On computing heteroclinic trajectories of non-autonomous maps

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January 2012, 17(1): 79-99. doi: 10.3934/dcdsb.2012.17.79

On computing heteroclinic trajectories of non-autonomous maps

Thorsten Hüls ^1, and Yongkui Zou ^2,

1.	Department of Mathematics, Bielefeld University, POB 100131, 33501 Bielefeld, Germany
2.	Department of Mathematics, Jilin University, Changchun 130012, China

Received January 2011 Revised June 2011 Published October 2011

Abstract
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We propose an adequate notion of a heteroclinic trajectory in non-autonomous systems that generalizes the notion of a heteroclinic orbit of an autonomous system. A heteroclinic trajectory connects two families of semi-bounded trajectories that are bounded in backward and forward time. We apply boundary value techniques for computing one representative of each family. These approximations allow the construction of projection boundary conditions that enable the calculation of a heteroclinic trajectory with high accuracy. The resulting algorithm is applied to non-autonomous toy models as well as to an example from mathematical biology.

Keywords: Non-autonomous discrete time dynamical systems, heteroclinic connection, numerical approximation, boundary value problems, error analysis..

Mathematics Subject Classification: Primary: 70K44, 37B55; Secondary: 34C3.

Citation: Thorsten Hüls, Yongkui Zou. On computing heteroclinic trajectories of non-autonomous maps. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 79-99. doi: 10.3934/dcdsb.2012.17.79

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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