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])
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media New York
About this chapter
Cite this chapter
Ikeda, K., Murota, K. (2002). Critical Points and Local Behavior. In: Imperfect Bifurcation in Structures and Materials. Applied Mathematical Sciences, vol 149. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3697-7_2
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3697-7_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-2989-1
Online ISBN: 978-1-4757-3697-7
eBook Packages: Springer Book Archive