# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 861-902. doi: 10.1088/0951-7715/4/3/013. [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), 199-229. doi: 10.1017/S0022112097005193. [3] A. J. Homburg and J. Knobloch, Multiple homoclinic orbits in conservative and reversible systems, Trans. Amer. Math. Soc., 358 (2006), 1715-1740. doi: 10.1090/S0002-9947-05-03793-1. [4] J. Härterich, Cascades of reversible homoclinic orbits to a saddle-focus equilibrium, Physica D, 112 (1998), 187-200. doi: 10.1016/S0167-2789(97)00210-8. [5] J. Knobloch, Bifurcation of degenerate homoclinics in reversible and conservative systems, J. Dyn. Diff. Equ., 9 (1997), 427-444. doi: 10.1007/BF02227489. [6] J. Klaus and J. Knobloch, Bifurcation of homoclinic orbits to a saddle-center in reversible systems, Inter. J. Bifu. Chaos, 13 (2003), 2603-2622. doi: 10.1142/S0218127403008119. [7] X. B. Lin, Using Melnikov's method to solve Shilnikov's problems, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 295-325. [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), 167-206. doi: 10.1016/S0167-2789(97)89488-2. [9] George R. Sell, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091. doi: 10.2307/2374346. [10] D.V. Turaev, Bifurcations of a homoclinic "figure eight" of a multidimensional saddle, Rus. Math. Surv., 43 (1988), 264-265. doi: 10.1070/RM1988v043n05ABEH001952. [11] T. Wagenknecht and A. R. Champneys, When gap solitons become embeded solitons: A generic unfolding, Physica D, 177 (2003), 50-70. doi: 10.1016/S0167-2789(02)00773-X. [12] S. Wiggins, "Introduction to Applied Nonlinear Dynamical System and Chaos," Springer-Verlag, New York, 1990. [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), 3689-3701. doi: 10.1142/S0218127408022652. [14] Y. C. Xu and D. M. Zhu, Bifurcations of heterodimensional cycles with one orbit flip and one inclination flip, Nonlinear Dynamics, 60 (2010), 1-13. doi: 10.1007/s11071-009-9575-z. [15] D. M. Zhu, Problems in homoclinic bifurcation with higher dimensions, Acta Math. Sinica, English Series, 14 (1998), 341-352. doi: 10.1007/BF02580437. [16] D. M. Zhu and Z. H. Xia, Bifurcation of heteroclinic loops, Sci. in China Series A, 41 (1998), 837-848. doi: 10.1007/BF02871667.

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), 861-902. doi: 10.1088/0951-7715/4/3/013. [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), 199-229. doi: 10.1017/S0022112097005193. [3] A. J. Homburg and J. Knobloch, Multiple homoclinic orbits in conservative and reversible systems, Trans. Amer. Math. Soc., 358 (2006), 1715-1740. doi: 10.1090/S0002-9947-05-03793-1. [4] J. Härterich, Cascades of reversible homoclinic orbits to a saddle-focus equilibrium, Physica D, 112 (1998), 187-200. doi: 10.1016/S0167-2789(97)00210-8. [5] J. Knobloch, Bifurcation of degenerate homoclinics in reversible and conservative systems, J. Dyn. Diff. Equ., 9 (1997), 427-444. doi: 10.1007/BF02227489. [6] J. Klaus and J. Knobloch, Bifurcation of homoclinic orbits to a saddle-center in reversible systems, Inter. J. Bifu. Chaos, 13 (2003), 2603-2622. doi: 10.1142/S0218127403008119. [7] X. B. Lin, Using Melnikov's method to solve Shilnikov's problems, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 295-325. [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), 167-206. doi: 10.1016/S0167-2789(97)89488-2. [9] George R. Sell, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091. doi: 10.2307/2374346. [10] D.V. Turaev, Bifurcations of a homoclinic "figure eight" of a multidimensional saddle, Rus. Math. Surv., 43 (1988), 264-265. doi: 10.1070/RM1988v043n05ABEH001952. [11] T. Wagenknecht and A. R. Champneys, When gap solitons become embeded solitons: A generic unfolding, Physica D, 177 (2003), 50-70. doi: 10.1016/S0167-2789(02)00773-X. [12] S. Wiggins, "Introduction to Applied Nonlinear Dynamical System and Chaos," Springer-Verlag, New York, 1990. [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), 3689-3701. doi: 10.1142/S0218127408022652. [14] Y. C. Xu and D. M. Zhu, Bifurcations of heterodimensional cycles with one orbit flip and one inclination flip, Nonlinear Dynamics, 60 (2010), 1-13. doi: 10.1007/s11071-009-9575-z. [15] D. M. Zhu, Problems in homoclinic bifurcation with higher dimensions, Acta Math. Sinica, English Series, 14 (1998), 341-352. doi: 10.1007/BF02580437. [16] D. M. Zhu and Z. H. Xia, Bifurcation of heteroclinic loops, Sci. in China Series A, 41 (1998), 837-848. doi: 10.1007/BF02871667.
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