Loop Numbers for the Stability of Homoclinic Loops of Planar Vector Fields

This paper is devoted to the study of stability and bifurcations of homoclinic loops for planar vector fields. For a given homoclinic loop, a sequence of loop numbers can be defined such that the s...

[1]  L. Chua,et al.  Methods of Qualitative Theory in Nonlinear Dynamics (Part II) , 2001 .

[2]  Christiane Rousseau,et al.  Saddle quantities and applications , 1989 .

[3]  Maoan Han,et al.  Bifurcation theory for finitely smooth planar autonomous differential systems , 2018 .

[4]  Zhenzhen Wang,et al.  Bifurcation of Rough heteroclinic Loop with Orbit Flips , 2012, Int. J. Bifurc. Chaos.

[5]  Jibin Li,et al.  Hilbert's 16th Problem and bifurcations of Planar Polynomial Vector Fields , 2003, Int. J. Bifurc. Chaos.

[6]  P. Joyal,et al.  Generalized Hopf Bifurcation and Its Dual Generalized Homoclinic Bifurcation , 1988 .

[7]  Robert Roussarie,et al.  On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields , 1986 .

[8]  Luo Hai-ying,et al.  What are the separatrix values named by Leontovich on homoclinic bifurcation , 2005 .

[9]  Huaiping Zhu,et al.  The loop quantities and bifurcations of homoclinic loops , 2007 .

[10]  Shouchuan Hu,et al.  On the stability of double homoclinic and heteroclinic cycles , 2003 .

[11]  李继彬,et al.  WHAT ARE THE SEPARATRIX VALUES NAMED BY LEONTOVICH ON HOMOCLINIC BIFURCATION , 2005 .

[12]  Deming Zhu,et al.  Bifurcations of heteroclinic loops , 1998 .

[13]  Martin Krupa,et al.  Asymptotic stability of heteroclinic cycles in systems with symmetry. II , 2004, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[14]  H. Dulac,et al.  Sur les cycles limites , 1923 .

[15]  Freddy Dumortier,et al.  Perturbations from an Elliptic Hamiltonian of Degree Four: I. Saddle Loop and Two Saddle Cycle☆ , 2001 .

[16]  Shuang Chen,et al.  Stability and Perturbations of Homoclinic Loops in a Class of Piecewise Smooth Systems , 2015, Int. J. Bifurc. Chaos.

[17]  Xingbo Liu,et al.  On the stability of homoclinic loops with higher dimension , 2012 .

[18]  Xingbo Liu,et al.  Degenerate bifurcations of Heterodimensional cycles with Orbit flip , 2013, Int. J. Bifurc. Chaos.

[19]  Maoan Han,et al.  The stability of double homoclinic loops , 2004, Appl. Math. Lett..

[20]  Melbourne,et al.  Asymptotic stability of heteroclinic cycles in systems with symmetry , 1995, Ergodic Theory and Dynamical Systems.

[21]  Liqin Zhao,et al.  The Separatrix Values of a Planar homoclinic Loop , 2009, Int. J. Bifurc. Chaos.