Quadratic models for generic local parameter bifurcations on the plane
Authors:
Freddy Dumortier and Peter Fiddelaers
Journal:
Trans. Amer. Math. Soc. 326 (1991), 101126
MSC:
Primary 58F14; Secondary 58F36
MathSciNet review:
1049864
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Abstract: The first chapter deals with singularities occurring in quadratic planar vector fields. We make distinction between singularities which as a general system are of finite codimension and singularities which are of infinite codimension in the sense that they are nonisolated, or Hamiltonian, or integrable, or that they have an axis of symmetry after a linear coordinate change or that they can be approximated by centers. In the second chapter we provide quadratic models for all the known versal parameter unfoldings with , except for the nilpotent focus which cannot occur as a quadratic system. We finally show that a certain type of elliptic points of codimension does not have a quadratic versal unfolding.
 [A.R.]
Ralph
Abraham and Joel
Robbin, Transversal mappings and flows, An appendix by Al
Kelley, W. A. Benjamin, Inc., New YorkAmsterdam, 1967. MR 0240836
(39 #2181)
 [B]
N.
N. Bautin, On the number of limit cycles appearing with variation
of the coefficients from an equilibrium state of the type of a focus or a
center, Mat. Sbornik N.S. 30(72) (1952),
181–196 (Russian). MR 0045893
(13,652a)
 [Br]
Henk
Broer, Formal normal form theorems for vector fields and some
consequences for bifurcations in the volume preserving case, Dynamical
systems and turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes
in Math., vol. 898, Springer, Berlin, 1981, pp. 54–74. MR 654883
(83j:58085)
 [C]
W.
A. Coppel, A survey of quadratic systems, J. Differential
Equations 2 (1966), 293–304. MR 0196182
(33 #4374)
 [D1]
Freddy
Dumortier, Singularities of vector fields on the plane, J.
Differential Equations 23 (1977), no. 1,
53–106. MR
0650816 (58 #31276)
 [D2]
, Singularities of vector fields, Monogr. Mat. 32, IMPA, Rio de Janeiro, 1978.
 [DRS1]
F.
Dumortier, R.
Roussarie, and J.
Sotomayor, Generic 3parameter families of vector fields on the
plane, unfolding a singularity with nilpotent linear part. The cusp case of
codimension 3, Ergodic Theory Dynam. Systems 7
(1987), no. 3, 375–413. MR 912375
(89g:58149), http://dx.doi.org/10.1017/S0143385700004119
 [DRS2]
F. Dumortier, R. Roussarie, and J. Sotomayor, Generic parameter families of planar vector fields, unfoldings of saddle, focus and elliptic singularities with nilpotent linear parts, Lecture Notes in Math., SpringerVerlag (to appear).
 [G.H.]
John
Guckenheimer and Philip
Holmes, Nonlinear oscillations, dynamical systems, and bifurcations
of vector fields, Applied Mathematical Sciences, vol. 42,
SpringerVerlag, New York, 1983. MR 709768
(85f:58002)
 [R]
Christiane
Rousseau, Example of a quadratic system with two cycles appearing
in a homoclinic loop bifurcation, J. Differential Equations
66 (1987), no. 1, 140–150. MR 871575
(88b:34041), http://dx.doi.org/10.1016/00220396(87)900441
 [R.A.]
Richard
H. Rand and Dieter
Armbruster, Perturbation methods, bifurcation theory and computer
algebra, Applied Mathematical Sciences, vol. 65, SpringerVerlag,
New York, 1987. MR 911274
(89a:58083)
 [S]
D. Schlomiuk, Personal Communication.
 [T1]
Floris
Takens, Singularities of vector fields, Inst. Hautes
Études Sci. Publ. Math. 43 (1974), 47–100. MR 0339292
(49 #4052)
 [T2]
Floris
Takens, Unfoldings of certain singularities of vectorfields:
generalized Hopf bifurcations, J. Differential Equations
14 (1973), 476–493. MR 0339264
(49 #4024)
 [Y]
Yan
Qian Ye, Sui
Lin Cai, Lan
Sun Chen, Ke
Cheng Huang, Ding
Jun Luo, Zhi
En Ma, Er
Nian Wang, Ming
Shu Wang, and Xin
An Yang, Theory of limit cycles, 2nd ed., Translations of
Mathematical Monographs, vol. 66, American Mathematical Society,
Providence, RI, 1986. Translated from the Chinese by Chi Y. Lo. MR 854278
(88e:58080)
 [A.R.]
 R. Abraham and J. Robbin, Transversal mappings and flows, Benjamin, New York, 1967. MR 0240836 (39:2181)
 [B]
 N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sb. 30(72) (1952), 181196; English transl., Amer. Math. Soc. Transl. No 100 (1954). MR 0045893 (13:652a)
 [Br]
 H. Broer, Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case, Dyn. Syst. and Turbulence, Warwick, 1980, Lecture Notes in Math., vol. 898, SpringerVerlag, 1981, pp. 5474. MR 654883 (83j:58085)
 [C]
 W. A. Coppel, A survey of quadratic systems, J. Differential Equations 2 (1966) 293304. MR 0196182 (33:4374)
 [D1]
 F. Dumortier, Singularities of vector fields on the plane, J. Differential Equations 23 (1977), 53106. MR 0650816 (58:31276)
 [D2]
 , Singularities of vector fields, Monogr. Mat. 32, IMPA, Rio de Janeiro, 1978.
 [DRS1]
 F. Dumortier, R. Roussarie, and J. Sotomayor, Generic parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case, Ergodic Theory Dynamical Systems 7 (1987), 375413. MR 912375 (89g:58149)
 [DRS2]
 F. Dumortier, R. Roussarie, and J. Sotomayor, Generic parameter families of planar vector fields, unfoldings of saddle, focus and elliptic singularities with nilpotent linear parts, Lecture Notes in Math., SpringerVerlag (to appear).
 [G.H.]
 J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Appl. Math. Sci. 42, SpringerVerlag, 1983. MR 709768 (85f:58002)
 [R]
 C. Rousseau, Example of a quadratic system with two cycles appearing in a homoclinic loop bifurcation, J. Differential Equations 66 (1987), 140150. MR 871575 (88b:34041)
 [R.A.]
 R. Rand and D. Armbruster, Perturbation methods, bifurcation theory and computer algebra, Appl. Math. Sci. 65, SpringerVerlag, 1987. MR 911274 (89a:58083)
 [S]
 D. Schlomiuk, Personal Communication.
 [T1]
 F. Takens, Singularities of vector fields, Publ. Math. IHES 43 (1974), 47100. MR 0339292 (49:4052)
 [T2]
 , Unfoldings of certain singularities of vector fields. Generalized Hopf bifurcations, J. Differential Equations 14 (1973), 476493. MR 0339264 (49:4024)
 [Y]
 Ye Yanqian et al., Theory of limit cycles, Transl. Math. Monos., vol. 66, Amer. Math. Soc., Providence, R.I., 1986. MR 854278 (88e:58080)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199110498640
PII:
S 00029947(1991)10498640
Keywords:
Quadratic planar vector fields,
singularities,
codimension,
versal unfoldings,
bifurcations
Article copyright:
© Copyright 1991 American Mathematical Society
