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Global stability and convergence rate of traveling waves for a nonlocal model in periodic media
Bifurcation of a heterodimensional cycle with weak inclination flip
1. | Department of Mathematics, North University of China, Taiyuan, 030051, China |
2. | Department of Mathematics, East China Normal University, Shanghai, 200241 |
3. | Institute of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018, China |
References:
[1] |
L. Diaz, Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcation, Nonlinearity, 8 (1995), 693-713.
doi: 10.1088/0951-7715/8/5/003. |
[2] |
F. Geng, D. Zhu and Y. Xu, Bifurcation of heterodimensional cycles with two saddle points, Chaos Solitons Fractals, 39 (2009), 2063-2075.
doi: 10.1016/j.chaos.2007.06.077. |
[3] |
R. George, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091. |
[4] |
Y. Jin and D. Zhu, Bifurcations of rough heteroclinic loop with two saddle points, Sci. China Ser. A, 46 (2003), 459-468. |
[5] |
J. Knobloch and T. Wagenknecht, Homoclinic snaking near a heteroclinic cycle in reversible systems, Phys. D, 206 (2005), 82-93.
doi: 10.1016/j.physd.2005.04.018. |
[6] |
J. Lamb, M.-A. Teixeira and K. N. Webster, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in $\mathbbR^3$, J. Differential Equations, 219 (2005), 78-115.
doi: 10.1016/j.jde.2005.02.019. |
[7] |
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. |
[8] |
D. Liu, S. Ruan and D. Zhu, Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbbR^4$, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 1511-1532.
doi: 10.3934/dcdss.2011.4.1511. |
[9] |
S. Newhouse and J. Palis, Bifurcations of Morse-Smale dynamical systems, in "Dynamical Systems,'' (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, (1973), 303-366. |
[10] |
S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.
doi: 10.1016/0040-9383(74)90034-2. |
[11] |
J. Palis, A global view of dynamics and a conjecture of the denseness of finitude of attractors, Astérisque, 261 (2000), 335-347. |
[12] |
S. Shui and D. Zhu, Codimension 3 non-reasonant bifurcations of rough heteroclinic loops with one orbit flip, Chinese Ann. Math. Ser. B, 27 (2006), 657-674.
doi: 10.1007/s11401-005-0472-6. |
[13] |
Q. Tian and D. Zhu, Bifurcations of nontwisted heteroclinic loop, Sci. China Ser. A, 43 (2000), 818-828.
doi: 10.1007/BF02884181. |
[14] |
L. Wen, Generic diffemorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 419-452. |
[15] |
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] |
L. Diaz, Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcation, Nonlinearity, 8 (1995), 693-713.
doi: 10.1088/0951-7715/8/5/003. |
[2] |
F. Geng, D. Zhu and Y. Xu, Bifurcation of heterodimensional cycles with two saddle points, Chaos Solitons Fractals, 39 (2009), 2063-2075.
doi: 10.1016/j.chaos.2007.06.077. |
[3] |
R. George, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091. |
[4] |
Y. Jin and D. Zhu, Bifurcations of rough heteroclinic loop with two saddle points, Sci. China Ser. A, 46 (2003), 459-468. |
[5] |
J. Knobloch and T. Wagenknecht, Homoclinic snaking near a heteroclinic cycle in reversible systems, Phys. D, 206 (2005), 82-93.
doi: 10.1016/j.physd.2005.04.018. |
[6] |
J. Lamb, M.-A. Teixeira and K. N. Webster, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in $\mathbbR^3$, J. Differential Equations, 219 (2005), 78-115.
doi: 10.1016/j.jde.2005.02.019. |
[7] |
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. |
[8] |
D. Liu, S. Ruan and D. Zhu, Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbbR^4$, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 1511-1532.
doi: 10.3934/dcdss.2011.4.1511. |
[9] |
S. Newhouse and J. Palis, Bifurcations of Morse-Smale dynamical systems, in "Dynamical Systems,'' (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, (1973), 303-366. |
[10] |
S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.
doi: 10.1016/0040-9383(74)90034-2. |
[11] |
J. Palis, A global view of dynamics and a conjecture of the denseness of finitude of attractors, Astérisque, 261 (2000), 335-347. |
[12] |
S. Shui and D. Zhu, Codimension 3 non-reasonant bifurcations of rough heteroclinic loops with one orbit flip, Chinese Ann. Math. Ser. B, 27 (2006), 657-674.
doi: 10.1007/s11401-005-0472-6. |
[13] |
Q. Tian and D. Zhu, Bifurcations of nontwisted heteroclinic loop, Sci. China Ser. A, 43 (2000), 818-828.
doi: 10.1007/BF02884181. |
[14] |
L. Wen, Generic diffemorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 419-452. |
[15] |
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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