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Homoclinic standing waves in focusing DNLS equations
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 (1988). 
[2] 
G. Binyamini, D. Novikov and S. Yakovenko, On the number of zeros of Abelian integrals,, Invent. Math., 181 (2010), 227. doi: 10.1007/s0022201002440. 
[3] 
L. Carleson and T. W. Gamelin, "Complex Dynamics,", Universitext: Tracts in Mathematics, (1993). 
[4] 
E. M. Chirka, "Complex Analytic Sets,", Mathematics and its Applications (Soviet Series), 46 (1989). 
[5] 
J. Conway, "Functions of One Complex Variable,", 2^{nd} edition, 11 (1978). 
[6] 
N. Dimitrov, Rapid evolution of complex limit cycles,, preprint, (). 
[7] 
N. Dimitrov, Rapid evolution of complex limit cycles,, Ph.D. thesis, (2009). 
[8] 
R. C. Gunning and H. Rossi, "Analytic Functions of Several Complex Variables,", PrenticeHall, (1965). 
[9] 
A. Hatcher, "Algebraic Topology,", Cambridge University Press, (2002). 
[10] 
M. W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, (1976). 
[11] 
Yu. Ilyashenko, Centennial history of Hilbert's 16th problem,, Bull. Amer. Math. Soc. (New Series), 39 (2002), 301. doi: 10.1090/S0273097902009461. 
[12] 
Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations,", Graduate Studies in Mathematics, 86 (2008). 
[13] 
J. Milnor, "Dynamics in One Complex Variable,", Third edition, 160 (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. 
[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. 
[16] 
L. S. Pontryagin, On dynamical systems that are close to integrable,, Zh. Eksp. Teor. Fiz., 4 (1934), 234. 
[17] 
W. Thurston, "ThreeDimensional Geometry and Topology,", Vol. I, 35 (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 (1988). 
[2] 
G. Binyamini, D. Novikov and S. Yakovenko, On the number of zeros of Abelian integrals,, Invent. Math., 181 (2010), 227. doi: 10.1007/s0022201002440. 
[3] 
L. Carleson and T. W. Gamelin, "Complex Dynamics,", Universitext: Tracts in Mathematics, (1993). 
[4] 
E. M. Chirka, "Complex Analytic Sets,", Mathematics and its Applications (Soviet Series), 46 (1989). 
[5] 
J. Conway, "Functions of One Complex Variable,", 2^{nd} edition, 11 (1978). 
[6] 
N. Dimitrov, Rapid evolution of complex limit cycles,, preprint, (). 
[7] 
N. Dimitrov, Rapid evolution of complex limit cycles,, Ph.D. thesis, (2009). 
[8] 
R. C. Gunning and H. Rossi, "Analytic Functions of Several Complex Variables,", PrenticeHall, (1965). 
[9] 
A. Hatcher, "Algebraic Topology,", Cambridge University Press, (2002). 
[10] 
M. W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, (1976). 
[11] 
Yu. Ilyashenko, Centennial history of Hilbert's 16th problem,, Bull. Amer. Math. Soc. (New Series), 39 (2002), 301. doi: 10.1090/S0273097902009461. 
[12] 
Yu. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations,", Graduate Studies in Mathematics, 86 (2008). 
[13] 
J. Milnor, "Dynamics in One Complex Variable,", Third edition, 160 (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. 
[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. 
[16] 
L. S. Pontryagin, On dynamical systems that are close to integrable,, Zh. Eksp. Teor. Fiz., 4 (1934), 234. 
[17] 
W. Thurston, "ThreeDimensional Geometry and Topology,", Vol. I, 35 (1997). 
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