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On the stability of homoclinic loops with higher dimension
1. | Department of Mathematics, East China Normal University, Shanghai, 200241 |
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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