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May  2012, 17(3): 915-932. doi: 10.3934/dcdsb.2012.17.915

On the stability of homoclinic loops with higher dimension

Xingbo Liu 1, and  Deming Zhu 1,

1. 

Department of Mathematics, East China Normal University, Shanghai, 200241

Received  September 2010 Revised  March 2011 Published  January 2012

In this paper the stability of homoclinic loops of saddle equilibrium states in high dimensional systems is analyzed. By constructing local moving frame along the unperturbed homoclinic orbit, the refined Poincaré map is well established, and simple criteria are given for the stability of the saddle homoclinic loop. Some known results are extended.
Keywords: homoclinic orbits, moving frame., Stability.
Mathematics Subject Classification: Primary: 34C23, 34C37; Secondary: 37C2.
Citation: Xingbo Liu, Deming Zhu. On the stability of homoclinic loops with higher dimension. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 915-932. doi: 10.3934/dcdsb.2012.17.915
References:
[1]

D. G.Aronson, S. A. van Gils and M. Krupa, Homoclinic twist bifurcations with $\mathbbZ_2$ symmetry, J. Non. Sci, 4 (1994), 195-219. doi: 10.1007/BF02430632.  Google Scholar

[2]

F. Battelli and M.Fečkan, Subharmonic solutions in singular systems, J. Diff. Eqs, 132 (1996), 21-45.  Google Scholar

[3]

A. R.Champneys, J. Härterich and B. Sandstede, A non-transverse homoclinic orbit to a saddle-node equilibrium, Ergodic Theory and Dynamical System, 16 (1996), 431-450. doi: 10.1017/S0143385700008919.  Google Scholar

[4]

M. Fečkan and J. Gruendler, Homoclinic-Hopf interaction: An autoparametric bifurcation, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 999-1015. doi: 10.1017/S0308210500000548.  Google Scholar

[5]

A. J. Homoburg and J. Knobloch, Multiple homoclinic orbits in conservative and reversible systems, Trans. Amer. Math. Soc, 358 (2006), 1715-1740. doi: 10.1090/S0002-9947-05-03793-1.  Google Scholar

[6]

J. Klaus and J. Knobloch, Bifurcation of homoclinic orbits to a saddle-center in reversible systems, Inter. J. Bifu Chaos Appl. Sci. Engrg., 13 (2003), 2603-2622. doi: 10.1142/S0218127403008119.  Google Scholar

[7]

J. S. W. Lamb, M.-A. Teixeira and K. N. Webster, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in $R^3$, J. Diff. Eqs., 219 (2005), 78-115. doi: 10.1016/j.jde.2005.02.019.  Google Scholar

[8]

F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergodic Theory Dynam. Systems, 7 (1987), 375-413. doi: 10.1017/S0143385700004119.  Google Scholar

[9]

J. D. M. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Diff. Equs., 218 (2005), 390-443. doi: 10.1016/j.jde.2005.03.016.  Google Scholar

[10]

K. Yagasaki, The method of Melnikov for perturbations of multi-degree-of-freedom Hamiltonian systems, Nonlinearity, 12 (1999), 799-822. doi: 10.1088/0951-7715/12/4/304.  Google Scholar

[11]

B. Krauskopf and B. E. Oldeman, Bifurcations of global reinjection orbits near a saddle-node hopf bifurcation, Nonlinearity, 19 (2006), 2149-2167. doi: 10.1088/0951-7715/19/9/010.  Google Scholar

[12]

M. A. Han, D. J. Luo and D. M. Zhu, Uniqueness of limit cycles bifurcating from a singular closed orbit. III, Acta Math. Sinica, 35 (1992), 673-684.  Google Scholar

[13]

M. A. Han, S. C. Hu and X. B. Liu, On the stability of double homoclinic and heteroclinic cycles, Nonlinear Analysis, 53 (2003), 701-713. doi: 10.1016/S0362-546X(02)00301-2.  Google Scholar

[14]

J. B. Li and B. Y. Feng, "Stability, Bifurcation and Chaos," Yun Nan Science Press, Kunming, 1995. Google Scholar

[15]

J. W. Reyn, A stability criterion for separatrix polygons in the phase plane, Nieuw Arch. Wisk. (3), 27 (1979), 238-254.  Google Scholar

[16]

P. Joyal, Generalized Hopf bifurcation and its dual generalized homoclinic bifurcation, SIAM J. Appl. Math., 48 (1988), 481-496. doi: 10.1137/0148027.  Google Scholar

[17]

S. Wiggins, "Global Bifurcation and Chaos. Analytical Methods," Applied Mathematical Sciences, 73, Springer-Verlag, New York, 1988.  Google Scholar

[18]

B. Y. Feng, Stability of homoclinic and heteroclinic cycles in space, Acta Math. Sinica (Chin. Ser.), 39 (1996), 649-658.  Google Scholar

[19]

E. Leontovič, On the generation of limit cycles from separatrices, Doklady Akad. Nauk SSSR (N.S.), 78 (1951), 641-644.  Google Scholar

[20]

L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, "Methods of Qualitative Theory in Nonlinear Dynamics," Part II. World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.  Google Scholar

[21]

M. V. Shashkov and D. V. Turaev, An existence theorem of smooth nonlocal center manifolds for systems close to a system with a homoclinic loop, J. Nonlinear Sci., 9 (1999), 525-573. doi: 10.1007/s003329900078.  Google Scholar

[22]

D. M. Zhu, Problems in homoclinic bifurcations with higher dimensions, Acta Math. Sinica (N.S.), 14 (1998), 341-352.  Google Scholar

[23]

D. M. Zhu and Z. H. Xia, Bifurcations of heteroclinic loops, Sci. China Ser A, 41 (1998), 837-848. doi: 10.1007/BF02871667.  Google Scholar

[24]

D. M. Zhu, Invariants of coordinate transformation,, J. of East China Normal Univ. Nat. Sci. Ed., 1998 (): 19.   Google Scholar

show all references

References:
[1]

D. G.Aronson, S. A. van Gils and M. Krupa, Homoclinic twist bifurcations with $\mathbbZ_2$ symmetry, J. Non. Sci, 4 (1994), 195-219. doi: 10.1007/BF02430632.  Google Scholar

[2]

F. Battelli and M.Fečkan, Subharmonic solutions in singular systems, J. Diff. Eqs, 132 (1996), 21-45.  Google Scholar

[3]

A. R.Champneys, J. Härterich and B. Sandstede, A non-transverse homoclinic orbit to a saddle-node equilibrium, Ergodic Theory and Dynamical System, 16 (1996), 431-450. doi: 10.1017/S0143385700008919.  Google Scholar

[4]

M. Fečkan and J. Gruendler, Homoclinic-Hopf interaction: An autoparametric bifurcation, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 999-1015. doi: 10.1017/S0308210500000548.  Google Scholar

[5]

A. J. Homoburg and J. Knobloch, Multiple homoclinic orbits in conservative and reversible systems, Trans. Amer. Math. Soc, 358 (2006), 1715-1740. doi: 10.1090/S0002-9947-05-03793-1.  Google Scholar

[6]

J. Klaus and J. Knobloch, Bifurcation of homoclinic orbits to a saddle-center in reversible systems, Inter. J. Bifu Chaos Appl. Sci. Engrg., 13 (2003), 2603-2622. doi: 10.1142/S0218127403008119.  Google Scholar

[7]

J. S. W. Lamb, M.-A. Teixeira and K. N. Webster, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in $R^3$, J. Diff. Eqs., 219 (2005), 78-115. doi: 10.1016/j.jde.2005.02.019.  Google Scholar

[8]

F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergodic Theory Dynam. Systems, 7 (1987), 375-413. doi: 10.1017/S0143385700004119.  Google Scholar

[9]

J. D. M. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Diff. Equs., 218 (2005), 390-443. doi: 10.1016/j.jde.2005.03.016.  Google Scholar

[10]

K. Yagasaki, The method of Melnikov for perturbations of multi-degree-of-freedom Hamiltonian systems, Nonlinearity, 12 (1999), 799-822. doi: 10.1088/0951-7715/12/4/304.  Google Scholar

[11]

B. Krauskopf and B. E. Oldeman, Bifurcations of global reinjection orbits near a saddle-node hopf bifurcation, Nonlinearity, 19 (2006), 2149-2167. doi: 10.1088/0951-7715/19/9/010.  Google Scholar

[12]

M. A. Han, D. J. Luo and D. M. Zhu, Uniqueness of limit cycles bifurcating from a singular closed orbit. III, Acta Math. Sinica, 35 (1992), 673-684.  Google Scholar

[13]

M. A. Han, S. C. Hu and X. B. Liu, On the stability of double homoclinic and heteroclinic cycles, Nonlinear Analysis, 53 (2003), 701-713. doi: 10.1016/S0362-546X(02)00301-2.  Google Scholar

[14]

J. B. Li and B. Y. Feng, "Stability, Bifurcation and Chaos," Yun Nan Science Press, Kunming, 1995. Google Scholar

[15]

J. W. Reyn, A stability criterion for separatrix polygons in the phase plane, Nieuw Arch. Wisk. (3), 27 (1979), 238-254.  Google Scholar

[16]

P. Joyal, Generalized Hopf bifurcation and its dual generalized homoclinic bifurcation, SIAM J. Appl. Math., 48 (1988), 481-496. doi: 10.1137/0148027.  Google Scholar

[17]

S. Wiggins, "Global Bifurcation and Chaos. Analytical Methods," Applied Mathematical Sciences, 73, Springer-Verlag, New York, 1988.  Google Scholar

[18]

B. Y. Feng, Stability of homoclinic and heteroclinic cycles in space, Acta Math. Sinica (Chin. Ser.), 39 (1996), 649-658.  Google Scholar

[19]

E. Leontovič, On the generation of limit cycles from separatrices, Doklady Akad. Nauk SSSR (N.S.), 78 (1951), 641-644.  Google Scholar

[20]

L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, "Methods of Qualitative Theory in Nonlinear Dynamics," Part II. World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.  Google Scholar

[21]

M. V. Shashkov and D. V. Turaev, An existence theorem of smooth nonlocal center manifolds for systems close to a system with a homoclinic loop, J. Nonlinear Sci., 9 (1999), 525-573. doi: 10.1007/s003329900078.  Google Scholar

[22]

D. M. Zhu, Problems in homoclinic bifurcations with higher dimensions, Acta Math. Sinica (N.S.), 14 (1998), 341-352.  Google Scholar

[23]

D. M. Zhu and Z. H. Xia, Bifurcations of heteroclinic loops, Sci. China Ser A, 41 (1998), 837-848. doi: 10.1007/BF02871667.  Google Scholar

[24]

D. M. Zhu, Invariants of coordinate transformation,, J. of East China Normal Univ. Nat. Sci. Ed., 1998 (): 19.   Google Scholar

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