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 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.

[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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems - B, 2017, 22 (5) : 2011-2051. doi: 10.3934/dcdsb.2017119

[20]

Radosław Czaja. Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021276

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]