Mathematics

Chaotic dynamical systems are ubiquitous in nature and modern technology, with applications ranging from secure communications and cryptography to the design of chaos-based sensors and modeling biological phenomena such as arrhythmias and neuronal behavior. Given their complexity, precise analysis of these systems is crucial for both theoretical understanding and practical implementation. The characterization of chaotic dynamical systems typically relies on conventional measures such as Lyapunov exponents and fractal dimensions. While these metrics are fundamental for describing dynamical behavior, they are often computationally expensive and may fail to capture subtle changes in the overall geometry of the attractor, limiting comparisons between systems with topologically similar structures and similar values in common chaos metrics such as the Lyapunov exponent. To address this limitation, this work proposes a geometric framework that treats chaotic attractors as spatial objects, using topological tools—specifically the $\alpha$-sphere—to quantify their shape and spatial extent. The proposed method was validated using Chua’s system (including two reported variations), the Rössler system (standard and piecewise-linear), and a fractional-order multi-scroll system. A parametric characterization of the Rössler system was also performed by varying parameter b. Experimental results show that this geometric approach successfully distinguishes between attractors where classical metrics reveal no perceptible differences, in addition to being computationally simpler. Notably, we observed geometric variations of up to 80% among attractors with similar dynamics and introduced a specific index to quantify these global discrepancies. Although this geometric analysis serves as a complement rather than a substitute for chaos detection, it provides a reliable and interpretable metric for differentiating systems and selecting attractors based on their spatial properties.

This work addresses the tracking control problem of nonstrict-feedback nonlinear systems affected by unmodeled dynamics and input delays, which significantly complicate controller design and degrade system performance. To overcome these challenges, a predefined-time adaptive control framework is developed. A command-filtered backstepping scheme is employed to reduce computational complexity, while an error compensation mechanism is introduced to counteract the inaccuracies caused by command filtering. The unknown nonlinear dynamics are approximated using radial basis function-based estimators, and a dynamic auxiliary signal is designed to mitigate the effects of unmodeled dynamics. Input delays are handled by integrating Padé approximation with an intermediate compensating variable. The proposed control strategy guarantees uniform boundedness of all closed-loop signals and ensures that the tracking error converges to a small neighborhood of the desired trajectory within a predefined time. Simulation results and comparative studies are provided to demonstrate the effectiveness and advantages of the proposed method.

This paper introduces a novel SEIRS-type differential model that incorporates significant real-world factors such as vaccination, hospitalization, and vital dynamics. The model is described by a system of nonlinear ordinary differential equations with time-dependent parameters and coefficients. First, fundamental biological properties of the model, including the existence, uniqueness, and non-negativity of its solution, are established. In addition, using official COVID-19 data from Bulgaria, a special inverse problem for the differential model is formulated and investigated through the construction of an appropriate family of time-discrete inverse problems. As a result, the model parameters are identified, and the model is validated using real-world data. The presented numerical experiments confirm that the proposed methodology performs well in real-world applications with actual data. A very good agreement between computed and officially reported data with respect to the $l_2$ and $l_{\infty }$ norms is obtained. The model and its simulation tools are adaptable and can be applied to datasets from other countries, provided suitable epidemiological data are available.

Modern Shape Analysis integrates mathematical and statistical methods to study morphology using geometric data from 2D and 3D digitization. Beyond visualization, local deformation analysis offers deeper insight into longitudinal changes like ontogenetic growth. This study explores strain tensors as analytical tools for shape transformations, evaluating Thin Plate Spline, Quadratic Trend, and Cubic Trend interpolants in estimating deformation tensors at landmarks. Simulated 2D data and 3D hominid skulls confirm that tensor-based multivariate analyses, such as Principal Component Analysis, yield results comparable to Parallel Transport methods. Thin Plate Spline more accurately recovers assigned tensors than least square interpolants, making it preferable for local deformation analysis. This method integrates traditional shape comparisons, proving particularly useful for studying ontogenetic trajectories and biomechanical deformations.