# American Institute of Mathematical Sciences

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

## Periodic perturbation of quadratic systems with two infinite heteroclinic cycles

 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.
Citation: Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881
##### References:
 [1] T. R. Blows and C. Rousseau, Bifurcation at infinity in polynomial vector fields, J. Differential Equations, 104 (1993), 215-242. [2] C. Chicone, "Ordinary Differential Equations with Applications," Texts in Appl. Math., 34, Springer-Verlag, New York, 1999. [3] C. Chicone and J. Sotomayor, On a class of complete polynomial vector fields in the plane, J. Differential Equations, 61 (1986), 398-418. doi: 10.1016/0022-0396(86)90113-0. [4] S. N. Chow, J. K. Hale and J. Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Differential Equations, 37 (1980), 351-373. doi: 10.1016/0022-0396(80)90104-7. [5] W. A. Coppel, A survey of quadratic systems, J. Differential Equations, 2 (1966), 293-304. doi: 10.1016/0022-0396(66)90070-2. [6] H. Dankowicz and P. Holmes, The existence of transverse homoclinic points in the Sitnikov problem, J. Differential Equations, 116 (1995), 468-483. doi: 10.1006/jdeq.1995.1044. [7] F. Dumortier, R. Roussarie and C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations, 110 (1994), 86-133. doi: 10.1006/jdeq.1994.1061. [8] F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Planar Differential Systems," Universitext, Springer-Verlag, Berlin, 2006. [9] A. Gasull, V. Mañosa and F. Mañosas, Stability of certain planar unbounded polycycles, J. Math. Anal. Appl., 269 (2002), 332-351. doi: 10.1016/S0022-247X(02)00027-6. [10] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Revised and corrected reprint of the 1983 original, Appl. Math. Sci., 42, Springer-Verlag, New York, 1990. [11] J. Hale and P. Táboas, Interaction of damping and forcing in a second order equation, Nonlinear Anal., 2 (1978), 77-84. doi: 10.1016/0362-546X(78)90043-3. [12] I. D. Iliev, Chengzhi Li and Jiang Yu, Bifurcations of limit cycles from quadratic non-Hamiltonian systems with two centres and two unbounded heteroclinic loops, Nonlinearity, 18 (2005), 305-330. doi: 10.1088/0951-7715/18/1/016. [13] V. K. Mel'nikov, On the stability of the center for time periodic perturbations, Trudy Moskov. Mat. Obšč., 12 (1963), 3-52. [14] M. Messias, Periodic perturbations of quadratic planar polynomial vector fields, An. Acad. Brasil. Ciênc., 74 (2002), 193-198. [15] M. Messias, Subharmonic bifurcations near infinity, Qual. Theory Dyn. Syst., 5 (2004), 301-336. doi: 10.1007/BF02972684. [16] C. Rousseau and H. Zhu, PP-graphics with a nilpotent elliptic singularity in quadratic systems and Hilbert's 16th problem, J. Differential Equations, 196 (2004), 169-208. [17] J. Sotomayor and R. Paterlini, Bifurcation of polynomial vector fields in the plane, in "Oscillations, Bifurcation and Chaos" (Toronto, Ont., 1986), CMS Conf. Proc., 8, Amer. Math. Soc., Providence, RI, (1987), 665-685. [18] P. Táboas, Periodic solutions of a forced Lotka-Volterra equation, J. Math. Anal. Appl., 124 (1987), 82-97. doi: 10.1016/0022-247X(87)90026-6. [19] S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Texts in Appl. Math., 2, Springer-Verlag, New York, 1990.

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##### References:
 [1] T. R. Blows and C. Rousseau, Bifurcation at infinity in polynomial vector fields, J. Differential Equations, 104 (1993), 215-242. [2] C. Chicone, "Ordinary Differential Equations with Applications," Texts in Appl. Math., 34, Springer-Verlag, New York, 1999. [3] C. Chicone and J. Sotomayor, On a class of complete polynomial vector fields in the plane, J. Differential Equations, 61 (1986), 398-418. doi: 10.1016/0022-0396(86)90113-0. [4] S. N. Chow, J. K. Hale and J. Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Differential Equations, 37 (1980), 351-373. doi: 10.1016/0022-0396(80)90104-7. [5] W. A. Coppel, A survey of quadratic systems, J. Differential Equations, 2 (1966), 293-304. doi: 10.1016/0022-0396(66)90070-2. [6] H. Dankowicz and P. Holmes, The existence of transverse homoclinic points in the Sitnikov problem, J. Differential Equations, 116 (1995), 468-483. doi: 10.1006/jdeq.1995.1044. [7] F. Dumortier, R. Roussarie and C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations, 110 (1994), 86-133. doi: 10.1006/jdeq.1994.1061. [8] F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Planar Differential Systems," Universitext, Springer-Verlag, Berlin, 2006. [9] A. Gasull, V. Mañosa and F. Mañosas, Stability of certain planar unbounded polycycles, J. Math. Anal. Appl., 269 (2002), 332-351. doi: 10.1016/S0022-247X(02)00027-6. [10] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Revised and corrected reprint of the 1983 original, Appl. Math. Sci., 42, Springer-Verlag, New York, 1990. [11] J. Hale and P. Táboas, Interaction of damping and forcing in a second order equation, Nonlinear Anal., 2 (1978), 77-84. doi: 10.1016/0362-546X(78)90043-3. [12] I. D. Iliev, Chengzhi Li and Jiang Yu, Bifurcations of limit cycles from quadratic non-Hamiltonian systems with two centres and two unbounded heteroclinic loops, Nonlinearity, 18 (2005), 305-330. doi: 10.1088/0951-7715/18/1/016. [13] V. K. Mel'nikov, On the stability of the center for time periodic perturbations, Trudy Moskov. Mat. Obšč., 12 (1963), 3-52. [14] M. Messias, Periodic perturbations of quadratic planar polynomial vector fields, An. Acad. Brasil. Ciênc., 74 (2002), 193-198. [15] M. Messias, Subharmonic bifurcations near infinity, Qual. Theory Dyn. Syst., 5 (2004), 301-336. doi: 10.1007/BF02972684. [16] C. Rousseau and H. Zhu, PP-graphics with a nilpotent elliptic singularity in quadratic systems and Hilbert's 16th problem, J. Differential Equations, 196 (2004), 169-208. [17] J. Sotomayor and R. Paterlini, Bifurcation of polynomial vector fields in the plane, in "Oscillations, Bifurcation and Chaos" (Toronto, Ont., 1986), CMS Conf. Proc., 8, Amer. Math. Soc., Providence, RI, (1987), 665-685. [18] P. Táboas, Periodic solutions of a forced Lotka-Volterra equation, J. Math. Anal. Appl., 124 (1987), 82-97. doi: 10.1016/0022-247X(87)90026-6. [19] S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Texts in Appl. Math., 2, Springer-Verlag, New York, 1990.
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