Plane autonomous systems with rational vector fields

Author:
Harold E. Benzinger

Journal:
Trans. Amer. Math. Soc. **326** (1991), 465-483

MSC:
Primary 58F25; Secondary 34A20, 58F08, 58F21, 65L99

MathSciNet review:
992604

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Abstract | References | Similar Articles | Additional Information

Abstract: The differential equation is studied, where is an arbitrary rational function. It is shown that the Riemann sphere is decomposed into finitely many open sets, on each of which the flow is analytic and, in each time direction, there is common long-term behavior. The boundaries of the open sets consist of those points for which the flow fails to be analytic in at least one time direction. The main idea is to express the differential equation as a continuous Newton method , where is an analytic function which can have branch points and essential singularities. A method is also given for the computer generation of phase plane portraits which shows the correct time parametrization and which is noniterative, thereby avoiding the problems associated with the iteration of rational functions.

**[1]**Harold Benzinger, Scott A. Burns, and Julian I. Palmore,*Chaotic complex dynamics and Newton’s method*, Phys. Lett. A**119**(1987), no. 9, 441–446. MR**879221**, 10.1016/0375-9601(87)90412-9**[2]**Earl A. Coddington and Norman Levinson,*Theory of ordinary differential equations*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR**0069338****[3]**Jean Écalle, Jean Martinet, Robert Moussu, and Jean-Pierre Ramis,*Non-accumulation des cycles-limites. I*, C. R. Acad. Sci. Paris Sér. I Math.**304**(1987), no. 13, 375–377 (French, with English summary). MR**889742****[4]**Jean Écalle, Jean Martinet, Robert Moussu, and Jean-Pierre Ramis,*Non-accumulation des cycles-limites. II*, C. R. Acad. Sci. Paris Sér. I Math.**304**(1987), no. 14, 431–434 (French, with English summary). MR**888240****[5]**Morris W. Hirsch and Stephen Smale,*On algorithms for solving 𝑓(𝑥)=0*, Comm. Pure Appl. Math.**32**(1979), no. 3, 281–313. MR**517937**, 10.1002/cpa.3160320302**[6]**Yu. S. Il'yashenko,*Theorems on the finiteness of limit cycles*, Uspekhi Mat. Nauk**42**(1987), no. 3, 223.**[7]**L. M. Perko,*On the accumulation of limit cycles*, Proc. Amer. Math. Soc.**99**(1987), no. 3, 515–526. MR**875391**, 10.1090/S0002-9939-1987-0875391-1**[8]**H. Poincaré,*Memoire sur les courbes définies par une equation différentielles*, J. Mathematiques**7**(1881), 375-422;*Ouevres*(1880-1890), Gauthier-Villars, Paris, pp. 1-221.**[9]**Song Ling Shi,*A concrete example of the existence of four limit cycles for plane quadratic systems*, Sci. Sinica**23**(1980), no. 2, 153–158. MR**574405****[10]**Michael Shub, David Tischler, and Robert F. Williams,*The Newtonian graph of a complex polynomial*, SIAM J. Math. Anal.**19**(1988), no. 1, 246–256. MR**924558**, 10.1137/0519018**[11]**Steve Smale,*A convergent process of price adjustment and global Newton methods*, J. Math. Econom.**3**(1976), no. 2, 107–120. MR**0411577****[12]**Steve Smale,*The fundamental theorem of algebra and complexity theory*, Bull. Amer. Math. Soc. (N.S.)**4**(1981), no. 1, 1–36. MR**590817**, 10.1090/S0273-0979-1981-14858-8**[13]**Steve Smale,*On the efficiency of algorithms of analysis*, Bull. Amer. Math. Soc. (N.S.)**13**(1985), no. 2, 87–121. MR**799791**, 10.1090/S0273-0979-1985-15391-1

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1991-0992604-1

Keywords:
Phase plane portraits,
analytic flows,
continuous Newton method,
computer graphics

Article copyright:
© Copyright 1991
American Mathematical Society