Abstract
A previously derived algorithm for the analysis of the Hopf bifurcation in functional differential equations is extended, allowing the elementary approximation of an existence and stability — determining scalar bifurcation function. With the assistance of the symbolic manipulation program MACSYMA [5], [9] this algorithm is used to implement the algorithm and to investigate the nature of nongeneric Hopf bifurcations in scalar delay — difference equations.
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References
Aboud, N. (1988): Contributions to the Computer-Aided Analysis of Functional Differential Equations. Master's Thesis, University of Minnesota
Aboud, N., Sathaye, A. Stech H. (1988): BIFDE: Software for the Investigation of the Hopf Bifurcation Problem in Functional Differential Equations. Proceedings of the 27th Conference on Decision and Control, IEEE, 821–824
Chow, S.-N, Mallet-Paret, J. (1977): Integral averaging and Hopf bifurcation. J. Differential Equations, 26, 112–159
Claeyssen, J. R. (1980): The integral-averaging bifurcation method and the general one-delay equation. J. Math. Anal. Appl, 78, 429–439
Drinkard, R. D., Sulinski N. (1981): MACSYMA: A Program For Computer Algebraic Manipulation (Demonstrations and Analysis). Naval Underwater Systems Center Technical Document 6401, Reprinted by SYMBOLICS
Franke, J. (1989): Symbolic Hopf Bifurcations for Functional Differential Equations. Master's Thesis, University of Minnesota
Hale, J. K. (1971): Functional Differential Equations. Applied Math. Sci., Vol. 3, Springer-Verlag, New York
Hassard, B., Kazarinoff, N., Wan, Y-H. (1981): Theory and Applications of Hopf Bifurcation. London Math. Soc. Lecture Notes, No. 41, Cambridge University Press, Cambridge
MACSYMA Reference Manual, Version 11 (1986), prepared by the MACSYMA group of SYMBOLICS. Inc. 11 Cambridge Center, Cambridge, MA 02142
Marsden, J. E., McCracken, M. (1976): The Hopf Bifurcation and its Applications. Applied Math. Sciences, Vol. 19, Springer-Verlag, New York
Sathaye, A. (1986): BIFDE: A Numerical Software Package for the Hopf Bifurcation Problem in Functional Differential Equations. Master's Thesis, Virginia Polytechic Institute and State University
Stech, H. W.: Generic Hopf bifurcations for a class of integro-differential equations. Submitted
Stech, H. W. (1985): Hopf bifurcation calculations for functional differential equations. Journal of Math Analysis and Applications, 109, No. 2, 472–491
Stech, H. W. (1985): Nongeneric Hopf Bifurcations in Functional Differential Equations. SIAM J. Appl. Math., 16, No. 6, 1134–1151
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Dedicated to Kenneth Cooke in Honor of his 65th Birthday
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© 1991 Springer-Verlag
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Franke, J.M., Stech, H.W. (1991). Extensions of an algorithm for the analysis of nongeneric Hopf bifurcations, with applications to delay-difference equations. In: Busenberg, S., Martelli, M. (eds) Delay Differential Equations and Dynamical Systems. Lecture Notes in Mathematics, vol 1475. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083488
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DOI: https://doi.org/10.1007/BFb0083488
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