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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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[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.

[23]

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

[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.

[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.

[26]

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

[27]

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

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[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.

[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.

[36]

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

[37]

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

[38]

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

[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.

[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.

[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.

[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.

[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.

[44]

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

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[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.

[23]

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

[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.

[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.

[26]

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

[27]

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

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[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.

[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.

[36]

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

[37]

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

[38]

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

[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.

[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.

[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.

[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.

[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.

[44]

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

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