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An example of rapid evolution of complex limit cycles
1.  Department of Mathematics and Statistics, McGill University, 805 Sherbrooke W. Montreal, QC H3A 2K6, Canada 
References:
[1] 
V. Arnol'd, S. GuseĭnZade and A. Varchenko, "Singularities of Differentiable Maps Vol. II, Monodromy and Asymptotic Integrals," Monographs in Mathematics, 83, Birkhäuser Boston, Inc., Boston, MA, 1988. 
[2] 
G. Binyamini, D. Novikov and S. Yakovenko, On the number of zeros of Abelian integrals, Invent. Math., 181 (2010), 227289. doi: 10.1007/s0022201002440. 
[3] 
L. Carleson and T. W. Gamelin, "Complex Dynamics," Universitext: Tracts in Mathematics, SpringerVerlag, New York, 1993. 
[4] 
E. M. Chirka, "Complex Analytic Sets," Mathematics and its Applications (Soviet Series), 46, Kluwer Academic Publishers Group, Dordrecht, 1989. 
[5] 
J. Conway, "Functions of One Complex Variable," 2^{nd} edition, Graduate Texts in Mathematics, 11, SpringerVerlag, New YorkBerlin, 1978. 
[6] 
N. Dimitrov, Rapid evolution of complex limit cycles, preprint, arXiv:1005.1697. 
[7] 
N. Dimitrov, Rapid evolution of complex limit cycles, Ph.D. thesis, Cornell University, 2009. 
[8] 
R. C. Gunning and H. Rossi, "Analytic Functions of Several Complex Variables," PrenticeHall, Inc., Englewood Cliffs, New Jersey, 1965. 
[9] 
A. Hatcher, "Algebraic Topology," Cambridge University Press, Cambridge, 2002. 
[10] 
M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, SpringerVerlag, New YorkHeidelberg, 1976. 
[11] 
Yu. Ilyashenko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math. Soc. (New Series), 39 (2002), 301354. doi: 10.1090/S0273097902009461. 
[12] 
Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations," Graduate Studies in Mathematics, 86, American Mathematical Society, Providence, RI, 2008. 
[13] 
J. Milnor, "Dynamics in One Complex Variable," Third edition, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, New Jersey, 2006. 
[14] 
I. G. Petrovskiĭ and E. M. Landis, On the number of limit cycles of the equation $dw$/$dz=$ $P(z,w)$/$Q(z,w)$, where $P$ and $Q$ are polynomials of 2nd degree, (in Russian), Matem. Sb. N. S., 37 (1955), 209250. 
[15] 
I. G. Petrovskiĭ and E. M. Landis, On the number of limit cycles of the equation $dw$/$dz=$ $P(z,w)$/$Q(z,w)$, where $P$ and $Q$ are polynomials, (in Russian), Matem. Sb. N. S., 43 (1957), 149168. 
[16] 
L. S. Pontryagin, On dynamical systems that are close to integrable, Zh. Eksp. Teor. Fiz., 4 (1934), 234238. 
[17] 
W. Thurston, "ThreeDimensional Geometry and Topology," Vol. I, Princeton Mathematical Series, 35, Princeton University Press, Princeton, New Jersey, 1997. 
show all references
References:
[1] 
V. Arnol'd, S. GuseĭnZade and A. Varchenko, "Singularities of Differentiable Maps Vol. II, Monodromy and Asymptotic Integrals," Monographs in Mathematics, 83, Birkhäuser Boston, Inc., Boston, MA, 1988. 
[2] 
G. Binyamini, D. Novikov and S. Yakovenko, On the number of zeros of Abelian integrals, Invent. Math., 181 (2010), 227289. doi: 10.1007/s0022201002440. 
[3] 
L. Carleson and T. W. Gamelin, "Complex Dynamics," Universitext: Tracts in Mathematics, SpringerVerlag, New York, 1993. 
[4] 
E. M. Chirka, "Complex Analytic Sets," Mathematics and its Applications (Soviet Series), 46, Kluwer Academic Publishers Group, Dordrecht, 1989. 
[5] 
J. Conway, "Functions of One Complex Variable," 2^{nd} edition, Graduate Texts in Mathematics, 11, SpringerVerlag, New YorkBerlin, 1978. 
[6] 
N. Dimitrov, Rapid evolution of complex limit cycles, preprint, arXiv:1005.1697. 
[7] 
N. Dimitrov, Rapid evolution of complex limit cycles, Ph.D. thesis, Cornell University, 2009. 
[8] 
R. C. Gunning and H. Rossi, "Analytic Functions of Several Complex Variables," PrenticeHall, Inc., Englewood Cliffs, New Jersey, 1965. 
[9] 
A. Hatcher, "Algebraic Topology," Cambridge University Press, Cambridge, 2002. 
[10] 
M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, SpringerVerlag, New YorkHeidelberg, 1976. 
[11] 
Yu. Ilyashenko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math. Soc. (New Series), 39 (2002), 301354. doi: 10.1090/S0273097902009461. 
[12] 
Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations," Graduate Studies in Mathematics, 86, American Mathematical Society, Providence, RI, 2008. 
[13] 
J. Milnor, "Dynamics in One Complex Variable," Third edition, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, New Jersey, 2006. 
[14] 
I. G. Petrovskiĭ and E. M. Landis, On the number of limit cycles of the equation $dw$/$dz=$ $P(z,w)$/$Q(z,w)$, where $P$ and $Q$ are polynomials of 2nd degree, (in Russian), Matem. Sb. N. S., 37 (1955), 209250. 
[15] 
I. G. Petrovskiĭ and E. M. Landis, On the number of limit cycles of the equation $dw$/$dz=$ $P(z,w)$/$Q(z,w)$, where $P$ and $Q$ are polynomials, (in Russian), Matem. Sb. N. S., 43 (1957), 149168. 
[16] 
L. S. Pontryagin, On dynamical systems that are close to integrable, Zh. Eksp. Teor. Fiz., 4 (1934), 234238. 
[17] 
W. Thurston, "ThreeDimensional Geometry and Topology," Vol. I, Princeton Mathematical Series, 35, Princeton University Press, Princeton, New Jersey, 1997. 
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