Abstract
This work is concerned with the rigorous analysis of the effects of small periodic forcing (perturbations) on the dynamical systems which present some interesting phenomena known as delayed bifurcations. We study the dynamical behavior of the system
whereu 0(I) is the solution off(u 0(I), I)=0 andI(t)=I i+εt is a slowly varying parameter that moves past a critical pointI_ of the system so that the linear stability aroundu 0(I) changes from stable to unstable atI_. General results are given with respect to the effects of the perturbation εg(u,I(t),ε, t) to several important types of dynamical systems
which present dynamical patterns that there exist persistent unstable solutions in the dynamical systems (delayed bifurcations) in contrast to bifurcations in the classical sense. It is shown that (1) the delayed bifurcations persist if the frequency ofg(…,…,…,t) ont is a constantΩ which is not a resonant frequency; (2) in case the frequency ofg(…,…,…,t) ont isΩ=Ω(I i+εt) that is slowly varying, the resonance frequencies where the delayed bifurcations might be destructed are shifted downward or upward depending onΩ′(I_)>0 orΩ′(I_)<0; and (3) delayed pitchfork (simple eigenvalue) bifurcations occur in a codimension one parameter family of periodic perturbations. (1) is a rigorous analysis of the results in [3], (2) is a new and interesting phenomenon, and (3) is a generalization of the results of Diener [8] and Schecter [19].
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ahlers, G. M., Cross, C., Hohenberg, P. C., and Safran, S. (1981). The amplitude equation near the convective threshold: application to time-dependent heating experiments.J. Fluid Mech. 110, 297–334.
Arnold, V. I. (1988).XVIIth International Conference in Theoretical and Applied Mechanics.
Baer, S. M., Erneux, T., and Rinzel, J. (1989). The slow passage through a Hopf bifurcation: Delay, memory effects and resonance.SIAM Appl. Math. 49, 55–71.
Candelpergher, B., Diener, F., and Diener, M. (1990).Bifurcations of Planar Vector Fields (edited by J. P. Francoise and R Roussarie), Springer-Verlag.
Chow, S.-N., and Hale, J. (1982).Methods of Bifurcation Theory, Springer-Verlag.
Coddington, E. A., and Levinson, N. (1955).Theory of Ordinary Differential Equations, McGraw-Hill, New York.
Cross, M. C., Hohenberg, P. C., and Lucke, M. (1983). Forcing of convection due to time-dependent heating near threshold.J. Fluid Mech. 136, 269–276.
Diener, F., and Diener, M. (1983). Sept formules relatives aux canards.C.R. Acad. Sci. Paris 297, 577–580.
Erneux, T., and Mandel, D. (1986). Imperfect bifurcation with a slowly-varying control parameter.SIAM J. Appl. Math. 46, 1–16.
Haberman, R. (1979). Slowly-varying jump and transition phenomena associated with algebraic bifurcation problems.SIAM J. Appl. Math. 37, 69–105.
Jakobsson, E., and Guttman, R. (1981).Biophysical Approach to Excitable Systems (eds., W. Adelman and D. Goldman), Plenum.
Kapila, A. K. (1981). Arrhenius system, dynamics of jump due to slow passage through criticality.SIAM J. Appl. Math. 41, 29–42.
Lebovitz, N. R., and Schaar, R. J. (1977). Exchange of stabilities in autonomous systems.Stud. Appl. Math. 54, 1–50.
Neishtadt, A. I. (1985). Asymptotical study of stability loss of equilibrium under slow transition of two eigenvalues through imaginary axes.Uspehi Math Nayk 40(5), 300–301.
Neishtadt, A. I. (1987). On delayed stability loss under dynamical bifurcations I.Diff. Eqs. 23, 1385–1390.
Neishtadt, A. I. (1988). On delayed stability loss under dynamical bifurcations II.Diff. Eqs. 24, 171–176.
Neishtadt, A. I. (1991).Proceedings of the International Congress of Mathematicians, Kyoto, Japan, The Mathematical Society of Japan.
Rinzel, J., and Baer, S. M. (1988). Firing threshold of the Hodgkin-Huxley model for a slow current ramp: A memory effect and its dependence on fluctuations.Biophys. J. 54, 551–555.
Schecter, S. (1985). Persistent unstable equilibria and closed orbits of a singularly perturbed equation.J. Diff. Esq. 60(1), 131–141.
Shishkova, M. A. (1973). Examination of a system of differential equations with a small parameter in the highest derivatives.Soviet Math. Dokl. 14(2), 384–387.
Su, J. (1990). Ph.D. thesis, University of Minnesota.
Su, J. (1993). Delayed oscillation phenomena in the Fitzhugh Nagumo equation.J. Diff. Eqs. 105(1), 180–215.
Su, J. (1994). On delayed oscillations in nonspatially uniform FitzHugh Nagumo equation.J. Diff. Eqs. 110(1), 38–52.
Su, J. (1996). Persistent unstable periodic motions.J. Math. Anal. Appl. 198, 796–825,199, 88–119.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Su, J. Effects of periodic forcing on delayed bifurcations. J Dyn Diff Equat 9, 561–625 (1997). https://doi.org/10.1007/BF02219398
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02219398