Generalized logistic map and its applications

This study examines the generalized logistic map, a mathematical model defined by a recurrence relation with parameters that govern its dynamics. By broadening the scope of the classical logistic map, this generalized form captures a wider range of nonlinear behaviors, making it applicable to systems with non-uniform growth and saturation dynamics. One key parameter controls the bifurcation structure of the system, dictating transitions between fixed points, periodic behavior, and chaotic regimes. Additional parameters introduce flexibility by influencing equilibrium states and stability boundaries. Analytical techniques are used to identify significant bifurcations, such as period-doubling cascades and transitions to chaos. Numerical simulations complement these analyses, showcasing the complex dynamics that arise as the parameters are varied. This work highlights the diverse dynamical behaviors inherent in the generalized logistic map and emphasizes its potential applications in modeling biological, physical, and economic systems characterized by growth and saturation processes following power-law relationships.