Systematic bibliometric and visualized analysis of global research trends, impact, emerging areas, and hotspots of the fractional-order system

Fractional-order calculus extends the concepts of integer-order calculus by considering derivatives and integrals of fractional order, and has found applications in many areas such as engineering, control theory, and computational sciences. This study aims to conduct a comprehensive bibliometric analysis of the global literature on fractional-order systems published from 2000 to 2024. We searched all the English-language publications in Scopus as the data source and analyzed the results with bibliometric tools (VOSviewer and the Bibliometrix R package). We used publication trends, citation data, authorship, productivity of institutions, international collaboration, and thematic development as indicators. A total of 13,838 publications were analyzed. The fractional-order research shows that the total citations are 345,070, with an annual growth rate of 22.6 % and a total h-index of 179. China was first in the world with output (41.6%), followed by India and Saudi Arabia. Key institutions represented are King Abdulaziz University (2.5%) and the Ministry of Education of China (2.5%). The best journal was Fractal and Fractional (3.6%). Jinde Cao (1.5%) was the top author. Stability analysis, synchronization, and neural networks were among the main topics of research. Trends that rose were in the field of machine learning, energy optimization, and biomedical applications. The field has grown from a theoretical beginning to interdisciplinary uses and is now strengthened by a growing international collaboration and changing theme focus. This study provides a strategic roadmap for future research directions in fractional-order systems.