Codimension-4 resonant homoclinic bifurcations with orbit flips and inclination flips

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• 作者：Tian Si Zhang ; De Ming Zhu
• 关键词：Orbit flip ; inclination flip ; resonant ; homoclinic ; doubling bifurcations ; 37C29 ; 34C23 ; 34C37
• 刊名：Acta Mathematica Sinica
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：31
• 期：8
• 页码：1359-1366
• 全文大小：231 KB
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  [2]Geng, F. J., Zhu, D. M.: Bifurcations of generic heteroclinic loop accompanied by transcritical bifurcation. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18, 1069-083 (2008)View Article MathSciNet MATH
  [3]Homburg, A. J., Kokubu, H., Naudot, V.: Homoclinic-doubling Cascades. Arch. Ration. Mech. Anal., 160(3), 195-43 (2001)View Article MathSciNet
  [4]Jin, Y. L., Zhu, D. M.: Bifurcation of rough heteroclinic loop with two saddle points. Sci. China, Ser. A, 46(4), 459-68 (2003)View Article MathSciNet MATH
  [5]Kokubu, H., Komuru, M., Oka, H.: Multiple homoclinic bifurcations from orbit flip I. Sucessive homoclinic doublings. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6, 833-50 (1996)View Article MathSciNet MATH
  [6]Liu, X. B.: Homoclinic flip bifurcations accompanied by transcritical bifurcation. Chinese Ann. Math. Ser. B, 6B, 905-16 (2011)View Article MATH
  [7]Lu, Q. Y., Qiao, Z. Q., Zhang, T. S., et al.: Heterodimensional cycle bifurcation with orbit-filp. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20(2), 491-08 (2010)View Article MathSciNet MATH
  [8]Morales, C. A., Pacifico, M. J.: Inclination-flip homoclinic orbits arising from orbit-flip. Nonlinearity, 14, 379-93 (2001)View Article MathSciNet
  [9]Oldeman, B. E., Krauskopf, B., Champneys, A. R.: Numerical unfoldings of codimension-three resonant homoclinic flip bifurcations. Nonlinearity, 14(3), 597-21 (2001)View Article MathSciNet MATH
  [10]Tian, Q. P., Zhu, D. M.: Bifurcation of nontwisted heteroclinic loop. Sci. China, Ser. A, 43(8), 818-28 (2000)View Article MathSciNet MATH
  [11]Yorke, J. A., Alligood, K. T.: Cascades of period doubling bifurcations: a prerequisite for horseshoes. Bull. Amer. Math. Soc. (N.S.), 9, 319-22 (1983)View Article MathSciNet MATH
  [12]Zhang, T. S., Zhu, D. M.: Homoclinic bifurcation of orbit flip with resonant principal eigenvalues. Acta Math. Sin., Engl. Series, 22(3), 855-64 (2006)View Article MATH
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  [14]Zhu, D. M., Xia, Z. H.: Bifurcation of heteroclinic loops. Sci. China, Ser. A, 41(8), 837-48 (1998)View Article MathSciNet
  
• 作者单位：Tian Si Zhang (1)
  De Ming Zhu (2)
  
  1. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, P. R. China
  2. Department of Mathematics, East China Normal University, Shanghai, 200062, P. R. China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Chinese Library of Science
  
• 出版者：Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, co-published
• ISSN：1439-7617

文摘

The paper studies a codimension-4 resonant homoclinic bifurcation with one orbit flip and two inclination flips, where the resonance takes place in the tangent direction of homoclinic orbit. Local active coordinate system is introduced to construct the Poincaré returning map, and also the associated successor functions. We prove the existence of the saddle-node bifurcation, the period-doubling bifurcation and the homoclinic-doubling bifurcation, and also locate the corresponding 1-periodic orbit, 1-homoclinic orbit, double periodic orbits and some 2 n -homoclinic orbits.