Identifying, utilizing and improving chaotic numerical instabilities in computational kinematics

Date of Completion

January 1997


Mathematics|Engineering, Mechanical




Very often, when dealing with computational methods in engineering analysis, the final state depends so sensitively on the system's precise initial conditions that the behavior is in effect unpredictable and cannot be distinguished from a random process. This phenomenon in computational kinematics, where the resulting systems of equations are highly non-linear, is a rule rather than exception. In this work, cases of instability are studied and it is shown that the nature of instability is deeply hidden chaos within the iteration schemes. This thesis offers techniques of remedying and improving these numerical instabilities.^ On the other hand, chaos is a deterministic feature which can be utilized for problems of finding global solutions in both nonlinear systems of equations as well as optimization. These two mathematically unresolved problems are the focus of this dissertation. It is shown that these two problems can be solved by using the theory of iteration of complex analytic functions. This novel approach results in locating the "gates" to all the solutions of nonlinear systems. The same approach is used for global optimization problems and leads to development of a new method--Chaotic Descent--based on descending to global minimums via regions that are the source of computational chaos. ^