Fractional-order models provide a powerful framework for capturing memory-dependent and viscoelastic dynamics in mechanical systems, which are often inadequately represented by classical integer-order characterizations. This study addresses the identification of dynamic parameters in both single-degree-of-freedom (1-DOF) and three-degree-of-freedom (3-DOF) Duffing oscillators with fractional damping, modeled using the Grünwald–Letnikov characterization. The 1-DOF system includes a cubic nonlinear restoring force and is excited by a harmonic input to induce steady-state oscillations. For both systems, time domain simulations are conducted to capture long-term responses, followed by Fourier decomposition to extract steady-state displacement, velocity, and acceleration signals. These components are combined with a GL-based fractional derivative approximation to construct structured regressor matrices. System parameters—including mass, stiffness, damping, and fractional-order effects—are then estimated using pseudoinverse techniques. The identified models are validated through a comparison of reconstructed and original trajectories in the phase space, demonstrating high accuracy in capturing the underlying dynamics. The proposed framework provides a consistent and interpretable approach for frequency domain system identification in fractional-order nonlinear systems, with relevance to applications such as mechanical vibration analysis, structural health monitoring, and smart material modeling.

The last years of the 20 th century and the beginning of the 21 th mark the facilitation trend of our real life due to the big development and progress of the computers and other intelligent devices. Algorithms based on artificial intelligence are basically a part of the software. The transmitted information by Internet or LAN arises continuously and it is expected that the protection of the data has been ensured. The aim of the present paper is to reveal false names of users' accounts as a result of hackers' attacks. The probability a given account to be either false or actual is calculated using a novel approach combining machine learning analysis (especially clusters' analysis) with chaos theory. The suspected account will be used as a pattern and by classification techniques clusters will be formed with a respective probability this name to be false. This investigation puts two main purposes: First, to determine if there exists a trend of appearance of the similar usernames, which arises during the creation of new accounts. Second, to detect the false usernames and to discriminate those from the real ones, independently of that if two types of accounts are generated with the same speed. These security systems are applied in different areas, where the security of the data in users' accounts is strictly required. For example, they can be used in on-line voting for balloting, in studying the social opinion by inquiries, in protection of the information in different user accounts of given system etc.