×

On infinite period bifurcations with an application to roll waves. (English) Zbl 0588.76024

By considering a model equation we are able to derive conditions under which a limit cycle, created (at small amplitude) by a Hopf bifurcation, can be destroyed (at finite amplitude) by an infinite period bifurcation, this latter appearing out of a homoclinic orbit formed by the separatrices of a saddle point equilibrium state. Further, we are able to extend the methods used for showing the existence of an infinite period bifurcation to calculate the amplitude of the limit cycle over its whole range of existence. These ideas are then applied to an equation arising in the theory of roll waves down an open inclined channel, extending previous work to include the case when the Reynolds number is large with the Froude number close to its critical value for the temporal instability of the uniform flow. Here the governing equation reduced to one similar in form to the model equation.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
34C25 Periodic solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Keener, J. P.: Infinite period bifurcation and global bifurcation branches. SIAM J. Appl. Math.41, 127-144 (1981). · Zbl 0523.34046
[2] Gray, P., Scott, S. K.: Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability. Chemical Engineering Science38, 29-43 (1983).
[3] Gray, P., Scott, S. K.: Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilites in the systemA+2B?3B;B?C. Chemical Engineering Science39, 1087-1097 (1984).
[4] Needham, D. J., Merkin, J. H.: On roll waves down an open inclined channel. Proc. Roy. Soc.A 394, 259-278 (1984). · Zbl 0553.76013
[5] Merkin, J. H., Needham, D. J.: An infinite period bifurcation arising in roll waves down an open inclined channel. Submitted for publication. · Zbl 0593.76024
[6] Jordan, D. W., Smith, P.: Nonlinear ordinary differential equations. Oxford: Clarendon Press 1977. · Zbl 0417.34002
[7] Andronov, A. A., Leontovich, E. A., Gordon, I. I., Maier, A. G.: Theory of bifurcations of dynamic systems on a plane. Jerusalem: Israel Program for Scientific Translations, 1971.
[8] Segel, L. A.: Mathematical models in molecular and cellular biology. Cambridge University Press 1980. · Zbl 0448.92001
[9] Golubitsky, M., Schaeffer, D. G.: Singularities and groups in bifurcation theory, Volume I, Applied Mathematical Sciences, Vol. 51. Springer 1984. · Zbl 0567.58004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.