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 & Continuous Dynamical Systems - A, 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.

[2]

C. Chicone, "Ordinary Differential Equations with Applications,", Texts in Appl. Math., 34 (1999).

[3]

C. Chicone and J. Sotomayor, On a class of complete polynomial vector fields in the plane,, J. Differential Equations, 61 (1986), 398. 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. doi: 10.1016/0022-0396(80)90104-7.

[5]

W. A. Coppel, A survey of quadratic systems,, J. Differential Equations, 2 (1966), 293. 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. 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. doi: 10.1006/jdeq.1994.1061.

[8]

F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Planar Differential Systems,", Universitext, (2006).

[9]

A. Gasull, V. Mañosa and F. Mañosas, Stability of certain planar unbounded polycycles,, J. Math. Anal. Appl., 269 (2002), 332. 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, 42 (1983).

[11]

J. Hale and P. Táboas, Interaction of damping and forcing in a second order equation,, Nonlinear Anal., 2 (1978), 77. 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. 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.

[14]

M. Messias, Periodic perturbations of quadratic planar polynomial vector fields,, An. Acad. Brasil. Ciênc., 74 (2002), 193.

[15]

M. Messias, Subharmonic bifurcations near infinity,, Qual. Theory Dyn. Syst., 5 (2004), 301. 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.

[17]

J. Sotomayor and R. Paterlini, Bifurcation of polynomial vector fields in the plane,, in, 8 (1987), 665.

[18]

P. Táboas, Periodic solutions of a forced Lotka-Volterra equation,, J. Math. Anal. Appl., 124 (1987), 82. doi: 10.1016/0022-247X(87)90026-6.

[19]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos,", Texts in Appl. Math., 2 (1990).

show all references

References:
[1]

T. R. Blows and C. Rousseau, Bifurcation at infinity in polynomial vector fields,, J. Differential Equations, 104 (1993), 215.

[2]

C. Chicone, "Ordinary Differential Equations with Applications,", Texts in Appl. Math., 34 (1999).

[3]

C. Chicone and J. Sotomayor, On a class of complete polynomial vector fields in the plane,, J. Differential Equations, 61 (1986), 398. 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. doi: 10.1016/0022-0396(80)90104-7.

[5]

W. A. Coppel, A survey of quadratic systems,, J. Differential Equations, 2 (1966), 293. 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. 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. doi: 10.1006/jdeq.1994.1061.

[8]

F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Planar Differential Systems,", Universitext, (2006).

[9]

A. Gasull, V. Mañosa and F. Mañosas, Stability of certain planar unbounded polycycles,, J. Math. Anal. Appl., 269 (2002), 332. 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, 42 (1983).

[11]

J. Hale and P. Táboas, Interaction of damping and forcing in a second order equation,, Nonlinear Anal., 2 (1978), 77. 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. 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.

[14]

M. Messias, Periodic perturbations of quadratic planar polynomial vector fields,, An. Acad. Brasil. Ciênc., 74 (2002), 193.

[15]

M. Messias, Subharmonic bifurcations near infinity,, Qual. Theory Dyn. Syst., 5 (2004), 301. 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.

[17]

J. Sotomayor and R. Paterlini, Bifurcation of polynomial vector fields in the plane,, in, 8 (1987), 665.

[18]

P. Táboas, Periodic solutions of a forced Lotka-Volterra equation,, J. Math. Anal. Appl., 124 (1987), 82. doi: 10.1016/0022-247X(87)90026-6.

[19]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos,", Texts in Appl. Math., 2 (1990).

[1]

Ling-Hao Zhang, Wei Wang. Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 591-604. doi: 10.3934/dcds.2017024

[2]

Wenjun Zhang, Bernd Krauskopf, Vivien Kirk. How to find a codimension-one heteroclinic cycle between two periodic orbits. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2825-2851. doi: 10.3934/dcds.2012.32.2825

[3]

Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114

[4]

Tifei Qian, Zhihong Xia. Heteroclinic orbits and chaotic invariant sets for monotone twist maps. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 69-95. doi: 10.3934/dcds.2003.9.69

[5]

Alexandre A. P. Rodrigues. Moduli for heteroclinic connections involving saddle-foci and periodic solutions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3155-3182. doi: 10.3934/dcds.2015.35.3155

[6]

Francesca Alessio, Carlo Carminati, Piero Montecchiari. Heteroclinic motions joining almost periodic solutions for a class of Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 569-584. doi: 10.3934/dcds.1999.5.569

[7]

Daniel Wilczak. Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1039-1055. doi: 10.3934/dcdsb.2009.11.1039

[8]

Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065

[9]

Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431

[10]

Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106

[11]

Zhanyuan Hou, Stephen Baigent. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4071-4093. doi: 10.3934/dcds.2013.33.4071

[12]

Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343

[13]

Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967

[14]

Fengjie Geng, Junfang Zhao, Deming Zhu, Weipeng Zhang. Bifurcations of a nongeneric heteroclinic loop with nonhyperbolic equilibria. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 133-145. doi: 10.3934/dcdsb.2013.18.133

[15]

Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097

[16]

Thorsten Hüls, Yongkui Zou. On computing heteroclinic trajectories of non-autonomous maps. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 79-99. doi: 10.3934/dcdsb.2012.17.79

[17]

Chao Wang, Dingbian Qian, Qihuai Liu. Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2305-2328. doi: 10.3934/dcds.2016.36.2305

[18]

Paolo Perfetti. An infinite-dimensional extension of a Poincaré's result concerning the continuation of periodic orbits. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 401-418. doi: 10.3934/dcds.1997.3.401

[19]

Mikhail Kamenskii, Boris Mikhaylenko. Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 437-452. doi: 10.3934/dcdsb.2013.18.437

[20]

Tiago de Carvalho, Rodrigo Donizete Euzébio, Jaume Llibre, Durval José Tonon. Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 1-11. doi: 10.3934/dcdsb.2016.21.1

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]