ZHOU, Y. Analysis of a not so well-known chaotic dynamical system [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2025.

Posudek vedoucího

Nechvátal, Luděk

The thesis investigates a chaotic system introduced in [Lü, Chen & Cheng, 2004]. This system depends on three (real) parameters, and although resembling the well-known Lorenz system, it exhibits somewhat richer dynamics. The aim was to explore as much as possible using the tools of nonlinear dynamics (the mentioned paper leaves space for such investigations). The goals were fulfilled. However, I feel the topic could be handled a bit better, particularly in the theoretical part. The individual concepts are just listed without a more profound logical continuity, comments/explanations, mutual connection, etc. Additional minor issues like ununified notation, misprints, unprecise formulations, and typographical imperfections do not help the overall impression either. Let me mention two issues as examples: 1. The Routh–Hurwitz stability criterion (an essential tool in the thesis) should be formulated more precisely. It is a statement about stability, but the stability of what? And also asymptotic stability or just stability? 2. Remark 2.37 should be formulated more cautiously. The so-called Perron effect says there can be an unstable equilibrium with negative Lyapunov exponents, and, vice versa, a stable equilibrium with positive Lyapunov exponents. So, this criterion is just an indicator (for most cases). The practical part is better. Even calculation of equlibria is a quite tedious task. Here, I can imagine a description of the Matlab routines used to calculate the solutions or the largest Lyapunov exponent (which is missing).

Posudek oponenta

Opluštil, Zdeněk

The diploma thesis focuses on the analysis of a not-well-known nonlinear chaotic system, specifically a three-dimensional nonlinear autonomous system proposed by Lü in 2004. Although this system exhibits interesting Lorenz-like dynamics, it has not been widely studied in the existing literature. The author’s work contributes to the understanding of this system, particularly in areas such as stability analysis, bifurcation mechanisms, and the occurrence of chaotic behavior. The thesis is divided into a theoretical and a practical part. In the theoretical section, the author provides an  introduction to the fundamental concepts of nonlinear dynamical systems, including stability, limit cycles, bifurcations, and chaos detection methods. However, it should be noted that not all topics introduced in the theoretical part were later directly applied or fully developed in the practical analysis. The core of the thesis lies in the detailed analytical study of the system’s equilibrium points and their stability, followed by a bifurcation analysis. The author applies the Routh-Hurwitz criterion to determine the stability of the equilibrium points and also investigates the occurrence of Hopf bifurcations. These analyses are mathematically demanding, especially in a system with the level of nonlinearity studied here. Numerical simulations of the solutions and calculations of the Lyapunov exponents were also performed, which confirmed the chaotic behavior of the investigated systems for certain parameter values. One of the strengths of the thesis is its graphical presentation. The bifurcation diagrams, phase portraits, and illustrative numerical simulations are clear, well-organized, and contribute significantly to the reader’s understanding of the system's dynamics. This increases the clarity of the analytical results and effectively demonstrates complex phenomena such as bifurcations and chaotic attractors. Some minor weaknesses should also be mentioned. The thesis contains a number of mathematical inaccuracies and imprecisions. However, their frequency and significance do not exceed the usual level found in works of comparable scope. There is also a lack of a stronger connection between Chapter 2 (theoretical results) and Chapters 3, 4 (numerical simulations). The submitted thesis demonstrates that the author is capable of independently solving mathematical problems, applying both analytical and numerical methods, and presenting the results in a comprehensible form. The main objectives of the diploma thesis were fulfilled. Despite minor shortcomings, I consider the thesis to be of  a good quality, I recommend it for defense, and I grade it B.

eVSKP id 165543