An example of rapid evolution of complex limit cycles

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September 2011, 31(3): 709-735. doi: 10.3934/dcds.2011.31.709

An example of rapid evolution of complex limit cycles

Nikolay Dimitrov ^1,

1.	Department of Mathematics and Statistics, McGill University, 805 Sherbrooke W. Montreal, QC H3A 2K6, Canada

Received May 2010 Revised June 2011 Published August 2011

Abstract
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In the current article we study complex cycles of higher multiplicity in a specific polynomial family of holomorphic foliations in the complex plane. The family in question is a perturbation of an exact polynomial one-form giving rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a nontrivial element of the fundamental group of a leaf from the foliation. In addition to that, we introduce the notion of a multi-fold cycle and show that in our example there exists a limit cycle of any multiplicity. Furthermore, such a cycle gives rise to a one-parameter family of cycles continuously depending on the perturbation parameter. As the parameter decreases in absolute value, the cycles from the continuous family escape from a very large subdomain of the complex plane.

Keywords: Poincaré map, fiber bundle., covering space, complex limit cycle, Holomorphic foliation.

Mathematics Subject Classification: Primary: 37F75, 37G15; Secondary: 57R22, 57M1.

Citation: Nikolay Dimitrov. An example of rapid evolution of complex limit cycles. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 709-735. doi: 10.3934/dcds.2011.31.709

References:

[1]	V. Arnol'd, S. Guseĭn-Zade 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), 227-289. doi: 10.1007/s00222-010-0244-0.
[3]	L. Carleson and T. W. Gamelin, "Complex Dynamics," Universitext: Tracts in Mathematics, Springer-Verlag, 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, Springer-Verlag, New York-Berlin, 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," Prentice-Hall, 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, Springer-Verlag, New York-Heidelberg, 1976.
[11]	Yu. Ilyashenko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math. Soc. (New Series), 39 (2002), 301-354. doi: 10.1090/S0273-0979-02-00946-1.
[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), 209-250.
[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), 149-168.
[16]	L. S. Pontryagin, On dynamical systems that are close to integrable, Zh. Eksp. Teor. Fiz., 4 (1934), 234-238.
[17]	W. Thurston, "Three-Dimensional 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. Guseĭn-Zade 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), 227-289. doi: 10.1007/s00222-010-0244-0.
[3]	L. Carleson and T. W. Gamelin, "Complex Dynamics," Universitext: Tracts in Mathematics, Springer-Verlag, 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, Springer-Verlag, New York-Berlin, 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," Prentice-Hall, 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, Springer-Verlag, New York-Heidelberg, 1976.
[11]	Yu. Ilyashenko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math. Soc. (New Series), 39 (2002), 301-354. doi: 10.1090/S0273-0979-02-00946-1.
[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), 209-250.
[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), 149-168.
[16]	L. S. Pontryagin, On dynamical systems that are close to integrable, Zh. Eksp. Teor. Fiz., 4 (1934), 234-238.
[17]	W. Thurston, "Three-Dimensional Geometry and Topology," Vol. I, Princeton Mathematical Series, 35, Princeton University Press, Princeton, New Jersey, 1997.

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