Fractional calculus (or arbitrary order calculus) refers to the integration and derivative operators of an order different than one and was developed in 1695. They have been widely used to study dynamical systems, especially chaotic systems, as the use of arbitrary-order operators broke the milestone of restricting autonomous continuous systems of order three to obtain chaotic behavior and triggered the study of fractional chaotic systems. In this paper, we study the chaotic behavior in fractional systems in more detail and characterize the geometric variations that the dynamics of the system undergo when using arbitrary-order operators by asking the following question: is the Lyapunov exponent sufficient to describe the dynamical variations in a chaotic system of fractional order? By quantifying the convex envelope generated by the 2D projection of the system into all its phase portraits, the changes in the area of the system, as well as the volume of the attractor, are characterized. The results are compared with standard metrics for the study of chaotic systems, such as the Kaplan–Yorke dimension and the fractal dimension, and we also evaluate the frequency fluctuations in the dynamical response. It is found that our methodology can better describe the changes occurring in the systems, while the traditional dimensions are limited to confirming chaotic behaviors; meanwhile, the frequency spectrum hardly changes. The results deepen the study of fractional-order chaotic systems, contribute to understanding the implications and effects observed in the dynamics of the systems, and provide a reference framework for decision-making when using arbitrary-order operators to model dynamical systems.

This study examines how generative AI is adopted and experienced across educational levels in Sonora, Mexico, and whether students’ perceived learning aligns with engagement behaviors during AI-assisted homework. We analyze survey data from 2024–2025 covering 1477 participants (high school and university students and teachers) from public and private institutions, including adoption, perceived learning and time savings, help-seeking preferences (teachers vs. ChatGPT vs. Google), and ethical concerns. To move beyond self-reports alone, we introduce a Learning Engagement Index (LEI; 0–1) based on three student behaviors when using ChatGPT to complete academic tasks: reading AI responses, modifying outputs, and integrating personal ideas. Adoption was widespread but consistently higher in university than in high school for both students and teachers. University students reported slightly higher perceived learning and greater time savings. LEI scores were generally high and higher among university students, indicating more frequent engagement behaviors such as reading and adapting AI outputs rather than copying them. However, perceived learning showed only weak alignment with LEI, suggesting that students’ self-assessments do not consistently track the engagement actions measured by the index. A complementary GitHub Copilot Free (version GPT-4) experiment (n = 16) indicated faster task completion and improved task completeness, while also highlighting the risk of reduced algorithmic reasoning when AI suggestions are used uncritically. Overall, the findings point to the need for pedagogical approaches that emphasize guided use, verification practices, and assessment designs that more directly evidence learning in AI-mediated settings.