Dynamics near the essential singularity of a class of entire vector fields

Hockett, Kevin; Ramamurti, Sita

doi:10.1090/S0002-9947-1994-1270665-5

Dynamics near the essential singularity of a class of entire vector fields
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by Kevin Hockett and Sita Ramamurti

Trans. Amer. Math. Soc. 345 (1994), 693-703

DOI: https://doi.org/10.1090/S0002-9947-1994-1270665-5

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Abstract:

We investigate the dynamics near the essential singularity at infinity for a class of zero-free entire vector fields of finite order, i.e., those of the form $f(z) = {e^{P(z)}}$ where $P(z) = {z^d}$ or $P(z) = a{z^2} + bz + c$. We show that the flow generated by such a vector field has a "bouquet of flowers" attached to the point at infinity.

References

Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
Harold E. Benzinger, Plane autonomous systems with rational vector fields, Trans. Amer. Math. Soc. 326 (1991), no. 2, 465–483. MR 992604, DOI 10.1090/S0002-9947-1991-0992604-1
Jack Carr, Applications of centre manifold theory, Applied Mathematical Sciences, vol. 35, Springer-Verlag, New York-Berlin, 1981. MR 635782
Robert L. Devaney, Singularities in classical mechanical systems, Ergodic theory and dynamical systems, I (College Park, Md., 1979–80), Progr. Math., vol. 10, Birkhäuser, Boston, Mass., 1981, pp. 211–333. MR 633766
Robert L. Devaney, Blowing up singularities in classical mechanical systems, Amer. Math. Monthly 89 (1982), no. 8, 535–552. MR 674521, DOI 10.2307/2320825
John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
Kevin Hockett, Global dynamical properties of Euler and backward Euler, Ergodic Theory Dynam. Systems 13 (1993), no. 4, 675–696. MR 1257029, DOI 10.1017/S0143385700007616