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May  2012, 32(5): 1881-1899. doi: 10.3934/dcds.2012.32.1881

Periodic perturbation of quadratic systems with two infinite heteroclinic cycles

Marcelo Messias 1,

1. 

Departamento de Matemática, Estatística e Computação, Faculdade de Ciências e Tecnologia, Univ Estadual Paulista - UNESP, Cx.Postal 266, 19060-900, Presidente Prudente, SP, Brazil

Received  November 2010 Revised  March 2011 Published  January 2012

We study periodic perturbations of planar quadratic vector fields having infinite heteroclinic cycles, consisting of an invariant straight line joining two saddle points at infinity and an arc of orbit also at infinity. The global study concerning the infinity of the perturbed system is performed by means of the Poincaré compactification in polar coordinates, from which we obtain a system defined on a set equivalent to a solid torus in $\mathbb{R}^3$, whose boundary plays the role of the infinity. It is shown that for certain type of periodic perturbation, there exist two differentiable curves in the parameter space for which the perturbed system presents heteroclinic tangencies and transversal intersections between the stable and unstable manifolds of two normally hyperbolic lines of singularities at infinity. The transversality of the manifolds is proved using the Melnikov method and implies, via the Birkhoff-Smale Theorem, in a complex dynamical behavior of the perturbed system solutions in a finite part of the phase space. Numerical simulations are performed for a particular example in order to illustrate this behavior, which could be called "the chaos arising from infinity", because it depends on the global structure of the quadratic system, including the points at infinity.
Keywords: infinite heteroclinic cycle, periodic perturbation, Poincaré compactification, Quadratic system, heteroclinic bifurcation, chaotic dynamics..
Mathematics Subject Classification: Primary: 37C29, 37D45; Secondary: 37B5.
Citation: Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881
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show all references

References:
[1]

J. Differential Equations, 104 (1993), 215-242.  Google Scholar

[2]

Texts in Appl. Math., 34, Springer-Verlag, New York, 1999.  Google Scholar

[3]

J. Differential Equations, 61 (1986), 398-418. doi: 10.1016/0022-0396(86)90113-0.  Google Scholar

[4]

J. Differential Equations, 37 (1980), 351-373. doi: 10.1016/0022-0396(80)90104-7.  Google Scholar

[5]

J. Differential Equations, 2 (1966), 293-304. doi: 10.1016/0022-0396(66)90070-2.  Google Scholar

[6]

J. Differential Equations, 116 (1995), 468-483. doi: 10.1006/jdeq.1995.1044.  Google Scholar

[7]

J. Differential Equations, 110 (1994), 86-133. doi: 10.1006/jdeq.1994.1061.  Google Scholar

[8]

Universitext, Springer-Verlag, Berlin, 2006.  Google Scholar

[9]

J. Math. Anal. Appl., 269 (2002), 332-351. doi: 10.1016/S0022-247X(02)00027-6.  Google Scholar

[10]

Revised and corrected reprint of the 1983 original, Appl. Math. Sci., 42, Springer-Verlag, New York, 1990.  Google Scholar

[11]

Nonlinear Anal., 2 (1978), 77-84. doi: 10.1016/0362-546X(78)90043-3.  Google Scholar

[12]

Nonlinearity, 18 (2005), 305-330. doi: 10.1088/0951-7715/18/1/016.  Google Scholar

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Trudy Moskov. Mat. Obšč., 12 (1963), 3-52.  Google Scholar

[14]

An. Acad. Brasil. Ciênc., 74 (2002), 193-198.  Google Scholar

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Qual. Theory Dyn. Syst., 5 (2004), 301-336. doi: 10.1007/BF02972684.  Google Scholar

[16]

J. Differential Equations, 196 (2004), 169-208.  Google Scholar

[17]

in "Oscillations, Bifurcation and Chaos" (Toronto, Ont., 1986), CMS Conf. Proc., 8, Amer. Math. Soc., Providence, RI, (1987), 665-685.  Google Scholar

[18]

J. Math. Anal. Appl., 124 (1987), 82-97. doi: 10.1016/0022-247X(87)90026-6.  Google Scholar

[19]

Texts in Appl. Math., 2, Springer-Verlag, New York, 1990.  Google Scholar

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