On System of Ordinary Differential Equations and the Trace-Determinant Plane
| dc.contributor.advisor | Taddese Abdi | |
| dc.contributor.author | Temesgen Bekele | |
| dc.date.accessioned | 2025-09-06T00:18:23Z | |
| dc.date.available | 2025-09-06T00:18:23Z | |
| dc.date.issued | 2016-08 | |
| dc.description.abstract | This thesis provides an advanced analysis of systems of ordinary differential equations (ODEs) using the trace-determinant plane as a central framework. The trace-determinant plane, defined by the trace and determinant of the Jacobean matrix of a linear system, serves as a powerful geometric tool for understanding and classifying the qualitative behavior of dynamical systems. The research begins with a detailed examination of the trace-determinant plane in the context of linear systems. By exploring the relationships between the trace (sum of Eigen-values) and determinant (product of Eigen-values) of the Jacobean matrix, the thesis elucidates the various types of equilibrium points and their stability characteristics. The trace-determinant plane is shown to map regions of distinct dynamic behaviors, including nodes, saddles, spirals, and centers, offering a comprehensive classification framework. Building on this foundation, the thesis extends the analysis to nonlinear systems by employing linearization techniques. It demonstrates how the trace-determinant plane can be utilized to approximate the local dynamics around equilibrium points and provides insights into the global behavior of nonlinear systems. The impact of parameter variations on the trace-determinant plane is also investigated, revealing how bifurcations and stability changes can be visualized and interpreted. A significant contribution of this thesis is the integration of computational methods with theoretical analysis. Algorithms for plotting the trace-determinant plane and analyzing system stability are developed and applied to a range of practical examples, including control systems, ecological models, and mechanical systems. These case studies illustrate the practical utility of the trace-determinant plane in both theoretical and applied contexts. | |
| dc.identifier.uri | https://etd.aau.edu.et/handle/123456789/7419 | |
| dc.language.iso | en_US | |
| dc.publisher | Addis Ababa University | |
| dc.subject | System of Ordinary Differential Equations | |
| dc.subject | Trace-Determinant Plane | |
| dc.title | On System of Ordinary Differential Equations and the Trace-Determinant Plane | |
| dc.type | Thesis |