Chaotic Behaviour of a Discrete Dynamical System With Geogebra

Objective: To explore the knowledge of Geogebra, Python and Matlab to solve graphical problems with animation of discrete dynamic systems. Also to investigate the chaotic behavior of a discrete dynamic system using the Lyapunov exponent. Theoretical Framework: This article deals with the chaotic behavior of a discrete dynamic system through the attractors and a parameter included in the model. First, the orbit and the equilibrium points are defined. Beyond that, according to Li-York there is a chaotic behavior. For this, the Lyapunov exponent plays a fundamental role. The chaotic behavior of an extension of a discrete dynamic system of population growth is analyzed dependent on a parameter “a”. To do this, first, the attractors for the discrete dynamic system are analyzed, together with the graph of the Lyapunov exponent versus the parameter a. Therefore, it is seen that the graph of the Lyapunov exponent function, where the parameter belongs to the interval (3,5), has a positive region, so it can be stated that the discrete dynamic system has a chaotic behavior. We will apply the Lyapunov exponent Method: The interval of chaotic behavior is located, using the Lyapounov exponent. Geogebra method for the graphs of the dynamics and analysis of chaos and location of the corresponding entropy. Results and Discussion: The Geogebra graphical method shows the Lyapunov exponent function, where the parameter belongs to the interval (3,5), has a positive region, so it can be stated that the discrete dynamic system (1) has a chaotic behavior. The work is similar to the Dynamic system, chaos and geogebra to model some phenomena of Costa (2023). Implications of the research: Apply in the first university years the application of geogebra to discrete dynamic systems in all university careers and specialties as part of the management of the tools of Information and Communication Technology (TIC). Originality/Value: This study contributes to the presentation of a chaotic dynamics analysis model with the corresponding entropy or disorder of the system.