Development and Comparative Analyses of Bifurcation Diagrams Versus Estimation of Feigenbaum Constant for Logistic Map and Nonlinear Excited Oscillators

Abstract

Aims: To develop bifurcation diagrams for educational purposes, characterize simulated steady-state solutions in periodic form, and analyse the distribution of distinct solutions across parameter levels in three nonlinear systems (the logistic model, nonlinear pendulum, and Duffing oscillator) using a periodic approach to simulate and validate solutions against literature standards. Study Design: Numerical simulation and analysis. Place and Duration of Study: Department of Mechanical Engineering, University of Ibadan, Ibadan (12 Months). Methodology: We analysed three nonlinear systems: the logistic model and harmonically excited oscillators (nonlinear pendulum and Duffing oscillator). Relevant numerical tools were employed to simulate steady-state solutions (32 steady iterations for the logistic map, and 256 for the nonlinear pendulum and Duffing oscillator), develop corresponding bifurcation diagrams, characterise solutions data in periodic form, and distribute solutions to distinct appearances for all studied parameter levels. Results were validated against established literature standards, with comparative analysis across different parameter levels and numerical tools. Results: Comparative results across parameter levels showed uniform as well as non-uniform distributions of distinct solutions at different bifurcation levels, contrary to the anticipated uniform distribution. Data on the number of distinct solutions per parameter level were essential for determining the Feigenbaum constant, a universal constant characterising nonlinear systems undergoing chaos via the period-doubling bifurcation route. No specific sample sizes, P-values, or confidence intervals were reported; findings were qualitative and validated against literature. Conclusion: Analysing distinct solutions in bifurcation diagrams reveals unexpected distributions and provides essential data for calculating the Feigenbaum constant, enhancing characterisation of nonlinear systems prone to chaos through period-doubling routes. It provides a pedagogical framework for studying transitions (through bifurcations) from periodic to chaotic response.