American Institute of Mathematical Sciences

Advanced
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

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 & 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. 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

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.  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

[1]

Alberto Cabada, J. Ángel Cid. Heteroclinic solutions for non-autonomous boundary value problems with singular $\Phi$-Laplacian operators. Conference Publications, 2009, 2009 (Special) : 118-122. doi: 10.3934/proc.2009.2009.118
[2]

Michael Dellnitz, Christian Horenkamp. The efficient approximation of coherent pairs in non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2012, 32 (9) : 3029-3042. doi: 10.3934/dcds.2012.32.3029
[3]

Barbara Bianconi, Francesca Papalini. Non-autonomous boundary value problems on the real line. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 759-776. doi: 10.3934/dcds.2006.15.759
[4]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703
[5]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
[6]

Noriaki Yamazaki. Global attractors for non-autonomous multivalued dynamical systems associated with double obstacle problems. Conference Publications, 2003, 2003 (Special) : 935-944. doi: 10.3934/proc.2003.2003.935
[7]

Pablo G. Barrientos, Abbas Fakhari. Ergodicity of non-autonomous discrete systems with non-uniform expansion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1361-1382. doi: 10.3934/dcdsb.2019231
[8]

Chantelle Blachut, Cecilia González-Tokman. A tale of two vortices: How numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems. Journal of Computational Dynamics, 2020, 7 (2) : 369-399. doi: 10.3934/jcd.2020015
[9]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[10]

Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211
[11]

Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120
[12]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
[13]

David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499
[14]

Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2635-2651. doi: 10.3934/dcdsb.2012.17.2635
[15]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[16]

Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569
[17]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1997-2013. doi: 10.3934/cpaa.2020088
[18]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281
[19]

Bixiang Wang. Multivalued non-autonomous random dynamical systems for wave equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2011-2051. doi: 10.3934/dcdsb.2017119
[20]

Javier Fernández, Sebastián Elías Graiff Zurita, Sergio Grillo. Error analysis of forced discrete mechanical systems. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021017

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences