On a Bifurcation Theorem of Hopf and Friedrichs*
Canadian mathematical bulletin, Tome 20 (1977) no. 1, pp. 95-102

Voir la notice de l'article provenant de la source Cambridge University Press

For the autonomous system x' = F(x,∊), the case where the second Hopf-Friedrichs condition fails is analyzed in that sufficient conditions for bifurcation of non-trivial periodic solutions to occur are given. An application to mathematical ecology is also discussed.
Freedman, H. I. On a Bifurcation Theorem of Hopf and Friedrichs*. Canadian mathematical bulletin, Tome 20 (1977) no. 1, pp. 95-102. doi: 10.4153/CMB-1977-016-x
@article{10_4153_CMB_1977_016_x,
     author = {Freedman, H. I.},
     title = {On a {Bifurcation} {Theorem} of {Hopf} and {Friedrichs*}},
     journal = {Canadian mathematical bulletin},
     pages = {95--102},
     year = {1977},
     volume = {20},
     number = {1},
     doi = {10.4153/CMB-1977-016-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1977-016-x/}
}
TY  - JOUR
AU  - Freedman, H. I.
TI  - On a Bifurcation Theorem of Hopf and Friedrichs*
JO  - Canadian mathematical bulletin
PY  - 1977
SP  - 95
EP  - 102
VL  - 20
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1977-016-x/
DO  - 10.4153/CMB-1977-016-x
ID  - 10_4153_CMB_1977_016_x
ER  - 
%0 Journal Article
%A Freedman, H. I.
%T On a Bifurcation Theorem of Hopf and Friedrichs*
%J Canadian mathematical bulletin
%D 1977
%P 95-102
%V 20
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1977-016-x/
%R 10.4153/CMB-1977-016-x
%F 10_4153_CMB_1977_016_x

[1] 1. Alexander, J. C. and Yorke, J. A., Global bifurcation of periodic orbits, Amer. J. Math, (to appear). Google Scholar

[2] 2. Chaffee, N., The bifurcation of one or more closed orbits from an equilibrium point of an autonomous differential system, J. Differential Equations 4 (1968), 661-679. Google Scholar

[3] 3. Chaffee, N., A bifurcation problem for a functional differential equation of finitely retarded type, J. Math. Anal. Appl. 35 (1971), 312-348. Google Scholar

[4] 4. Freedman, H. I., The implicit function theorem in the scalar case, Canad. Math. Bull. 12 (1969), 721-732. Google Scholar

[5] 5. Freedman, H. I., Graphical stability, enrichment, and pest control by a natural enemy, Math. Biosci, 31, (1976) 207-225. Google Scholar

[6] 6. Freedman, H. I. and Waltman, P., Perturbation of two-dimensional, predator-prey equations, SIAM J. Appl. Math. 28 (1975), 1-10. Google Scholar

[7] 7. Friedrichs, K. O., Advanced Ordinary Differential Equations, Gordon and Breach, 1965. Google Scholar

[8] 8. Hopf, E., Abzweigung einer periodischer Lösung von einer stationären Lösung eines Differentialsystems, Ber. Verh. Sachs. Akad. Wiss. Leipzig. Math-Nat. Kl 95 (1943), 3-22. Google Scholar

[9] 9. Hsu, I. D., A higher order Hopf bifurcation formula and its application to Fitzhugh's nerve conduction equations, J. Math. Anal. Appl. (to appear). Google Scholar

[10] 10. Joseph, D. D. and Sattinger, D. H., Bifurcating time periodic solutions and their stability, Arch. Rational Mech. Anal., 45 (1972), 79-109. Google Scholar

[11] 11. Loud, W. S., Periodic solutions of perturbed second-order autonomous equations, Mem. Amer. Math. Soc. #47, 1964. Google Scholar

[12] 12. Loud, W. S., Behavior of the period of solutions of certain plane autonomous systems near centers, in "Contributions to Differential Equations, Vol III", Wiley-Interscience, New York (1964), 21-36. Google Scholar

[13] 13. Pimbley, G. H. Jr., Periodic solutions of predator-prey equations simulating an immune response I, Math. Biosci. 20 (1974), 27-51. Google Scholar

[14] 14. Poore, A. B., On the theory and application of the Hopf-Friedrichs bifurcation theory, Arch. Rat. Mech. Anal. 60 (1976) 371-393. Google Scholar

[15] 15. Takens, F., Unfolding of certain singularities of vectorfields: generalized Hopf bifurcations, J. Differential Equations 14 (1973), 476-493. Google Scholar

[16] 16. Utz, W. R. and Waltman, P. E., Periodicity and boundedness of solutions of generalized differential equations of growth, Bull. Math. Biophys. 25 (1963), 75-93. Google Scholar

[17] 17. Waltman, P. E., The equations of growth, Bull. Math. Biophys. 26 (1964), 39-43. Google Scholar

Cité par Sources :