Summary
The application of bifurcation theory to nonlinear physical problems is reviewed. It is shown that the topological singularities and bifurcation processes deriving from the concept of structural stability determine the most significant phenomena observed in both structure formation and structure recognition. From this emerges a unifying geometrical framework for the description of nonlinear physical systems which, when passing through instabilities, exhibit analogous critical behavior both at the microscopic and macroscopic levels. After a survey on the basic concepts of singularity and bifurcation theory some new developments are outlined. These include nonlinear conservation laws in various physical fields, the relation between analytical and topological singularities in the inverse scattering problem and in phonon focusing, interacting Hopf and steady-state bifurcations in nonlinear evolution equations and applications to optical bistability and neuronal activity.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
G. Nicolis and I. Prigogine, “Self-organization in nonequilibrium systems,” Wiley (1977).
H. Haken, “Synergetics” (2nd Edition), Springer (1982) and “Advanced Synergetics,” Springer (1983).
R. Thorn, “Structural Stability and Morphogenesis,” Benjamin (1975).
T. Poston and I. Stewart, “Catastrophe Theory and Applications,” Pitman, London (1978). I. Stewart, Physica 2D: 245 (1981).
E. C. Zeeman, “Catastrophe Theory,” Addison-Wesley (1977). R. Gilmore, “Catastrophe Theory,” Wiley (1981).
M. V. Berry, “Les Houches Lectures”, (1980).
W. Güttinger and H. Eikemeier (Eds.), “Structural Stability in Physics,” Springer (1979).
V. I. Arnold, “Mathematical Methods of Classical Mechanics,” Springer (1978).
M. Golubitsky and D. Schaeffer, Comm. Pure Appl. Math. 32:21 (1979). M. Golubitsky and W. F. Langford, J. Diff. Equs. 41:375 (1981). D. G. Schaeffer and M. Golubitsky, Arch. Rat. Mech. Anal. 75:315
M. Golubitsky and D. Schaeffer, Comm. Math. Phys. 67:205 (1979).
D. H. Sattinger, Bull. Am. Math. Soc. 3:779 (1980).
G. Dangelmayr and D. Armbruster, Proc. London Math. Soc. 46 (3):517 (1983).
M. Golubitsky, B. L. Keyfitz and D. Schaeffer, Comm. Pure Appl. Math. 34:433 (1981).
I. S. Labouriau, “Applications of Singularity Theory to Neurobiology,” Univ. of Warwick preprint (1983).
I. S. Labouriau, “Applications of Singularity Theory to Neurobiology,” Univ. of Warwick preprint (1983).
Geiger and W. Giittinger, in preparation (1984).
Geiger and W. Giittinger, in preparation (1984).
Armbruster and W. Güttinger, in preparation (1984).
Dangelmayr and W. Güttinger, Geophys, J. Roy. Astro. Soc. 71:79 (1982)
W. Güttinger and F. J. Wright, Topological Approach to inverse scattering in remote sensing, NATO-ARW proceedings, Reidel (1984).
W. Güttinger and F. J. Wright, Topological Approach to inverse scattering in remote sensing, NATO-ARW proceedings, Reidel (1984).
N. Bleistein and J. Cohen, US Nav. Res. Rep. MS-R-7806 (1978).
J. B. Keller and I. Papadakis, “Wave Propagation and underwater acoustics,” Springer (1977).
J. B. Keller and I. Papadakis, “Wave Propagation and underwater acoustics,” Springer (1977).
Baltes, “Inverse Scattering in Optics,” Springer (1980).
W. M. Boerner in [24].
H. Eisenmenger, K. Lassmann and S. Döttinger (Eds.), “Phonon Scattering in Condensed Matter,” Springer (1984).
F. J. Wright, preprint (1984), see also [20].
F. J. Wright, preprint (1984), see also [20].
M. V. Berry and C. Upstill in: “Progress in Optics,” E. Wolf (Ed.), (1980).
T. Poston and I. Stewart in [4].
F. J. Wright in [28], G. Dangelmayr and W. Giittinger in [19].
J. P. Wolfe, Phys. Today, 33: 44 (1980).
P. Taborek and D. Goodstein, Solid State Comm., 33:1191 (1980). Cf. also Ref. [26].
D. Armbruster and G. Dangelmayr, Z. Phys. B, 52:87 (1983). D. Armbruster, G. Dangelmayr and W. Güttinger in [26]
D. Armbruster, G. Dangelmayr and W. Giittinger, Imperfection Sensitivity of interacting Hopf and Steady-State Bifurcations and their Classification, Physica D (1984).
D. Armbruster in Ref. T[5].
J. Guckenheimer and P. Holmes, “Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,” Springer (1983).
R. Bonifacio (Ed.), “Dissipative Systems in Quantum Optics,” Springer (1982).
See Ref. [15].
F. Takens, Singularities of vector fields, Publications of the IHES, 43: 47 (1974).
J. P. Eckmann, Rev. Mod. Phys., 53:643 (1981). E. C. Zeeman, in: “New Directions in Applied Mathematics,” Case Western Reserve University (1980).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1986 Plenum Press, New York
About this chapter
Cite this chapter
Güttinger, W. (1986). Bifurcation Geometry in Physics. In: Moore, G.T., Scully, M.O. (eds) Frontiers of Nonequilibrium Statistical Physics. NATO ASI Series, vol 135. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-2181-1_4
Download citation
DOI: https://doi.org/10.1007/978-1-4613-2181-1_4
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4612-9284-5
Online ISBN: 978-1-4613-2181-1
eBook Packages: Springer Book Archive