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Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbb{R}^4$
1. | Department of Mathematics, Shanghai Normal University, Shanghai 200234, China |
2. | Department of Mathematics, The University of Miami, P.O. Box 249085, Coral Gables, Florida 33124 |
3. | Department of Mathematics, East China Normal University, Shanghai 200062 |
References:
[1] |
V. V. Bykov, Orbit structure in a neighborhood of a separatrix cycle containing two saddle-foci, Amer. Math. Soc. Transl., 200 (2000), 87-97. |
[2] |
S.-N. Chow and X. Lin, Bifurcation of a homoclinic orbit with a saddle node equilibrium, Differential Integral Equations, 3 (1990), 435-466. |
[3] |
B. Deng, Sil'nikov problem, exponential expansion, strong $\lambda$-Lemma, $C^1$-linearization and homoclinic bifurcation, J. Differential Equations, 79 (1989), 189-231.
doi: 10.1016/0022-0396(89)90100-9. |
[4] |
G. Deng and D. Zhu, A codimension 3 bifurcation of heteroclinic contour involving a hyperbolic and a nonhyperbolic saddle-foci, (Chinese), Chin. Ann. Math. Ser. A, 28 (2007), 667-678. |
[5] |
L. J. Díaz, Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcations, Nonlinearity, 8 (1995), 693-713.
doi: 10.1088/0951-7715/8/5/003. |
[6] |
F. Geng, "Bifurcations of Heterodimensional Cycles and Heteroclinic Loop and BVPS of Dynamic Equations on Time Scales," Ph.D thesis, East China Normal University, 2007. |
[7] |
P. Hartman, "Ordinary Differential Equations," 2nd edition, Birkhauser, Boston, 1982. |
[8] |
Y. Jin and D. Zhu, Bifurcations of rough heteroclinic loops with three saddle points, Acta Math. Sinica Eng. Ser., 18 (2002), 199-208.
doi: 10.1007/s101140100139. |
[9] |
J. S. W. Lamb, M. A. Teixeira and N. W. Kevin, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in $R^3$, J. Differential Equations, 219 (2005), 78-115.
doi: 10.1016/j.jde.2005.02.019. |
[10] |
D. Liu, F. Geng and D. Zhu, Degenerate bifurcations of nontwisted heterodimensional cycles with codimension 3, Nonlinear Anal., 68 (2008), 2813-2827.
doi: 10.1016/j.na.2007.02.028. |
[11] |
S. E. Newhouse and J. Palis, Bifurcations of Morse-Smale dynamical systems, in "Dynamical Systems," Academic Press, (1973), 303-366. |
[12] |
K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256.
doi: 10.1016/0022-0396(84)90082-2. |
[13] |
J. D. M. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Differential Equations, 218 (2005), 390-443.
doi: 10.1016/j.jde.2005.03.016. |
[14] |
S. Shui and D. Zhu, Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips, Sci. China Ser. A, 48 (2005), 248-260.
doi: 10.1360/03ys0201. |
[15] |
J. Sun, Bifurcations of heteroclinic loop with nonhyperbolic critical points in $\mathbb{R}^{N}$, Sci. China Ser. A, 24 (1994), 1145-1151. |
[16] |
S. Wiggins, "Global Bifurcations and Chaos-Analytical Methods," Springer-Verlag, New York, 1988. |
[17] |
P. A. Worfolk, An equivariant, inclination-flip, heteroclinic bifurcation, Nonlinearity, 9 (1996), 631-647.
doi: 10.1088/0951-7715/9/3/002. |
[18] |
T. Zhang and D. Zhu, Bifurcations of homoclinic orbit connecting two nonleading eigendirections, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 823-836.
doi: 10.1142/S0218127407017574. |
[19] |
D. Zhu, Problems in homoclinic bifurcation with higher dimensions, Acta Math. Sinica Eng. Ser., 14 (1998), 341-352.
doi: 10.1007/BF02580437. |
[20] |
D. Zhu and Z. Xia, Bifurcations of heteroclinic loops, Sci. China Ser. A, 41 (1998), 837-848.
doi: 10.1007/BF02871667. |
show all references
References:
[1] |
V. V. Bykov, Orbit structure in a neighborhood of a separatrix cycle containing two saddle-foci, Amer. Math. Soc. Transl., 200 (2000), 87-97. |
[2] |
S.-N. Chow and X. Lin, Bifurcation of a homoclinic orbit with a saddle node equilibrium, Differential Integral Equations, 3 (1990), 435-466. |
[3] |
B. Deng, Sil'nikov problem, exponential expansion, strong $\lambda$-Lemma, $C^1$-linearization and homoclinic bifurcation, J. Differential Equations, 79 (1989), 189-231.
doi: 10.1016/0022-0396(89)90100-9. |
[4] |
G. Deng and D. Zhu, A codimension 3 bifurcation of heteroclinic contour involving a hyperbolic and a nonhyperbolic saddle-foci, (Chinese), Chin. Ann. Math. Ser. A, 28 (2007), 667-678. |
[5] |
L. J. Díaz, Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcations, Nonlinearity, 8 (1995), 693-713.
doi: 10.1088/0951-7715/8/5/003. |
[6] |
F. Geng, "Bifurcations of Heterodimensional Cycles and Heteroclinic Loop and BVPS of Dynamic Equations on Time Scales," Ph.D thesis, East China Normal University, 2007. |
[7] |
P. Hartman, "Ordinary Differential Equations," 2nd edition, Birkhauser, Boston, 1982. |
[8] |
Y. Jin and D. Zhu, Bifurcations of rough heteroclinic loops with three saddle points, Acta Math. Sinica Eng. Ser., 18 (2002), 199-208.
doi: 10.1007/s101140100139. |
[9] |
J. S. W. Lamb, M. A. Teixeira and N. W. Kevin, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in $R^3$, J. Differential Equations, 219 (2005), 78-115.
doi: 10.1016/j.jde.2005.02.019. |
[10] |
D. Liu, F. Geng and D. Zhu, Degenerate bifurcations of nontwisted heterodimensional cycles with codimension 3, Nonlinear Anal., 68 (2008), 2813-2827.
doi: 10.1016/j.na.2007.02.028. |
[11] |
S. E. Newhouse and J. Palis, Bifurcations of Morse-Smale dynamical systems, in "Dynamical Systems," Academic Press, (1973), 303-366. |
[12] |
K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256.
doi: 10.1016/0022-0396(84)90082-2. |
[13] |
J. D. M. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Differential Equations, 218 (2005), 390-443.
doi: 10.1016/j.jde.2005.03.016. |
[14] |
S. Shui and D. Zhu, Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips, Sci. China Ser. A, 48 (2005), 248-260.
doi: 10.1360/03ys0201. |
[15] |
J. Sun, Bifurcations of heteroclinic loop with nonhyperbolic critical points in $\mathbb{R}^{N}$, Sci. China Ser. A, 24 (1994), 1145-1151. |
[16] |
S. Wiggins, "Global Bifurcations and Chaos-Analytical Methods," Springer-Verlag, New York, 1988. |
[17] |
P. A. Worfolk, An equivariant, inclination-flip, heteroclinic bifurcation, Nonlinearity, 9 (1996), 631-647.
doi: 10.1088/0951-7715/9/3/002. |
[18] |
T. Zhang and D. Zhu, Bifurcations of homoclinic orbit connecting two nonleading eigendirections, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 823-836.
doi: 10.1142/S0218127407017574. |
[19] |
D. Zhu, Problems in homoclinic bifurcation with higher dimensions, Acta Math. Sinica Eng. Ser., 14 (1998), 341-352.
doi: 10.1007/BF02580437. |
[20] |
D. Zhu and Z. Xia, Bifurcations of heteroclinic loops, Sci. China Ser. A, 41 (1998), 837-848.
doi: 10.1007/BF02871667. |
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