Abstract
Non linear interactions between a Hopf bifurcation and a pitchfork-type stationary bifurcation can produce secondary bifurcations of periodic solutions, and tertiary bifurcations of periodic or aperiodic solutions lying on an invariant torus. A complete classification of the resulting bifurcation diagrams is presented, with emphasis on the cases which exhibit tertiary bifurcation. Calculations involving successive transformations to polar normal forms lead to existence theorems for the secondary and tertiary solutions and asymptotic formulae for the invariant torus.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ashkenazi M. and Othmer H.G. Spacial Patterns in coupled biochemical oscillators. J, Math. Biology 5, 305–350, (1978).
Bauer L., Keller H.B. and Reiss E.L. Multiple eigenvalues lead to secondary bifurcation. SIAM Review 17, 101–122 (1975).
Baxter R., Eiscrike H. and Stokes A. A pictorial study of an invariant torus in phase space of four dimensions. Ordinary Differential Equations, NRL-MRC Conference. Academic Press (1972)
Bouc R., Defilippi M. and Iooss G. On a problem of forced nonlinear oscillations. Nonlinear Analysis 2, 211–224 (1978).
Chow S.N. and Mallet-Paret J. Integral averaging and bifurcation. J. Differential Equations 26, 112–159 (1977).
Cronin J. Bifurcation of periodic solutions. J. Math. Anal. and Appl. 68, 130–151 (1979).
Golubitsky M. and Schaeffer D. Imperfect bifurcation in the presence of symmetry. Comm. Math. Phys. 67, 205–232 (1979).
Guckenheimer J. On a codimension two bifurcation. University of California at Santa (ruz, preprint (1979).
Holmes P. Unfolding a degenerate nonlinear oscillator: a codimension two bifurcation. New York Academy of Sciences, to appear.
Iooss G. Bifurcation of Maps and Applications. North-Holland (1979).
Iooss G. and Joseph D.D. Elementary stability and Bifurcation Theory. To appear.
Iooss G. and Langford W.F. Conjectures on the routes to turbulence via bifurcations. New York Academy of Sciences, to appear.
Iooss G. and Langford W.F. On the interactions of two Hopf bifurcations. In preparation.
Keener J.P. Secondary bifurcation in nonlinear diffusion reaction equations. Studies in Appl. Math. 55, 187–211 (1976).
Keener J.P. Infinite period bifurcation and global bifurcation branches. University of Utah, preprint (1979).
Langford W.F. Periodic and steady-state mode interactions lead to tori. SIAM J. Appl. Math. 37, 22–48 (1979).
Langford W.F., Arneodo A., Coullet P., Tresser C. and Coste J. A mechanism for a soft mode instability. Submitted to Phys. Lett.
Lin J. and Kahn P.B. Qualitative dynamics of three species predator-prey systems. J. Math. Biol. 5, 257–268 (1978).
Marsden J.E. and Mc Cracken M. The Hopf Bifurcation and its Applications. Springer-Verlag, New-York (1976).
Schaeffer D. and Golubitsky M. Boundary conditions and mode jumping in the buckling of a rectangular plate. Comm. Math. Phys. 69, 209–236 (1979).
Shearer M. Coincident bifurcation of equilibrium and periodic solutions of evolution equations. Preprint (1979).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1980 Springer Basel AG
About this chapter
Cite this chapter
Langford, W.F., Iooss, G. (1980). Interactions of Hopf and Pitchfork Bifurcations. In: Mittelmann, H.D., Weber, H. (eds) Bifurcation Problems and their Numerical Solution. ISNM: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 54. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6294-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-0348-6294-3_5
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-7643-1204-6
Online ISBN: 978-3-0348-6294-3
eBook Packages: Springer Book Archive