On the stability of homoclinic loops with higher dimension

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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

Abstract
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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 and 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.
[2]	F. Battelli and M.Fečkan, Subharmonic solutions in singular systems, J. Diff. Eqs, 132 (1996), 21-45.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	J. B. Li and B. Y. Feng, "Stability, Bifurcation and Chaos," Yun Nan Science Press, Kunming, 1995.
[15]	J. W. Reyn, A stability criterion for separatrix polygons in the phase plane, Nieuw Arch. Wisk. (3), 27 (1979), 238-254.
[16]	P. Joyal, Generalized Hopf bifurcation and its dual generalized homoclinic bifurcation, SIAM J. Appl. Math., 48 (1988), 481-496. doi: 10.1137/0148027.
[17]	S. Wiggins, "Global Bifurcation and Chaos. Analytical Methods," Applied Mathematical Sciences, 73, Springer-Verlag, New York, 1988.
[18]	B. Y. Feng, Stability of homoclinic and heteroclinic cycles in space, Acta Math. Sinica (Chin. Ser.), 39 (1996), 649-658.
[19]	E. Leontovič, On the generation of limit cycles from separatrices, Doklady Akad. Nauk SSSR (N.S.), 78 (1951), 641-644.
[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.
[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.
[22]	D. M. Zhu, Problems in homoclinic bifurcations with higher dimensions, Acta Math. Sinica (N.S.), 14 (1998), 341-352.
[23]	D. M. Zhu and Z. H. Xia, Bifurcations of heteroclinic loops, Sci. China Ser A, 41 (1998), 837-848. doi: 10.1007/BF02871667.
[24]	D. M. Zhu, Invariants of coordinate transformation,, J. of East China Normal Univ. Nat. Sci. Ed., 1998 (): 19.

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.
[2]	F. Battelli and M.Fečkan, Subharmonic solutions in singular systems, J. Diff. Eqs, 132 (1996), 21-45.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	J. B. Li and B. Y. Feng, "Stability, Bifurcation and Chaos," Yun Nan Science Press, Kunming, 1995.
[15]	J. W. Reyn, A stability criterion for separatrix polygons in the phase plane, Nieuw Arch. Wisk. (3), 27 (1979), 238-254.
[16]	P. Joyal, Generalized Hopf bifurcation and its dual generalized homoclinic bifurcation, SIAM J. Appl. Math., 48 (1988), 481-496. doi: 10.1137/0148027.
[17]	S. Wiggins, "Global Bifurcation and Chaos. Analytical Methods," Applied Mathematical Sciences, 73, Springer-Verlag, New York, 1988.
[18]	B. Y. Feng, Stability of homoclinic and heteroclinic cycles in space, Acta Math. Sinica (Chin. Ser.), 39 (1996), 649-658.
[19]	E. Leontovič, On the generation of limit cycles from separatrices, Doklady Akad. Nauk SSSR (N.S.), 78 (1951), 641-644.
[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.
[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.
[22]	D. M. Zhu, Problems in homoclinic bifurcations with higher dimensions, Acta Math. Sinica (N.S.), 14 (1998), 341-352.
[23]	D. M. Zhu and Z. H. Xia, Bifurcations of heteroclinic loops, Sci. China Ser A, 41 (1998), 837-848. doi: 10.1007/BF02871667.
[24]	D. M. Zhu, Invariants of coordinate transformation,, J. of East China Normal Univ. Nat. Sci. Ed., 1998 (): 19.

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