Bifurcations associated with a three-fold zero eigenvalue
Authors:
Pei Yu and K. Huseyin
Journal:
Quart. Appl. Math. 46 (1988), 193-216
MSC:
Primary 58F14; Secondary 34C15, 94C05
DOI:
https://doi.org/10.1090/qam/950597
MathSciNet review:
950597
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The bifurcation and instability behavior of a nonlinear autonomous system in the vicinity of a compound critical point is studied in detail. The critical point is characterized by a triple zero of index one eigenvalue, and the system is described by three independent parameters. The analysis is carried out via a unification technique, leading to a simple set of differential equations for the analysis of local behavior. Incipient and secondary bifurcations as well as bifurcations into invariant tori are discussed, and the explicit asymptotic results concerning periodic solutions are presented. Moreover, the criteria leading to a sequence of bifurcations into a family of two-dimensional tori are established. An electrical network is analyzed to illustrate the analytical results.
- Floris Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 47–100. MR 339292
- V. Mandadi and K. Huseyin, Nonlinear bifurcation analysis of nongradient system, Internat. J. Non-Linear Mech. 15 (1980), no. 3, 159–172 (English, with French and German summaries). MR 585210, DOI https://doi.org/10.1016/0020-7462%2880%2990038-4
- Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 251, Springer-Verlag, New York-Berlin, 1982. MR 660633
- John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768
- Koncay Huseyin, Multiple parameter stability theory and its applications, Oxford Engineering Science Series, vol. 18, The Clarendon Press, Oxford University Press, New York, 1986. Bifurcations, catastrophes, instabilities; Oxford Science Publications. MR 870153
- Pei Yu and K. Huseyin, Static and dynamic bifurcations associated with a double-zero eigenvalue, Dynam. Stability Systems 1 (1986), no. 1, 73–86. MR 908374, DOI https://doi.org/10.1080/02681118608806005
K. Huseyin, The multiple-parameter perturbation technique for the analysis of non-linear systems, Internat. J. Nonlinear Mech. 8, 431–433 (1973)
- K. Huseyin and A. S. Atadan, On the analysis of Hopf bifurcations, Internat. J. Engrg. Sci. 21 (1983), no. 3, 247–262. MR 691437, DOI https://doi.org/10.1016/0020-7225%2883%2990026-5
K. Huseyin and V. Mandadi, On the instability of multiple-parameter systems, Sectional Lecture, Proceedings of the Fifteenth International Congress of Theoretical and Applied Mechanics, Toronto (eds. F. P. J. Rimort and B. Tabarrok; North-Holland, Amsterdam), pp. 281–294, 1980
- Milan Medveď, On a codimension three bifurcation, Časopis Pěst. Mat. 109 (1984), no. 1, 3–26 (English, with Russian summary). MR 741206
- Paul Glendinning and Colin Sparrow, Local and global behavior near homoclinic orbits, J. Statist. Phys. 35 (1984), no. 5-6, 645–696. MR 749842, DOI https://doi.org/10.1007/BF01010828
- A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, Asymptotic chaos, Phys. D 14 (1985), no. 3, 327–347. MR 793709, DOI https://doi.org/10.1016/0167-2789%2885%2990093-4
F. Takens, Singularities of vector fields, Publications Mathématiques 43 (Institut des Hautes Etudes Scientifiques, Paris), 47–100 (1974)
V. Mandadi and K. Huseyin, Nonlinear bifurcation analysis of non-gradient systems, Internat. Journal of Nonlinear Mech. 15, 159–172 (1980)
S. N. Chow and J. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York (1983)
K. Huseyin, Multiple-parameter Stability Theory and Its Applications, Oxford University Press, 1986
P. Yu and K. Huseyin, Static and dynamic bifurcations associated with a double zero eigenvalue, Dynamics and Stability of Systems 1, 73–86 (1986)
K. Huseyin, The multiple-parameter perturbation technique for the analysis of non-linear systems, Internat. J. Nonlinear Mech. 8, 431–433 (1973)
K. Huseyin and A. S. Atadan, On the analysis of Hopf bifurcations, Internat. Journal of Engineering Science 21, 247–262 (1983)
K. Huseyin and V. Mandadi, On the instability of multiple-parameter systems, Sectional Lecture, Proceedings of the Fifteenth International Congress of Theoretical and Applied Mechanics, Toronto (eds. F. P. J. Rimort and B. Tabarrok; North-Holland, Amsterdam), pp. 281–294, 1980
M. Medved, On a codimension three bifurcation, Casopis propestovani matematiky, roc. 109, 3–26 (1984)
P. Glendinning and C. Sparrow, Local and global behaviour near homoclinic orbits, J. Stat. Phys. 35, 645–695 (1984)
A. Arneodo, P. Coullet, E. Spiegel and C. Tresser, Asymptotic chaos, Physica 14D, 327–347 (1985)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
58F14,
34C15,
94C05
Retrieve articles in all journals
with MSC:
58F14,
34C15,
94C05
Additional Information
Article copyright:
© Copyright 1988
American Mathematical Society