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January  2014, 1(1): 71-109. doi: 10.3934/jcd.2014.1.71

Continuation and collapse of homoclinic tangles

Wolf-Jürgen Beyn 1, and  Thorsten Hüls 2,

1. 

Department of Mathematics, Bielefeld University, P.O. Box 100131, 33501 Bielefeld
2. 

Department of Mathematics, Bielefeld University, POB 100131, 33501 Bielefeld

Received  February 2011 Revised  June 2012 Published  April 2014

By a classical theorem transversal homoclinic points of maps lead to shift dynamics on a maximal invariant set, also referred to as a homoclinic tangle. In this paper we study the fate of homoclinic tangles in parameterized systems from the viewpoint of numerical continuation and bifurcation theory. The new bifurcation result shows that the maximal invariant set near a homoclinic tangency, where two homoclinic tangles collide, can be characterized by a system of bifurcation equations that is indexed by a symbolic sequence. These bifurcation equations consist of a finite or infinite set of hilltop normal forms known from singularity theory. For the Hénon family we determine numerically the connected components of branches with multi-humped homoclinic orbits that pass through several tangencies. The homoclinic network found by numerical continuation is explained by combining our bifurcation result with graph-theoretical arguments.
Keywords: bifurcation of homoclinic orbits., Homoclinic tangency, symbolic dynamics, numerical continuation.
Mathematics Subject Classification: 37N30, 65P20, 65P3.
Citation: 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
References:
[1]

E. L. Allgower and K. Georg, Numerical Continuation Methods, Springer-Verlag, Berlin, 1990, An introduction. doi: 10.1007/978-3-642-61257-2.  Google Scholar

[2]

W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal., 10 (1990), 379-405. doi: 10.1093/imanum/10.3.379.  Google Scholar

[3]

W.-J. Beyn and J.-M. Kleinkauf, The numerical computation of homoclinic orbits for maps, SIAM J. Numer. Anal., 34 (1997), 1207-1236. doi: 10.1137/S0036142995281693.  Google Scholar

[4]

W.-J. Beyn, T. Hüls, J.-M. Kleinkauf and Y. Zou, Numerical analysis of degenerate connecting orbits for maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 3385-3407. doi: 10.1142/S0218127404011405.  Google Scholar

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C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, vol. 102 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2005, A global geometric and probabilistic perspective, Mathematical Physics, III.  Google Scholar

[6]

H. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity, 11 (1998), 667-770. doi: 10.1088/0951-7715/11/3/015.  Google Scholar

[7]

P. Collins, Symbolic dynamics from homoclinic tangles, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 605-617. doi: 10.1142/S0218127402004565.  Google Scholar

[8]

P. Collins, Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits, Dyn. Syst., 19 (2004), 1-39. doi: 10.1080/14689360310001623421.  Google Scholar

[9]

P. Collins and B. Krauskopf, Entropy and bifurcations in a chaotic laser, Phys. Rev. E (3), 66 (2002), 056201, 8pp. doi: 10.1103/PhysRevE.66.056201.  Google Scholar

[10]

D. W. Decker and H. B. Keller, Path following near bifurcation, Comm. Pure Appl. Math., 34 (1981), 149-175. doi: 10.1002/cpa.3160340202.  Google Scholar

[11]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edition, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1989.  Google Scholar

[12]

J. P. England, B. Krauskopf and H. M. Osinga, Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse, SIAM J. Appl. Dyn. Syst., 3 (2004), 161-190 (electronic). doi: 10.1137/030600131.  Google Scholar

[13]

R. K. Ghaziani, W. Govaerts, Y. A. Kuznetsov and H. G. E. Meijer, Numerical continuation of connecting orbits of maps in MATLAB, J. Difference Equ. Appl., 15 (2009), 849-875. doi: 10.1080/10236190802357677.  Google Scholar

[14]

M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory. Vol. I, vol. 51 of Applied Mathematical Sciences, Springer-Verlag, New York, 1985.  Google Scholar

[15]

S. V. Gonchenko, D. V. Turaev and L. P. Shilnikov, On the dynamic properties of diffeomorphisms with homoclinic tangencies, Sovrem. Mat. Prilozh., 7 (2003), 91-117, J. Math. Sci. (N.Y.) 126 (2005), 1317-1343. doi: 10.1007/s10958-005-0107-1.  Google Scholar

[16]

V. S. Gonchenko, Y. A. Kuznetsov and H. G. E. Meijer, Generalized Hénon map and bifurcations of homoclinic tangencies, SIAM J. Appl. Dyn. Syst., 4 (2005), 407-436. doi: 10.1137/04060487X.  Google Scholar

[17]

W. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719543.  Google Scholar

[18]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer-Verlag, New York, 1990.  Google Scholar

[19]

J. K. Hale and H. Koçak, Dynamics and Bifurcations, vol. 3 of Texts in Applied Mathematics, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4426-4.  Google Scholar

[20]

M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77. doi: 10.1007/BF01608556.  Google Scholar

[21]

A. J. Homburg and B. Krauskopf, Resonant homoclinic flip bifurcations, J. Dynam. Differential Equations, 12 (2000), 807-850. doi: 10.1023/A:1009046621861.  Google Scholar

[22]

A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in Handbook of Dynamical Systems III, (eds. H. Broer, F. Takens and B. Hasselblatt), Elsevier, 2010, 379-524. doi: 10.1016/S1874-575X(10)00316-4.  Google Scholar

[23]

T. Hüls, Homoclinic trajectories of non-autonomous maps, J. Difference Equ. Appl., 17 (2011), 9-31. doi: 10.1080/10236190902932742.  Google Scholar

[24]

M. C. Irwin, Smooth Dynamical Systems, vol. 17 of Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co. Inc., River Edge, NJ, 2001, Reprint of the 1980 original, With a foreword by R. S. MacKay. doi: 10.1142/9789812810120.  Google Scholar

[25]

H. B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, in Applications of bifurcation theory (Proc. Advanced Sem., Univ. Wisconsin, Madison, Wis., 1976), Academic Press, New York, 1977, 359-384. Publ. Math. Res. Center, No. 38.  Google Scholar

[26]

J.-M. Kleinkauf, The Numerical Computation and Geometrical Analysis of Heteroclinic Tangencies, Technical Report 98-048, SFB 343, 1998. Google Scholar

[27]

J.-M. Kleinkauf, Numerische Analyse Tangentialer Homokliner Orbits, PhD thesis, Universität Bielefeld, 1998, Shaker Verlag, Aachen. Google Scholar

[28]

J. Knobloch, Chaotic behaviour near non-transversal homoclinic points with quadratic tangency, J. Difference Equ. Appl., 12 (2006), 1037-1056. doi: 10.1080/10236190600986644.  Google Scholar

[29]

J. Knobloch and T. Rieş, Lin's method for heteroclinic chains involving periodic orbits, Nonlinearity, 23 (2010), 23-54. doi: 10.1088/0951-7715/23/1/002.  Google Scholar

[30]

B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791. doi: 10.1142/S0218127405012533.  Google Scholar

[31]

B. Krauskopf and T. Rieş, A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits, Nonlinearity, 21 (2008), 1655-1690. doi: 10.1088/0951-7715/21/8/001.  Google Scholar

[32]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.  Google Scholar

[33]

C. Mira, Chaotic Dynamics, World Scientific Publishing Co., Singapore, 1987, From the one-dimensional endomorphism to the two-dimensional diffeomorphism.  Google Scholar

[34]

B. E. Oldeman, A. R. Champneys and B. Krauskopf, Homoclinic branch switching: A numerical implementation of Lin's method, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2977-2999. doi: 10.1142/S0218127403008326.  Google Scholar

[35]

J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, vol. 35 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1993.  Google Scholar

[36]

K. Palmer, Shadowing in Dynamical Systems, vol. 501 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2000, Theory and applications.  Google Scholar

[37]

K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, in Dynamics reported, Teubner, Stuttgart, 1 (1988), 265-306.  Google Scholar

[38]

S. Y. Pilyugin, Shadowing in Dynamical Systems, vol. 1706 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1999.  Google Scholar

[39]

J. D. M. Rademacher, Lyapunov-Schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies, J. Differential Equations, 249 (2010), 305-348. doi: 10.1016/j.jde.2010.04.007.  Google Scholar

[40]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.  Google Scholar

[41]

B. Sandstede, Constructing dynamical systems having homoclinic bifurcation points of codimension two, J. Dynam. Differential Equations, 9 (1997), 269-288. doi: 10.1007/BF02219223.  Google Scholar

[42]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987, With the collaboration of Albert Fathi and Rémi Langevin, Translated from the French by Joseph Christy.  Google Scholar

[43]

L. P. Šil'nikov, Existence of a countable set of periodic motions in a neighborhood of a homoclinic curve, Dokl. Akad. Nauk SSSR, 172 (1967), 298-301, Soviet Math. Dokl. 8 (1967), 102-106.  Google Scholar

[44]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

show all references

References:
[1]

E. L. Allgower and K. Georg, Numerical Continuation Methods, Springer-Verlag, Berlin, 1990, An introduction. doi: 10.1007/978-3-642-61257-2.  Google Scholar

[2]

W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal., 10 (1990), 379-405. doi: 10.1093/imanum/10.3.379.  Google Scholar

[3]

W.-J. Beyn and J.-M. Kleinkauf, The numerical computation of homoclinic orbits for maps, SIAM J. Numer. Anal., 34 (1997), 1207-1236. doi: 10.1137/S0036142995281693.  Google Scholar

[4]

W.-J. Beyn, T. Hüls, J.-M. Kleinkauf and Y. Zou, Numerical analysis of degenerate connecting orbits for maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 3385-3407. doi: 10.1142/S0218127404011405.  Google Scholar

[5]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, vol. 102 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2005, A global geometric and probabilistic perspective, Mathematical Physics, III.  Google Scholar

[6]

H. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity, 11 (1998), 667-770. doi: 10.1088/0951-7715/11/3/015.  Google Scholar

[7]

P. Collins, Symbolic dynamics from homoclinic tangles, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 605-617. doi: 10.1142/S0218127402004565.  Google Scholar

[8]

P. Collins, Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits, Dyn. Syst., 19 (2004), 1-39. doi: 10.1080/14689360310001623421.  Google Scholar

[9]

P. Collins and B. Krauskopf, Entropy and bifurcations in a chaotic laser, Phys. Rev. E (3), 66 (2002), 056201, 8pp. doi: 10.1103/PhysRevE.66.056201.  Google Scholar

[10]

D. W. Decker and H. B. Keller, Path following near bifurcation, Comm. Pure Appl. Math., 34 (1981), 149-175. doi: 10.1002/cpa.3160340202.  Google Scholar

[11]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edition, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1989.  Google Scholar

[12]

J. P. England, B. Krauskopf and H. M. Osinga, Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse, SIAM J. Appl. Dyn. Syst., 3 (2004), 161-190 (electronic). doi: 10.1137/030600131.  Google Scholar

[13]

R. K. Ghaziani, W. Govaerts, Y. A. Kuznetsov and H. G. E. Meijer, Numerical continuation of connecting orbits of maps in MATLAB, J. Difference Equ. Appl., 15 (2009), 849-875. doi: 10.1080/10236190802357677.  Google Scholar

[14]

M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory. Vol. I, vol. 51 of Applied Mathematical Sciences, Springer-Verlag, New York, 1985.  Google Scholar

[15]

S. V. Gonchenko, D. V. Turaev and L. P. Shilnikov, On the dynamic properties of diffeomorphisms with homoclinic tangencies, Sovrem. Mat. Prilozh., 7 (2003), 91-117, J. Math. Sci. (N.Y.) 126 (2005), 1317-1343. doi: 10.1007/s10958-005-0107-1.  Google Scholar

[16]

V. S. Gonchenko, Y. A. Kuznetsov and H. G. E. Meijer, Generalized Hénon map and bifurcations of homoclinic tangencies, SIAM J. Appl. Dyn. Syst., 4 (2005), 407-436. doi: 10.1137/04060487X.  Google Scholar

[17]

W. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719543.  Google Scholar

[18]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer-Verlag, New York, 1990.  Google Scholar

[19]

J. K. Hale and H. Koçak, Dynamics and Bifurcations, vol. 3 of Texts in Applied Mathematics, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4426-4.  Google Scholar

[20]

M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77. doi: 10.1007/BF01608556.  Google Scholar

[21]

A. J. Homburg and B. Krauskopf, Resonant homoclinic flip bifurcations, J. Dynam. Differential Equations, 12 (2000), 807-850. doi: 10.1023/A:1009046621861.  Google Scholar

[22]

A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in Handbook of Dynamical Systems III, (eds. H. Broer, F. Takens and B. Hasselblatt), Elsevier, 2010, 379-524. doi: 10.1016/S1874-575X(10)00316-4.  Google Scholar

[23]

T. Hüls, Homoclinic trajectories of non-autonomous maps, J. Difference Equ. Appl., 17 (2011), 9-31. doi: 10.1080/10236190902932742.  Google Scholar

[24]

M. C. Irwin, Smooth Dynamical Systems, vol. 17 of Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co. Inc., River Edge, NJ, 2001, Reprint of the 1980 original, With a foreword by R. S. MacKay. doi: 10.1142/9789812810120.  Google Scholar

[25]

H. B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, in Applications of bifurcation theory (Proc. Advanced Sem., Univ. Wisconsin, Madison, Wis., 1976), Academic Press, New York, 1977, 359-384. Publ. Math. Res. Center, No. 38.  Google Scholar

[26]

J.-M. Kleinkauf, The Numerical Computation and Geometrical Analysis of Heteroclinic Tangencies, Technical Report 98-048, SFB 343, 1998. Google Scholar

[27]

J.-M. Kleinkauf, Numerische Analyse Tangentialer Homokliner Orbits, PhD thesis, Universität Bielefeld, 1998, Shaker Verlag, Aachen. Google Scholar

[28]

J. Knobloch, Chaotic behaviour near non-transversal homoclinic points with quadratic tangency, J. Difference Equ. Appl., 12 (2006), 1037-1056. doi: 10.1080/10236190600986644.  Google Scholar

[29]

J. Knobloch and T. Rieş, Lin's method for heteroclinic chains involving periodic orbits, Nonlinearity, 23 (2010), 23-54. doi: 10.1088/0951-7715/23/1/002.  Google Scholar

[30]

B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791. doi: 10.1142/S0218127405012533.  Google Scholar

[31]

B. Krauskopf and T. Rieş, A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits, Nonlinearity, 21 (2008), 1655-1690. doi: 10.1088/0951-7715/21/8/001.  Google Scholar

[32]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302.  Google Scholar

[33]

C. Mira, Chaotic Dynamics, World Scientific Publishing Co., Singapore, 1987, From the one-dimensional endomorphism to the two-dimensional diffeomorphism.  Google Scholar

[34]

B. E. Oldeman, A. R. Champneys and B. Krauskopf, Homoclinic branch switching: A numerical implementation of Lin's method, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2977-2999. doi: 10.1142/S0218127403008326.  Google Scholar

[35]

J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, vol. 35 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1993.  Google Scholar

[36]

K. Palmer, Shadowing in Dynamical Systems, vol. 501 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2000, Theory and applications.  Google Scholar

[37]

K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, in Dynamics reported, Teubner, Stuttgart, 1 (1988), 265-306.  Google Scholar

[38]

S. Y. Pilyugin, Shadowing in Dynamical Systems, vol. 1706 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1999.  Google Scholar

[39]

J. D. M. Rademacher, Lyapunov-Schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies, J. Differential Equations, 249 (2010), 305-348. doi: 10.1016/j.jde.2010.04.007.  Google Scholar

[40]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.  Google Scholar

[41]

B. Sandstede, Constructing dynamical systems having homoclinic bifurcation points of codimension two, J. Dynam. Differential Equations, 9 (1997), 269-288. doi: 10.1007/BF02219223.  Google Scholar

[42]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987, With the collaboration of Albert Fathi and Rémi Langevin, Translated from the French by Joseph Christy.  Google Scholar

[43]

L. P. Šil'nikov, Existence of a countable set of periodic motions in a neighborhood of a homoclinic curve, Dokl. Akad. Nauk SSSR, 172 (1967), 298-301, Soviet Math. Dokl. 8 (1967), 102-106.  Google Scholar

[44]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

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