American Institute of Mathematical Sciences

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

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

Received  January 2011 Revised  June 2011 Published  October 2011

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.
Citation: Thorsten Hüls, Yongkui Zou. On computing heteroclinic trajectories of non-autonomous maps. Discrete & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] R. L. 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Osinga, Bifurcations of stable sets in noninvertible planar maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 891-904. doi: 10.1142/S0218127405012466.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] J. K. Hale and H. Koçak, "Dynamics and Bifurcations," Texts in Applied Mathematics, 3, Springer-Verlag, New York, 1991.  Google Scholar [15] M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77. doi: 10.1007/BF01608556.  Google Scholar [16] M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [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.  Google Scholar [18] T. Hüls, Computing Sacker-Sell spectra in discrete time dynamical systems, SIAM J. Numer. Anal., 48 (2010), 2043-2064. doi: 10.1137/090754509.  Google Scholar [19] T. Hüls, Homoclinic trajectories of non-autonomous maps, J. Difference Equ. Appl., 17 (2011), 9-31.  Google Scholar [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. Google Scholar [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.  Google Scholar [22] C. Mira, "Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism," World Scientific Publishing Co., Singapore, 1987.  Google Scholar [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.  Google Scholar [24] G. Papaschinopoulos, Exponential dichotomy for almost periodic linear difference equations, Ann. Soc. Sci. Bruxelles Sér. I, 102 (1988), 19-28.  Google Scholar [25] C. Pötzsche and S. Siegmund, $C^m$ -smoothness of invariant fiber bundles, Topol. Methods Nonlinear Anal., 24 (2004), 107-145.  Google Scholar [26] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] S. Elaydi and R. J. Sacker, Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures, J. Difference Equ. Appl., 11 (2005), 337-346.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] J. K. Hale and H. Koçak, "Dynamics and Bifurcations," Texts in Applied Mathematics, 3, Springer-Verlag, New York, 1991.  Google Scholar [15] M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77. doi: 10.1007/BF01608556.  Google Scholar [16] M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [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.  Google Scholar [18] T. Hüls, Computing Sacker-Sell spectra in discrete time dynamical systems, SIAM J. Numer. Anal., 48 (2010), 2043-2064. doi: 10.1137/090754509.  Google Scholar [19] T. Hüls, Homoclinic trajectories of non-autonomous maps, J. Difference Equ. Appl., 17 (2011), 9-31.  Google Scholar [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. Google Scholar [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.  Google Scholar [22] C. Mira, "Chaotic Dynamics. From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism," World Scientific Publishing Co., Singapore, 1987.  Google Scholar [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.  Google Scholar [24] G. Papaschinopoulos, Exponential dichotomy for almost periodic linear difference equations, Ann. Soc. Sci. Bruxelles Sér. I, 102 (1988), 19-28.  Google Scholar [25] C. Pötzsche and S. Siegmund, $C^m$ -smoothness of invariant fiber bundles, Topol. Methods Nonlinear Anal., 24 (2004), 107-145.  Google Scholar [26] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  Google Scholar [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.  Google Scholar
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