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On computing heteroclinic trajectories of non-autonomous maps
1. | Department of Mathematics, Bielefeld University, POB 100131, 33501 Bielefeld, Germany |
2. | Department of Mathematics, Jilin University, Changchun 130012, China |
References:
[1] |
A. I. Alonso, J. Hong and R. Obaya, Exponential dichotomy and trichotomy for difference equations, Comput. Math. Appl., 38 (1999), 41-49.
doi: 10.1016/S0898-1221(99)00167-4. |
[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 T. Hüls, Error estimates for approximating non-hyperbolic heteroclinic orbits of maps, Numer. Math., 99 (2004), 289-323.
doi: 10.1007/s00211-004-0563-4. |
[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] |
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. |
[6] |
R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," Second edition, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. |
[7] |
A. Dhooge, W. Govaerts and Y. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2003), 141-164.
doi: 10.1145/779359.779362. |
[8] |
L. Dieci, C. Elia and E. Van Vleck, Exponential dichotomy on the real line: SVD and QR methods, J. Differential Equations, 248 (2010), 287-308. |
[9] |
S. Elaydi and R. J. Sacker, Global stability of periodic orbits of non-autonomous difference equations and population biology, J. Differential Equations, 208 (2005), 258-273. |
[10] |
S. Elaydi and R. J. Sacker, Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures, J. Difference Equ. Appl., 11 (2005), 337-346. |
[11] |
J. P. England, B. Krauskopf and H. M. Osinga, Bifurcations of stable sets in noninvertible planar maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 891-904.
doi: 10.1142/S0218127405012466. |
[12] |
D. Fundinger, Toward the calculation of higher-dimensional stable manifolds and stable sets for noninvertible and piecewise-smooth maps, J. Nonlinear Sci., 18 (2008), 391-413.
doi: 10.1007/s00332-007-9016-4. |
[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. |
[14] |
J. K. Hale and H. Koçak, "Dynamics and Bifurcations," Texts in Applied Mathematics, 3, Springer-Verlag, New York, 1991. |
[15] |
M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77.
doi: 10.1007/BF01608556. |
[16] |
M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. |
[17] |
T. Hüls, Numerical computation of dichotomy rates and projectors in discrete time, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 109-131.
doi: 10.3934/dcdsb.2009.12.109. |
[18] |
T. Hüls, Computing Sacker-Sell spectra in discrete time dynamical systems, SIAM J. Numer. Anal., 48 (2010), 2043-2064.
doi: 10.1137/090754509. |
[19] |
T. Hüls, Homoclinic trajectories of non-autonomous maps, J. Difference Equ. Appl., 17 (2011), 9-31. |
[20] |
Y. Kang and H. Smith, Global dynamics of a discrete two-species Lottery-Ricker competition model, To appear in Journal of Biological Dynamics, Feb. 2011, arXiv:1102.2286. |
[21] |
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. |
[22] |
C. Mira, "Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism," World Scientific Publishing Co., Singapore, 1987. |
[23] |
K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, in "Dynamics Reported, Vol. 1," 265-306, Dynam. Report. Ser. Dynam. Systems Appl., 1, Wiley, Chichester, 1988. |
[24] |
G. Papaschinopoulos, Exponential dichotomy for almost periodic linear difference equations, Ann. Soc. Sci. Bruxelles Sér. I, 102 (1988), 19-28. |
[25] |
C. Pötzsche and S. Siegmund, $C^m$ -smoothness of invariant fiber bundles, Topol. Methods Nonlinear Anal., 24 (2004), 107-145. |
[26] |
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. |
[27] |
S. Wiggins, "Normally Hyperbolic Invariant Manifolds in Dynamical Systems," With the assistance of György Haller and Igor Mezić, Applied Mathematical Sciences, 105, Springer-Verlag, New York, 1994. |
show all references
References:
[1] |
A. I. Alonso, J. Hong and R. Obaya, Exponential dichotomy and trichotomy for difference equations, Comput. Math. Appl., 38 (1999), 41-49.
doi: 10.1016/S0898-1221(99)00167-4. |
[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 T. Hüls, Error estimates for approximating non-hyperbolic heteroclinic orbits of maps, Numer. Math., 99 (2004), 289-323.
doi: 10.1007/s00211-004-0563-4. |
[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] |
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. |
[6] |
R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," Second edition, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. |
[7] |
A. Dhooge, W. Govaerts and Y. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2003), 141-164.
doi: 10.1145/779359.779362. |
[8] |
L. Dieci, C. Elia and E. Van Vleck, Exponential dichotomy on the real line: SVD and QR methods, J. Differential Equations, 248 (2010), 287-308. |
[9] |
S. Elaydi and R. J. Sacker, Global stability of periodic orbits of non-autonomous difference equations and population biology, J. Differential Equations, 208 (2005), 258-273. |
[10] |
S. Elaydi and R. J. Sacker, Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures, J. Difference Equ. Appl., 11 (2005), 337-346. |
[11] |
J. P. England, B. Krauskopf and H. M. Osinga, Bifurcations of stable sets in noninvertible planar maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 891-904.
doi: 10.1142/S0218127405012466. |
[12] |
D. Fundinger, Toward the calculation of higher-dimensional stable manifolds and stable sets for noninvertible and piecewise-smooth maps, J. Nonlinear Sci., 18 (2008), 391-413.
doi: 10.1007/s00332-007-9016-4. |
[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. |
[14] |
J. K. Hale and H. Koçak, "Dynamics and Bifurcations," Texts in Applied Mathematics, 3, Springer-Verlag, New York, 1991. |
[15] |
M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77.
doi: 10.1007/BF01608556. |
[16] |
M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. |
[17] |
T. Hüls, Numerical computation of dichotomy rates and projectors in discrete time, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 109-131.
doi: 10.3934/dcdsb.2009.12.109. |
[18] |
T. Hüls, Computing Sacker-Sell spectra in discrete time dynamical systems, SIAM J. Numer. Anal., 48 (2010), 2043-2064.
doi: 10.1137/090754509. |
[19] |
T. Hüls, Homoclinic trajectories of non-autonomous maps, J. Difference Equ. Appl., 17 (2011), 9-31. |
[20] |
Y. Kang and H. Smith, Global dynamics of a discrete two-species Lottery-Ricker competition model, To appear in Journal of Biological Dynamics, Feb. 2011, arXiv:1102.2286. |
[21] |
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. |
[22] |
C. Mira, "Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism," World Scientific Publishing Co., Singapore, 1987. |
[23] |
K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, in "Dynamics Reported, Vol. 1," 265-306, Dynam. Report. Ser. Dynam. Systems Appl., 1, Wiley, Chichester, 1988. |
[24] |
G. Papaschinopoulos, Exponential dichotomy for almost periodic linear difference equations, Ann. Soc. Sci. Bruxelles Sér. I, 102 (1988), 19-28. |
[25] |
C. Pötzsche and S. Siegmund, $C^m$ -smoothness of invariant fiber bundles, Topol. Methods Nonlinear Anal., 24 (2004), 107-145. |
[26] |
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. |
[27] |
S. Wiggins, "Normally Hyperbolic Invariant Manifolds in Dynamical Systems," With the assistance of György Haller and Igor Mezić, Applied Mathematical Sciences, 105, Springer-Verlag, New York, 1994. |
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