Notes on Normal Form
نویسنده
چکیده
A favorite parlor game of us engineers, scientists and mathematicians is to find the exact solution for a given nonlinear problem. It is generally difficult to come up with a technique to solve a given nonlinear problem exactly. A traditional analysis approach is to approximate the given nonlinear system by a first order linear system about its nominal point and hope that solution of approximate system will not be very different from the exact solution of actual nonlinear system. But validity of first order linear approximation depend upon the relative size of higher order term or in other words “How nonlinear” is our actual system. In many cases, such qualitative discussions fail to focus on an important underlying truth: Nonlinearity is not an inherent attribute of a physical system, but rather is heavily dependent upon our mathematical description of the system’s geometry, kinematics, and evolution dynamics. With fixed physical model assumptions, an infinity of coordinate choices is typically possible, and it is apparent that the issue of analyst decisions (which are frequently made subjectively!) cannot be dismissed as immaterial to the system nonlinearity. For many cases, we can simplify our actual nonlinear system by change of variables. In late 18th century Poincare introduced the theory of Normal form for change of variables to consider the higher order terms in the expansion of the nonlinear system about its nominal point. In this report we will discuss that how to put equations in their normal form so that linear approximation is more valid.
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تاریخ انتشار 2003