Nongeneric bifurcations near heterodimensional cycles with inclination flip in \mathbb{R}^4

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December 2011, 4(6): 1511-1532. doi: 10.3934/dcdss.2011.4.1511

Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbb{R}^4$

Dan Liu ^1, , Shigui Ruan ^2, and Deming Zhu ^3,

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

Received April 2009 Revised October 2009 Published December 2010

Abstract
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Nongeneric bifurcation analysis near rough heterodimensional cycles associated to two saddles in $\mathbb{R}^4$ is presented under inclination flip. By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles, we construct a Poincaré return map under the nongeneric conditions and further obtain the bifurcation equations. Coexistence of a heterodimensional cycle and a unique periodic orbit is proved after perturbations. New features produced by the inclination flip that heterodimensional cycles and homoclinic orbits coexist on the same bifurcation surface are shown. It is also conjectured that homoclinic orbits associated to different equilibria coexist.

Keywords: Bifurcation, dichotomy, heterodimensional cycle, homolinic orbit, inclination flip, Poincaré map..

Mathematics Subject Classification: Primary: 34C23, 37G25; Secondary: 34D0.

Citation: Dan Liu, Shigui Ruan, Deming Zhu. Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbb{R}^4$. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1511-1532. doi: 10.3934/dcdss.2011.4.1511

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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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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