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Instabilities and Self-Organization in MaterialsVolume I: Fundamentals of Nanoscience, Volume II: Applications in Materials Design and Nanotechnology$
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Nasr Ghoniem and Daniel Walgraef

Print publication date: 2008

Print ISBN-13: 9780199298686

Published to Oxford Scholarship Online: May 2008

DOI: 10.1093/acprof:oso/9780199298686.001.0001

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BIFURCATIONS AND INSTABILITIES

BIFURCATIONS AND INSTABILITIES

Chapter:
(p.256) 7 BIFURCATIONS AND INSTABILITIES
Source:
Instabilities and Self-Organization in Materials
Author(s):

Nasr M. Ghoniem

Daniel D. Walgraef

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199298686.003.0007

This chapter focuses on the mathematical structures underlying the notions of stability, bifurcation, and instabilities in complex nonlinear dynamical systems described by sets of ordinary or partial differential equations. It begins by presenting the basic ideas of stability analysis in systems described by ordinary differential equations, introducing Lyapunov functions and their utilization in stability analysis. The stability of systems described by partial differential equations is discussed, emphasizing some of the basic ideas that allow quantitative descriptions of patterns. Specific models illustrating the concepts behind Hopf and Turing instabilities are also discussed.

Keywords:   stability, bifurcation, instability, partial differential equations

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