# American Institute of Mathematical Sciences

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

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

 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

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.
Citation: Dan Liu, Shigui Ruan, Deming Zhu. Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbb{R}^4$. Discrete & 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.  Google Scholar [2] S.-N. Chow and X. Lin, Bifurcation of a homoclinic orbit with a saddle node equilibrium, Differential Integral Equations, 3 (1990), 435-466.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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. Google Scholar [7] P. Hartman, "Ordinary Differential Equations," 2nd edition, Birkhauser, Boston, 1982.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] S. E. Newhouse and J. Palis, Bifurcations of Morse-Smale dynamical systems, in "Dynamical Systems," Academic Press, (1973), 303-366.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] J. Sun, Bifurcations of heteroclinic loop with nonhyperbolic critical points in $\mathbbR^n$, Sci. China Ser. A, 24 (1994), 1145-1151. Google Scholar [16] S. Wiggins, "Global Bifurcations and Chaos-Analytical Methods," Springer-Verlag, New York, 1988. Google Scholar [17] P. A. Worfolk, An equivariant, inclination-flip, heteroclinic bifurcation, Nonlinearity, 9 (1996), 631-647. doi: 10.1088/0951-7715/9/3/002.  Google Scholar [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.  Google Scholar [19] D. Zhu, Problems in homoclinic bifurcation with higher dimensions, Acta Math. Sinica Eng. Ser., 14 (1998), 341-352. doi: 10.1007/BF02580437.  Google Scholar [20] D. Zhu and Z. Xia, Bifurcations of heteroclinic loops, Sci. China Ser. A, 41 (1998), 837-848. doi: 10.1007/BF02871667.  Google Scholar

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.  Google Scholar [2] S.-N. Chow and X. Lin, Bifurcation of a homoclinic orbit with a saddle node equilibrium, Differential Integral Equations, 3 (1990), 435-466.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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. Google Scholar [7] P. Hartman, "Ordinary Differential Equations," 2nd edition, Birkhauser, Boston, 1982.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] S. E. Newhouse and J. Palis, Bifurcations of Morse-Smale dynamical systems, in "Dynamical Systems," Academic Press, (1973), 303-366.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] J. Sun, Bifurcations of heteroclinic loop with nonhyperbolic critical points in $\mathbbR^n$, Sci. China Ser. A, 24 (1994), 1145-1151. Google Scholar [16] S. Wiggins, "Global Bifurcations and Chaos-Analytical Methods," Springer-Verlag, New York, 1988. Google Scholar [17] P. A. Worfolk, An equivariant, inclination-flip, heteroclinic bifurcation, Nonlinearity, 9 (1996), 631-647. doi: 10.1088/0951-7715/9/3/002.  Google Scholar [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.  Google Scholar [19] D. Zhu, Problems in homoclinic bifurcation with higher dimensions, Acta Math. Sinica Eng. Ser., 14 (1998), 341-352. doi: 10.1007/BF02580437.  Google Scholar [20] D. Zhu and Z. Xia, Bifurcations of heteroclinic loops, Sci. China Ser. A, 41 (1998), 837-848. doi: 10.1007/BF02871667.  Google Scholar
 [1] Zhiqin Qiao, Deming Zhu, Qiuying Lu. Bifurcation of a heterodimensional cycle with weak inclination flip. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1009-1025. doi: 10.3934/dcdsb.2012.17.1009 [2] Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173 [3] Sebastián Ferrer, Francisco Crespo. Alternative angle-based approach to the $\mathcal{KS}$-Map. An interpretation through symmetry and reduction. Journal of Geometric Mechanics, 2018, 10 (3) : 359-372. doi: 10.3934/jgm.2018013 [4] Andrus Giraldo, Bernd Krauskopf, Hinke M. Osinga. Computing connecting orbits to infinity associated with a homoclinic flip bifurcation. Journal of Computational Dynamics, 2020, 7 (2) : 489-510. doi: 10.3934/jcd.2020020 [5] Thai Son Doan, Martin Rasmussen, Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 875-887. doi: 10.3934/dcdsb.2015.20.875 [6] Fang Wu, Lihong Huang, Jiafu Wang. Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021264 [7] Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293 [8] Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295 [9] Ming Zhao, Cuiping Li, Jinliang Wang, Zhaosheng Feng. Bifurcation analysis of the three-dimensional Hénon map. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 625-645. doi: 10.3934/dcdss.2017031 [10] Andrus Giraldo, Neil G. R. Broderick, Bernd Krauskopf. Chaotic switching in driven-dissipative Bose-Hubbard dimers: When a flip bifurcation meets a T-point in $\mathbb{R}^4$. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021217 [11] Bourama Toni. Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization. Conference Publications, 2005, 2005 (Special) : 846-853. doi: 10.3934/proc.2005.2005.846 [12] Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129 [13] Mikhail Kamenskii, Boris Mikhaylenko. Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 437-452. doi: 10.3934/dcdsb.2013.18.437 [14] Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron. Electronic Research Archive, 2021, 29 (5) : 2987-3015. doi: 10.3934/era.2021023 [15] Brian Ryals, Robert J. Sacker. Bifurcation in the almost periodic $2$D Ricker map. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021089 [16] Christian Pötzsche. Dichotomy spectra of triangular equations. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 423-450. doi: 10.3934/dcds.2016.36.423 [17] Shin Kiriki, Yusuke Nishizawa, Teruhiko Soma. Heterodimensional tangencies on cycles leading to strange attractors. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 285-300. doi: 10.3934/dcds.2010.27.285 [18] V. Afraimovich, J. Schmeling, Edgardo Ugalde, Jesús Urías. Spectra of dimensions for Poincaré recurrences. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 901-914. doi: 10.3934/dcds.2000.6.901 [19] Eva Miranda, Romero Solha. A Poincaré lemma in geometric quantisation. Journal of Geometric Mechanics, 2013, 5 (4) : 473-491. doi: 10.3934/jgm.2013.5.473 [20] Lorenzo J. Díaz, Jorge Rocha. How do hyperbolic homoclinic classes collide at heterodimensional cycles?. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 589-627. doi: 10.3934/dcds.2007.17.589

2020 Impact Factor: 2.425