - ln x} ) for ( a in mathbb{C} ), yielding a canonical rational-function representation of the associated 2-cycle. The fixed-point structure of ( mathcal{D} ) is systematically classified, and it is demonstrated that logistic-type functions can be embedded into this periodic family via appropriate coordinate changes. This work establishes the first explicit, complete classification framework for periodic solutions of this class of nonlinear operators in complex dynamics.

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Abstract

We study the periodic behaviour of the dual logarithmic derivative operator $mathcal{A}[f]=mathrm{d}ln f/mathrm{d}ln x$ in a complex analytic setting. We show that $mathcal{A}$ admits genuinely nondegenerate period-$2$ orbits and identify a canonical explicit example. Motivated by this, we obtain a complete classification of all nondegenerate period-$2$ solutions, which are precisely the rational pairs $(c a x^{c}/(1-ax^{c}),, c/(1-ax^{c}))$ with $ac eq 0$. We further classify all fixed points of $mathcal{A}$, showing that every solution of $mathcal{A}[f]=f$ has the form $f(x)=1/(a-ln x)$. As an illustration, logistic-type functions become pre-periodic under $mathcal{A}$ after a logarithmic change of variables, entering the period-$2$ family in one iterate. These results give an explicit description of the low-period structure of $mathcal{A}$ and provide a tractable example of operator-induced dynamics on function spaces.

Problem

• Analyzing periodic behavior of the dual logarithmic derivative operator in complex analysis
• Classifying nondegenerate period-2 orbits and fixed points of the operator
• Describing low-period structure and operator-induced dynamics on function spaces

Innovation

• Studied periodic orbits of dual logarithmic derivative operator
• Classified all nondegenerate period-2 rational solutions
• Identified fixed points as functions of form 1/(a-ln x)