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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, (1990).
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.
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.
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.
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, (2005).
|
[6] |
H. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms,, Nonlinearity, 11 (1998), 667.
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.
doi: 10.1142/S0218127402004565. |
[8] |
P. Collins, Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits,, Dyn. Syst., 19 (2004), 1.
doi: 10.1080/14689360310001623421. |
[9] |
P. Collins and B. Krauskopf, Entropy and bifurcations in a chaotic laser,, Phys. Rev. E (3), 66 (2002).
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.
doi: 10.1002/cpa.3160340202. |
[11] |
R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, 2nd edition, (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.
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.
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, (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.
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.
doi: 10.1137/04060487X. |
[17] |
W. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria,, Society for Industrial and Applied Mathematics (SIAM), (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, (1990).
|
[19] |
J. K. Hale and H. Koçak, Dynamics and Bifurcations,, vol. 3 of Texts in Applied Mathematics, (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.
doi: 10.1007/BF01608556. |
[21] |
A. J. Homburg and B. Krauskopf, Resonant homoclinic flip bifurcations,, J. Dynam. Differential Equations, 12 (2000), 807.
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, (2010), 379.
doi: 10.1016/S1874-575X(10)00316-4. |
[23] |
T. Hüls, Homoclinic trajectories of non-autonomous maps,, J. Difference Equ. Appl., 17 (2011), 9.
doi: 10.1080/10236190902932742. |
[24] |
M. C. Irwin, Smooth Dynamical Systems,, vol. 17 of Advanced Series in Nonlinear Dynamics, (2001).
doi: 10.1142/9789812810120. |
[25] |
H. B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems,, in Applications of bifurcation theory (Proc. Advanced Sem., (1976), 359.
|
[26] |
J.-M. Kleinkauf, The Numerical Computation and Geometrical Analysis of Heteroclinic Tangencies,, Technical Report 98-048, (1998), 98. Google Scholar |
[27] |
J.-M. Kleinkauf, Numerische Analyse Tangentialer Homokliner Orbits,, PhD thesis, (1998). Google Scholar |
[28] |
J. Knobloch, Chaotic behaviour near non-transversal homoclinic points with quadratic tangency,, J. Difference Equ. Appl., 12 (2006), 1037.
doi: 10.1080/10236190600986644. |
[29] |
J. Knobloch and T. Rieş, Lin's method for heteroclinic chains involving periodic orbits,, Nonlinearity, 23 (2010), 23.
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.
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.
doi: 10.1088/0951-7715/21/8/001. |
[32] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (1995).
doi: 10.1017/CBO9780511626302. |
[33] |
C. Mira, Chaotic Dynamics,, World Scientific Publishing Co., (1987).
|
[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.
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, (1993).
|
[36] |
K. Palmer, Shadowing in Dynamical Systems, vol. 501 of Mathematics and its Applications,, Kluwer Academic Publishers, (2000).
|
[37] |
K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points,, in Dynamics reported, 1 (1988), 265.
|
[38] |
S. Y. Pilyugin, Shadowing in Dynamical Systems, vol. 1706 of Lecture Notes in Mathematics,, Springer-Verlag, (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.
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.
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.
doi: 10.1007/BF02219223. |
[42] |
M. Shub, Global Stability of Dynamical Systems,, Springer-Verlag, (1987).
|
[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.
|
[44] |
S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.
doi: 10.1090/S0002-9904-1967-11798-1. |
show all references
References:
[1] |
E. L. Allgower and K. Georg, Numerical Continuation Methods,, Springer-Verlag, (1990).
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.
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.
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.
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, (2005).
|
[6] |
H. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms,, Nonlinearity, 11 (1998), 667.
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.
doi: 10.1142/S0218127402004565. |
[8] |
P. Collins, Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits,, Dyn. Syst., 19 (2004), 1.
doi: 10.1080/14689360310001623421. |
[9] |
P. Collins and B. Krauskopf, Entropy and bifurcations in a chaotic laser,, Phys. Rev. E (3), 66 (2002).
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.
doi: 10.1002/cpa.3160340202. |
[11] |
R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, 2nd edition, (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.
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.
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, (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.
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.
doi: 10.1137/04060487X. |
[17] |
W. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria,, Society for Industrial and Applied Mathematics (SIAM), (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, (1990).
|
[19] |
J. K. Hale and H. Koçak, Dynamics and Bifurcations,, vol. 3 of Texts in Applied Mathematics, (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.
doi: 10.1007/BF01608556. |
[21] |
A. J. Homburg and B. Krauskopf, Resonant homoclinic flip bifurcations,, J. Dynam. Differential Equations, 12 (2000), 807.
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, (2010), 379.
doi: 10.1016/S1874-575X(10)00316-4. |
[23] |
T. Hüls, Homoclinic trajectories of non-autonomous maps,, J. Difference Equ. Appl., 17 (2011), 9.
doi: 10.1080/10236190902932742. |
[24] |
M. C. Irwin, Smooth Dynamical Systems,, vol. 17 of Advanced Series in Nonlinear Dynamics, (2001).
doi: 10.1142/9789812810120. |
[25] |
H. B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems,, in Applications of bifurcation theory (Proc. Advanced Sem., (1976), 359.
|
[26] |
J.-M. Kleinkauf, The Numerical Computation and Geometrical Analysis of Heteroclinic Tangencies,, Technical Report 98-048, (1998), 98. Google Scholar |
[27] |
J.-M. Kleinkauf, Numerische Analyse Tangentialer Homokliner Orbits,, PhD thesis, (1998). Google Scholar |
[28] |
J. Knobloch, Chaotic behaviour near non-transversal homoclinic points with quadratic tangency,, J. Difference Equ. Appl., 12 (2006), 1037.
doi: 10.1080/10236190600986644. |
[29] |
J. Knobloch and T. Rieş, Lin's method for heteroclinic chains involving periodic orbits,, Nonlinearity, 23 (2010), 23.
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.
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.
doi: 10.1088/0951-7715/21/8/001. |
[32] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (1995).
doi: 10.1017/CBO9780511626302. |
[33] |
C. Mira, Chaotic Dynamics,, World Scientific Publishing Co., (1987).
|
[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.
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, (1993).
|
[36] |
K. Palmer, Shadowing in Dynamical Systems, vol. 501 of Mathematics and its Applications,, Kluwer Academic Publishers, (2000).
|
[37] |
K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points,, in Dynamics reported, 1 (1988), 265.
|
[38] |
S. Y. Pilyugin, Shadowing in Dynamical Systems, vol. 1706 of Lecture Notes in Mathematics,, Springer-Verlag, (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.
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.
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.
doi: 10.1007/BF02219223. |
[42] |
M. Shub, Global Stability of Dynamical Systems,, Springer-Verlag, (1987).
|
[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.
|
[44] |
S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.
doi: 10.1090/S0002-9904-1967-11798-1. |
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