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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:
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
34C25Periodic solutions of ODE
References:
[1]Keener, J. P.: Infinite period bifurcation and global bifurcation branches. SIAM J. Appl. Math.41, 127-144 (1981). · Zbl 0523.34046 · doi:10.1137/0141010
[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). · doi:10.1016/0009-2509(83)80132-8
[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). · doi:10.1016/0009-2509(84)87017-7
[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 · doi:10.1098/rspa.1984.0079
[5]Merkin, J. H., Needham, D. J.: An infinite period bifurcation arising in roll waves down an open inclined channel. Submitted for publication.
[6]Jordan, D. W., Smith, P.: Nonlinear ordinary differential equations. Oxford: Clarendon Press 1977.
[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.
[9]Golubitsky, M., Schaeffer, D. G.: Singularities and groups in bifurcation theory, Volume I, Applied Mathematical Sciences, Vol. 51. Springer 1984.