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The computation of convex invariant sets via Newton's method
Continuation and collapse of homoclinic tangles
1. | Department of Mathematics, Bielefeld University, P.O. Box 100131, 33501 Bielefeld |
2. | Department of Mathematics, Bielefeld University, POB 100131, 33501 Bielefeld |
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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