August  2011, 30(3): 945-963. doi: 10.3934/dcds.2011.30.945

Bifurcations of multiple homoclinics in general dynamical systems

1. 

Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

2. 

Department of Mathematics, East China Normal University, Shanghai, 200241, China, China

Received  December 2009 Revised  November 2010 Published  March 2011

By using the local active coordinates consisting of tangent vectors of the invariant subspaces, as well as the Silnikov coordinates, the simple normal form is established in the neighborhood of the double homoclinic loops with bellows configuration in a general system, then the dynamics near the homoclinic bellows is investigated, and the existence, uniqueness of the homoclinic orbits and periodic orbits with various patterns bifurcated from the primary orbits are demonstrated, and the corresponding bifurcation curves (or surfaces) and existence regions are located.
Citation: Yancong Xu, Deming Zhu, Xingbo Liu. Bifurcations of multiple homoclinics in general dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 945-963. doi: 10.3934/dcds.2011.30.945
References:
[1]

D. G. Aronson, M. Golubitsky and M. Krupa, Coupled arrays of Josephson junctions and bifurcations of maps with $S_N$ symmetry,, Nonlinearity, 4 (1991), 861. doi: 10.1088/0951-7715/4/3/013. Google Scholar

[2]

A. R. Champneys and M. D. Groves, A global investigation of solitary-wave solutions to a two-parameter model equation for water waves,, J. Fluid Mechanics, 342 (1997), 199. doi: 10.1017/S0022112097005193. Google Scholar

[3]

A. J. Homburg and J. Knobloch, Multiple homoclinic orbits in conservative and reversible systems,, Trans. Amer. Math. Soc., 358 (2006), 1715. doi: 10.1090/S0002-9947-05-03793-1. Google Scholar

[4]

J. Härterich, Cascades of reversible homoclinic orbits to a saddle-focus equilibrium,, Physica D, 112 (1998), 187. doi: 10.1016/S0167-2789(97)00210-8. Google Scholar

[5]

J. Knobloch, Bifurcation of degenerate homoclinics in reversible and conservative systems,, J. Dyn. Diff. Equ., 9 (1997), 427. doi: 10.1007/BF02227489. Google Scholar

[6]

J. Klaus and J. Knobloch, Bifurcation of homoclinic orbits to a saddle-center in reversible systems,, Inter. J. Bifu. Chaos, 13 (2003), 2603. doi: 10.1142/S0218127403008119. Google Scholar

[7]

X. B. Lin, Using Melnikov's method to solve Shilnikov's problems,, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 295. Google Scholar

[8]

B. Sandstede, C. K. R. T. Jones and J. C. Alexander, Existence and stability of N-pulses on optical fibres with phase-sensitive amplifiers,, Physica D, 106 (1997), 167. doi: 10.1016/S0167-2789(97)89488-2. Google Scholar

[9]

George R. Sell, Smooth linearization near a fixed point,, Amer. J. Math., 107 (1985), 1035. doi: 10.2307/2374346. Google Scholar

[10]

D.V. Turaev, Bifurcations of a homoclinic "figure eight" of a multidimensional saddle,, Rus. Math. Surv., 43 (1988), 264. doi: 10.1070/RM1988v043n05ABEH001952. Google Scholar

[11]

T. Wagenknecht and A. R. Champneys, When gap solitons become embeded solitons: A generic unfolding,, Physica D, 177 (2003), 50. doi: 10.1016/S0167-2789(02)00773-X. Google Scholar

[12]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical System and Chaos,", Springer-Verlag, (1990). Google Scholar

[13]

Y. C. Xu, D. M. Zhu and F. J. Geng, Codimension 3 heteroclinic bifurcations with orbit and inclination flips in reversible systems,, Inter. J. Bifu. Chaos, 18 (2008), 3689. doi: 10.1142/S0218127408022652. Google Scholar

[14]

Y. C. Xu and D. M. Zhu, Bifurcations of heterodimensional cycles with one orbit flip and one inclination flip,, Nonlinear Dynamics, 60 (2010), 1. doi: 10.1007/s11071-009-9575-z. Google Scholar

[15]

D. M. Zhu, Problems in homoclinic bifurcation with higher dimensions,, Acta Math. Sinica, 14 (1998), 341. doi: 10.1007/BF02580437. Google Scholar

[16]

D. M. Zhu and Z. H. Xia, Bifurcation of heteroclinic loops,, Sci. in China Series A, 41 (1998), 837. doi: 10.1007/BF02871667. Google Scholar

show all references

References:
[1]

D. G. Aronson, M. Golubitsky and M. Krupa, Coupled arrays of Josephson junctions and bifurcations of maps with $S_N$ symmetry,, Nonlinearity, 4 (1991), 861. doi: 10.1088/0951-7715/4/3/013. Google Scholar

[2]

A. R. Champneys and M. D. Groves, A global investigation of solitary-wave solutions to a two-parameter model equation for water waves,, J. Fluid Mechanics, 342 (1997), 199. doi: 10.1017/S0022112097005193. Google Scholar

[3]

A. J. Homburg and J. Knobloch, Multiple homoclinic orbits in conservative and reversible systems,, Trans. Amer. Math. Soc., 358 (2006), 1715. doi: 10.1090/S0002-9947-05-03793-1. Google Scholar

[4]

J. Härterich, Cascades of reversible homoclinic orbits to a saddle-focus equilibrium,, Physica D, 112 (1998), 187. doi: 10.1016/S0167-2789(97)00210-8. Google Scholar

[5]

J. Knobloch, Bifurcation of degenerate homoclinics in reversible and conservative systems,, J. Dyn. Diff. Equ., 9 (1997), 427. doi: 10.1007/BF02227489. Google Scholar

[6]

J. Klaus and J. Knobloch, Bifurcation of homoclinic orbits to a saddle-center in reversible systems,, Inter. J. Bifu. Chaos, 13 (2003), 2603. doi: 10.1142/S0218127403008119. Google Scholar

[7]

X. B. Lin, Using Melnikov's method to solve Shilnikov's problems,, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 295. Google Scholar

[8]

B. Sandstede, C. K. R. T. Jones and J. C. Alexander, Existence and stability of N-pulses on optical fibres with phase-sensitive amplifiers,, Physica D, 106 (1997), 167. doi: 10.1016/S0167-2789(97)89488-2. Google Scholar

[9]

George R. Sell, Smooth linearization near a fixed point,, Amer. J. Math., 107 (1985), 1035. doi: 10.2307/2374346. Google Scholar

[10]

D.V. Turaev, Bifurcations of a homoclinic "figure eight" of a multidimensional saddle,, Rus. Math. Surv., 43 (1988), 264. doi: 10.1070/RM1988v043n05ABEH001952. Google Scholar

[11]

T. Wagenknecht and A. R. Champneys, When gap solitons become embeded solitons: A generic unfolding,, Physica D, 177 (2003), 50. doi: 10.1016/S0167-2789(02)00773-X. Google Scholar

[12]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical System and Chaos,", Springer-Verlag, (1990). Google Scholar

[13]

Y. C. Xu, D. M. Zhu and F. J. Geng, Codimension 3 heteroclinic bifurcations with orbit and inclination flips in reversible systems,, Inter. J. Bifu. Chaos, 18 (2008), 3689. doi: 10.1142/S0218127408022652. Google Scholar

[14]

Y. C. Xu and D. M. Zhu, Bifurcations of heterodimensional cycles with one orbit flip and one inclination flip,, Nonlinear Dynamics, 60 (2010), 1. doi: 10.1007/s11071-009-9575-z. Google Scholar

[15]

D. M. Zhu, Problems in homoclinic bifurcation with higher dimensions,, Acta Math. Sinica, 14 (1998), 341. doi: 10.1007/BF02580437. Google Scholar

[16]

D. M. Zhu and Z. H. Xia, Bifurcation of heteroclinic loops,, Sci. in China Series A, 41 (1998), 837. doi: 10.1007/BF02871667. Google Scholar

[1]

François Monard. Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms. Inverse Problems & Imaging, 2018, 12 (2) : 433-460. doi: 10.3934/ipi.2018019

[2]

Leonardo Mora. Homoclinic bifurcations, fat attractors and invariant curves. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1133-1148. doi: 10.3934/dcds.2003.9.1133

[3]

Eleonora Catsigeras, Marcelo Cerminara, Heber Enrich. Simultaneous continuation of infinitely many sinks at homoclinic bifurcations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 693-736. doi: 10.3934/dcds.2011.29.693

[4]

Enrique R. Pujals. On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphisms in attracting regions. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 179-226. doi: 10.3934/dcds.2006.16.179

[5]

Enrique R. Pujals. Density of hyperbolicity and homoclinic bifurcations for attracting topologically hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 335-405. doi: 10.3934/dcds.2008.20.335

[6]

Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021

[7]

Xiao-Biao Lin, Changrong Zhu. Saddle-node bifurcations of multiple homoclinic solutions in ODES. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1435-1460. doi: 10.3934/dcdsb.2017069

[8]

Thorsten Riess. Numerical study of secondary heteroclinic bifurcations near non-reversible homoclinic snaking. Conference Publications, 2011, 2011 (Special) : 1244-1253. doi: 10.3934/proc.2011.2011.1244

[9]

Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803

[10]

Antonio Pumariño, José Ángel Rodríguez, Joan Carles Tatjer, Enrique Vigil. Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 523-541. doi: 10.3934/dcdsb.2014.19.523

[11]

Robert M. Strain. Coordinates in the relativistic Boltzmann theory. Kinetic & Related Models, 2011, 4 (1) : 345-359. doi: 10.3934/krm.2011.4.345

[12]

Robert Schippa. Sharp Strichartz estimates in spherical coordinates. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2047-2051. doi: 10.3934/cpaa.2017100

[13]

François Monard. Efficient tensor tomography in fan-beam coordinates. Inverse Problems & Imaging, 2016, 10 (2) : 433-459. doi: 10.3934/ipi.2016007

[14]

Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067

[15]

Bernard Brighi, Tewfik Sari. Blowing-up coordinates for a similarity boundary layer equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 929-948. doi: 10.3934/dcds.2005.12.929

[16]

Xiao Wen. Structurally stable homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1693-1707. doi: 10.3934/dcds.2016.36.1693

[17]

Victoria Rayskin. Homoclinic tangencies in $R^n$. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 465-480. doi: 10.3934/dcds.2005.12.465

[18]

Christian Bonatti, Shaobo Gan, Dawei Yang. On the hyperbolicity of homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1143-1162. doi: 10.3934/dcds.2009.25.1143

[19]

Wolf-Jürgen Beyn, Thorsten Hüls. Continuation and collapse of homoclinic tangles. Journal of Computational Dynamics, 2014, 1 (1) : 71-109. doi: 10.3934/jcd.2014.1.71

[20]

Amadeu Delshams, Pere Gutiérrez, Tere M. Seara. Exponentially small splitting for whiskered tori in Hamiltonian systems: flow-box coordinates and upper bounds. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 785-826. doi: 10.3934/dcds.2004.11.785

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]