Development of stability indices to determine the geometrical features of autonomous dynamical systems
There are plenty of definitions about stability of a dynamic system. Also a nonlinear system response can be classified geometrically as equilibrium point, periodic solution, strange attractors, homoclinic orbits, etc. In this thesis firstly, sector stability has been defined and a theorem is presented to provide the stability conditions. By using sector stability a proper framework for the geometrical representation of equilibrium point in addition to stability analysis and also a classification for the equilibrium point of second order autonomous nonlinear systems is presented. Then, by defining rotational region concept some theorems about existence of periodic orbit and homoclinic orbit has been proved. Also, by using this concept through a theorem, asymptotic stability of the equilibrium point of a second order autonomous system is generalized to global asymptotic stability. Then, a new definition of algorithm in the stability problem is stated. In the sense presented in this thesis, an algorithm in stability problem in not limites just to result in a stable or unstable solution, but it may lead to functions that can be used to analyze dynamic response of the system. In this regard, a basic theorem about the use of such functions is introduced. This theorem is important in two aspects; firstly, in this method there is no need for positive definite functions in stability analysis and this greatly simplifies the operation. Secondly, in addition to stability it can assess instability of the equilibrium point. Furthermore, definition of the eigenfunction and its connection with analysis of dynamic behavior of an autonomous system through invariant sets is introduced. By using these concepts an eigenfunction for a linear system is presented which can be used to analyze stability condition of hyperbolic or non-hyperbolic linear systems. Finally, by using a different definition of eigenvalue problem comparing to other references, the eigenfunctions of a system is calculated. Along the thesis, different case studies and complex dynamic system, have been studied and analyzed by the proposed methods, in order to transfer the required expertise to use this theorem to the reader.
|2013||M.Sc.||Dynamical Systems Analysis and Control|