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

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

Received  May 2010 Revised  June 2011 Published  August 2011

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.
Citation: Nikolay Dimitrov. An example of rapid evolution of complex limit cycles. Discrete & Continuous Dynamical Systems - A, 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 (1988).   Google Scholar

[2]

G. Binyamini, D. Novikov and S. Yakovenko, On the number of zeros of Abelian integrals,, Invent. Math., 181 (2010), 227.  doi: 10.1007/s00222-010-0244-0.  Google Scholar

[3]

L. Carleson and T. W. Gamelin, "Complex Dynamics,", Universitext: Tracts in Mathematics, (1993).   Google Scholar

[4]

E. M. Chirka, "Complex Analytic Sets,", Mathematics and its Applications (Soviet Series), 46 (1989).   Google Scholar

[5]

J. Conway, "Functions of One Complex Variable,", 2nd edition, 11 (1978).   Google Scholar

[6]

N. Dimitrov, Rapid evolution of complex limit cycles,, preprint, ().   Google Scholar

[7]

N. Dimitrov, Rapid evolution of complex limit cycles,, Ph.D. thesis, (2009).   Google Scholar

[8]

R. C. Gunning and H. Rossi, "Analytic Functions of Several Complex Variables,", Prentice-Hall, (1965).   Google Scholar

[9]

A. Hatcher, "Algebraic Topology,", Cambridge University Press, (2002).   Google Scholar

[10]

M. W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, (1976).   Google Scholar

[11]

Yu. Ilyashenko, Centennial history of Hilbert's 16th problem,, Bull. Amer. Math. Soc. (New Series), 39 (2002), 301.  doi: 10.1090/S0273-0979-02-00946-1.  Google Scholar

[12]

Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations,", Graduate Studies in Mathematics, 86 (2008).   Google Scholar

[13]

J. Milnor, "Dynamics in One Complex Variable,", Third edition, 160 (2006).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[16]

L. S. Pontryagin, On dynamical systems that are close to integrable,, Zh. Eksp. Teor. Fiz., 4 (1934), 234.   Google Scholar

[17]

W. Thurston, "Three-Dimensional Geometry and Topology,", Vol. I, 35 (1997).   Google Scholar

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 (1988).   Google Scholar

[2]

G. Binyamini, D. Novikov and S. Yakovenko, On the number of zeros of Abelian integrals,, Invent. Math., 181 (2010), 227.  doi: 10.1007/s00222-010-0244-0.  Google Scholar

[3]

L. Carleson and T. W. Gamelin, "Complex Dynamics,", Universitext: Tracts in Mathematics, (1993).   Google Scholar

[4]

E. M. Chirka, "Complex Analytic Sets,", Mathematics and its Applications (Soviet Series), 46 (1989).   Google Scholar

[5]

J. Conway, "Functions of One Complex Variable,", 2nd edition, 11 (1978).   Google Scholar

[6]

N. Dimitrov, Rapid evolution of complex limit cycles,, preprint, ().   Google Scholar

[7]

N. Dimitrov, Rapid evolution of complex limit cycles,, Ph.D. thesis, (2009).   Google Scholar

[8]

R. C. Gunning and H. Rossi, "Analytic Functions of Several Complex Variables,", Prentice-Hall, (1965).   Google Scholar

[9]

A. Hatcher, "Algebraic Topology,", Cambridge University Press, (2002).   Google Scholar

[10]

M. W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, (1976).   Google Scholar

[11]

Yu. Ilyashenko, Centennial history of Hilbert's 16th problem,, Bull. Amer. Math. Soc. (New Series), 39 (2002), 301.  doi: 10.1090/S0273-0979-02-00946-1.  Google Scholar

[12]

Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations,", Graduate Studies in Mathematics, 86 (2008).   Google Scholar

[13]

J. Milnor, "Dynamics in One Complex Variable,", Third edition, 160 (2006).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[16]

L. S. Pontryagin, On dynamical systems that are close to integrable,, Zh. Eksp. Teor. Fiz., 4 (1934), 234.   Google Scholar

[17]

W. Thurston, "Three-Dimensional Geometry and Topology,", Vol. I, 35 (1997).   Google Scholar

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