Zobecněná logistická zobrazení

Abstract

The logistic map is a classical model used to describe population dynamics in discrete time. A natural extension of this model is the generalized logistic map, which introduces additional flexibility through an extra exponent parameter. In this study, we focus on a particular case of the generalized logistic map known as the Richard's logistic map, which is closely related to Richard’s difference equation. To support this analysis, we begin by reviewing the foundational concepts of differential and difference equations, providing a basis for understanding the behavior of systems in both continuous and discrete time settings. We analyze the properties of these generalized logistic maps with emphasis on the special case, which simplifies to the Richard's logistic model. Through detailed algebraic manipulation, equilibrium points and their stability are investigated using derivative tests and the Schwarzian derivative. The maximum value of the growth parameter u is also computed to identify when chaotic behavior begins. The work further examines period-2 cycles and their bifurcation behavior as the growth parameter increases. Numerical simulations, cobweb plots, and bifurcation diagrams are implemented in Python to visualize transitions from stability to periodicity and chaos. The analysis reveals that for the Richard's case studied, the map becomes chaotic when u exceeds approximately 2.59, which is notably lower than the simple logistic case. Lastly, comparisons with the classical logistic map demonstrate that increasing the nonlinearity accelerates convergence to equilibrium and narrows the range of u that results in bounded behavior. These findings underline how small changes in model structure can significantly affect long-term dynamics, contributing to a deeper understanding of discrete chaos in nonlinear systems.

The logistic map is a classical model used to describe population dynamics in discrete time. A natural extension of this model is the generalized logistic map, which introduces additional flexibility through an extra exponent parameter. In this study, we focus on a particular case of the generalized logistic map known as the Richard's logistic map, which is closely related to Richard’s difference equation. To support this analysis, we begin by reviewing the foundational concepts of differential and difference equations, providing a basis for understanding the behavior of systems in both continuous and discrete time settings. We analyze the properties of these generalized logistic maps with emphasis on the special case, which simplifies to the Richard's logistic model. Through detailed algebraic manipulation, equilibrium points and their stability are investigated using derivative tests and the Schwarzian derivative. The maximum value of the growth parameter u is also computed to identify when chaotic behavior begins. The work further examines period-2 cycles and their bifurcation behavior as the growth parameter increases. Numerical simulations, cobweb plots, and bifurcation diagrams are implemented in Python to visualize transitions from stability to periodicity and chaos. The analysis reveals that for the Richard's case studied, the map becomes chaotic when u exceeds approximately 2.59, which is notably lower than the simple logistic case. Lastly, comparisons with the classical logistic map demonstrate that increasing the nonlinearity accelerates convergence to equilibrium and narrows the range of u that results in bounded behavior. These findings underline how small changes in model structure can significantly affect long-term dynamics, contributing to a deeper understanding of discrete chaos in nonlinear systems.