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Topological Hopf bifurcation in the plane

Published online by Cambridge University Press:  24 October 2008

Dieter Erle
Dieter Erle
Affiliation:
Universitüt Dortmund, Fed. Rep., Germany
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Extract

Classical bifurcation theorems for a 1 -parameter family of plane dynamical systems

assert the presence of closed orbits clustering at some distinguished parameter value (∈ = 0, say). Here, for any ∈, the origin is the only stationary point. The topological content of the mostly analytic hypotheses imposed is some change in the stability behaviour of the origin at ∈ = 0, roughly the passing of a kind of stability to a kind of instability. Topologically speaking, e.g. some of the conditions demanded are asymptotic stability of the origin for the negative system at ∈ > 0 and asymptotic stability of the origin for at ∈ < 0 (Hopf (8), Ruelle and Takens(11)) or ∈ = 0 (Chafee(2)).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 1 , January 1983 , pp. 113 - 119
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

REFERENCES

(1)Alexander, J. C. A primer on connectivity. In Fixed point theory. Lect. Notes Math. 886 (Springer, Berlin, 1981), pp. 455–483.CrossRefGoogle Scholar
(2)Chafee, N.The bifurcation of one or more closed orbits from an equilibrium point of an autonomous differential system. J. Diff. Eq. 4 (1968), 661–679.CrossRefGoogle Scholar
(3)Coddington, E. A. and Levinson, N.Theory of ordinary differential equations (McGraw-Hill, New York, 1955).Google Scholar
(4)Dold, A.Lectures on algebraic topology (Springer, Berlin, 1972).CrossRefGoogle Scholar
(5)Erle, D.Stable closed orbits in plane autonomous dynamical systems. J. reine angew. Math. 305 (1979), 136–139.Google Scholar
(6)Giesecke, B.Simpliziale Zerlegung abzählbarer analytischer Räume. Math. Z. 83 (1964), 177–213.CrossRefGoogle Scholar
(7)Hirsch, M. and Smale, S.Differential equations, dynamical systems, and linear algebra (Acad. Press, New York, 1974).Google Scholar
(8)Hopf, E.Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber. Verh. Sächs. Akad. Wise. Leipzig. Math. Nat. Kl. 95 (1942), 3–22.Google Scholar
(9)Massera, J.Contributions to stability theory. Annals of Math. 64 (1956), 182–206.CrossRefGoogle Scholar
(10)Renz, P.Equivalent flows on smooth Banach manifolds. Indiana Univ. Math. J. 20 (1971), 695–698.CrossRefGoogle Scholar
(11)Ruelle, D. and Takens, F.On the nature of turbulence. Commun. Math.Phys. 20 (1971), 167–192; 23 (1971), 343–344.CrossRefGoogle Scholar
(12)Zubov, V. I.Methods of A.M. Lyapunov and their application (Noordhoff, Groningen, 1964).Google Scholar