A New Look at the Fractional Initial Value Problem: The Aberration Phenomenon

Abstract

A typical phenomenon of the fractional order system is presented to describe the initial value problem from a brand-new perspective in this paper. Several simulation examples are given to introduce the named aberration phenomenon, which reflects the complexity and the importance of the initial value problem. Then, generalizations on the infinite dimensional property and the long memory property are proposed to reveal the nature of the phenomenon. As a result, the relationship between the pseudo state-space model and the infinite dimensional exact state-space model is demonstrated. It shows the inborn defects of the initial values of the fractional order system. Afterward, the pre-initial process and the initialization function are studied. Finally, specific methods to estimate exact state-space models and fit initialization functions are proposed.