Critical Points and Local Behavior

Abstract

As we have seen with the examples in the previous chapter, the criticality of the Jacobian matrix of the system under consideration yields the nonuniqueness of the solutions. The “bifurcation equation” is a standard means to describe the behavior of a system undergoing bifurcation. In particular, in the neighborhood of a simple critical point, a set of equilibrium equations reduces to a single bifurcation equation, which retains important bifurcation behavioral characteristics. By virtue of this reduction, the influence of a number of independent variables is condensed into a single scalar variable and, in turn, can be dealt with in a much simpler manner. Similar reduction can be conducted on a system with a large number of initial imperfection parameters to arrive at the bifurcation equation for an imperfect system. In nonlinear mathematics, the process of deriving this equation is called the “Liapunov—Schmidt reduction” (Sattinger, 1979 [159]; Chow and Hale, 1982 [28]; Golubitsky and Schaeffer, 1985 [57]) or, sometimes, the “Liapunov—Schmidt—Koiter reduction” (e.g., Peek and Kheyrkhahan, 1993 [145]). It is called the “elimination of passive coordinates” in the static perturbation method (Thompson and Hunt, 1973 [174]; Thompson, 1982 [173]; El Naschie, 1990 [50]; Godoy, 2000 [56])