-
Previous Article
An equation-free approach to coarse-graining the dynamics of networks
- JCD Home
- This Issue
-
Next Article
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. |
[1] |
Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203 |
[2] |
Andrus Giraldo, Bernd Krauskopf, Hinke M. Osinga. Computing connecting orbits to infinity associated with a homoclinic flip bifurcation. Journal of Computational Dynamics, 2020, 7 (2) : 489-510. doi: 10.3934/jcd.2020020 |
[3] |
Flaviano Battelli, Claudio Lazzari. On the bifurcation from critical homoclinic orbits in n-dimensional maps. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 289-303. doi: 10.3934/dcds.1997.3.289 |
[4] |
Amadeu Delshams, Pere Gutiérrez. Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 757-783. doi: 10.3934/dcds.2004.11.757 |
[5] |
Eleonora Catsigeras, Marcelo Cerminara, Heber Enrich. Simultaneous continuation of infinitely many sinks at homoclinic bifurcations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 693-736. doi: 10.3934/dcds.2011.29.693 |
[6] |
Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1097-1108. doi: 10.3934/dcdsb.2019210 |
[7] |
Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2855-2878. doi: 10.3934/cpaa.2019128 |
[8] |
Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419 |
[9] |
S. Secchi, C. A. Stuart. Global bifurcation of homoclinic solutions of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1493-1518. doi: 10.3934/dcds.2003.9.1493 |
[10] |
Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1 |
[11] |
Addolorata Salvatore. Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 778-787. doi: 10.3934/proc.2003.2003.778 |
[12] |
Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929 |
[13] |
Jianquan Li, Yanni Xiao, Yali Yang. Global analysis of a simple parasite-host model with homoclinic orbits. Mathematical Biosciences & Engineering, 2012, 9 (4) : 767-784. doi: 10.3934/mbe.2012.9.767 |
[14] |
Yingxiang Xu, Yongkui Zou. Preservation of homoclinic orbits under discretization of delay differential equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 275-299. doi: 10.3934/dcds.2011.31.275 |
[15] |
John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851 |
[16] |
W.R. Derrick, P. van den Driessche. Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 299-309. doi: 10.3934/dcdsb.2003.3.299 |
[17] |
Christian Bonatti, Lorenzo J. Díaz, Todd Fisher. Super-exponential growth of the number of periodic orbits inside homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 589-604. doi: 10.3934/dcds.2008.20.589 |
[18] |
Chen-Chang Peng, Kuan-Ju Chen. Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1181-1197. doi: 10.3934/dcdsb.2010.14.1181 |
[19] |
Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807 |
[20] |
Flaviano Battelli, Ken Palmer. A remark about Sil'nikov saddle-focus homoclinic orbits. Communications on Pure & Applied Analysis, 2011, 10 (3) : 817-830. doi: 10.3934/cpaa.2011.10.817 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]