A Dual State Variable Formulation for Ordinary Differential Equations,

Doctoral Dissertation, A. Post, University of Hawaii, UMI, Inc.1996. 

 Abstract 

          This dissertation defines a new state variable formulation for ordinary differential equations. The formulation allows the systematic identification of eigenvalues* for any ordinary differential equation, and leads to parallels with other concepts  from linear algebra as well. Furthermore, the eigenvalues described here are generally defined by ordinary differential equations, and as such,  the proposed state variable formulation can be reapplied to them. This results in the identification of nested, subsidiary eigenvalues.

          As a simple example of its utility, the formulation is applied to the oscillatory motion of the nonlinear pendulum. By modeling the behavior of the eigenvalues for this equation, an approximate solution can be obtained for the period of the pendulum and for its motion. The results are excellent when compared to those of other non-numerical approximation methods.

*Technical Clarification:

            The term eigenvalue is no longer used in connection with this work, because it implies the ability to construct solutions as vectors in a vector space. Functions are found or modeled that are analogous to eigenvalues and have the same physical meaning, but because they apply generally to nonlinear equations, no vector space exists, and they cannot be used to construct additional solutions.