Sensitivity analysis and parametric evaluation of the behavior of solutions in dynamic systems

The behavior of solutions in dynamical systems emerges as an important area of research in a wide range of disciplines, including engineering, biology, economics, and epidemiology. These systems, which describe time-varying states through mathematical models, may exhibit different behaviors depending on external influences, initial conditions, or changes in the system's input elements (input parameters). The long-term behavior of the solutions is directly related to the stability properties of the system. The concepts of the motion, stability properties, and stability quality of solutions of linear dynamical systems are well known. Therefore, there are parameters called stability parameters (Schur stability and Hurwitz stability parameters, 𝜔 and 𝜅, respectively) that provide information about the motion and stability quality of the system without calculating the solutions. Depending on the stability parameters, the stability properties of these systems and their resistance to external influences are determined by sensitivity analysis; for this, theorems generally known as continuity theorems are used in the literature. The investigation of the conditions necessary for a problem to maintain its desired properties is also known as an inverse problem. As an inverse problem, addressing the stability properties of system behavior, determining stability intervals for parametric input elements, and expanding these intervals as much as possible—in other words, establishing stability regions—is an important research problem. In the literature, the behavior of solutions is generally determined by the spectral properties of the matrices associated with the systems. These matrices appear as coefficient matrices if the systems are linear, and as Jacobian matrices of the system at the equilibrium points if the systems are nonlinear. However, it is well known that spectral analysis of non-symmetric matrices is an ill-posed concept. In this thesis, considering the poorly constructed spectral analysis of non-symmetric matrices, stability analyses of some real-world problems are performed using results obtained on the stability parameters 𝜔 and 𝜅. These analyses focus on two important objectives. As an inverse problem, the aim is to find (1) equilibrium points and (2) stability regions of parametric input elements in mathematical models and to expand these regions using appropriate procedures. For these purposes, the stability analyses of a type of two-economy macroeconomic model, a type of Lotka-Volterra model, and an epidemic disease model have been studied in detail. Keywords: Dynamical systems, mathematical models, equilibrium points, stability-asymptotic stability, linearization, Jacobian matrix, stability according to Lyapunov, Schur stability, Hurwitz stability, stability parameters, nonlinear dynamical systems, perturbed systems, sensitivity of stability