An example of rapid evolution of complex limit cycles

Previous Article
Homoclinic standing waves in focusing DNLS equations
DCDS Home
This Issue
Next Article
Minimal Følner foliations are amenable

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
Related Papers

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, ().
[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, ().
[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.

[1]	Oliver Butterley, Carlangelo Liverani. Robustly invariant sets in fiber contracting bundle flows. Journal of Modern Dynamics, 2013, 7 (2) : 255-267. doi: 10.3934/jmd.2013.7.255
[2]	M. Jotz. The leaf space of a multiplicative foliation. Journal of Geometric Mechanics, 2012, 4 (3) : 313-332. doi: 10.3934/jgm.2012.4.313
[3]	Marco Abate, Jasmin Raissy. Formal Poincaré-Dulac renormalization for holomorphic germs. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1773-1807. doi: 10.3934/dcds.2013.33.1773
[4]	Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129
[5]	Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439
[6]	Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447
[7]	Jeffrey K. Lawson, Tanya Schmah, Cristina Stoica. Euler-Poincaré reduction for systems with configuration space isotropy. Journal of Geometric Mechanics, 2011, 3 (2) : 261-275. doi: 10.3934/jgm.2011.3.261
[8]	Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure and Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1
[9]	Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123
[10]	Stijn Luca, Freddy Dumortier, Magdalena Caubergh, Robert Roussarie. Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1081-1108. doi: 10.3934/dcds.2009.25.1081
[11]	Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047
[12]	Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure and Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257
[13]	Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893
[14]	Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295
[15]	Jussi Behrndt, A. F. M. ter Elst. The Dirichlet-to-Neumann map for Schrödinger operators with complex potentials. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 661-671. doi: 10.3934/dcdss.2017033
[16]	Jiaxi Huang, Youde Wang, Lifeng Zhao. Equivariant Schrödinger map flow on two dimensional hyperbolic space. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4379-4425. doi: 10.3934/dcds.2020184
[17]	Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann. The Dirichlet to Neumann map - An application to the Stokes problem in half space. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 221-230. doi: 10.3934/dcdss.2010.3.221
[18]	Elvise Berchio, Debdip Ganguly. Improved higher order poincaré inequalities on the hyperbolic space via Hardy-type remainder terms. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1871-1892. doi: 10.3934/cpaa.2016020
[19]	Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803
[20]	Bourama Toni. Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization. Conference Publications, 2005, 2005 (Special) : 846-853. doi: 10.3934/proc.2005.2005.846

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Nikolay Dimitrov
on Google Scholar
Nikolay Dimitrov

[Back to Top]