January  2016, 3(1): 93-112. doi: 10.3934/jcd.2016005

On the computation of attractors for delay differential equations

1. 

Institute for Mathematics, University of Paderborn, D-33095 Paderborn

2. 

Department of Mathematics, Paderborn University, 33095 Paderborn, Germany, Germany

Received  September 2015 Revised  March 2016 Published  October 2016

In this work we present a novel framework for the computation of finite dimensional invariant sets of infinite dimensional dynamical systems. It extends a classical subdivision technique [7] for the computation of such objects of finite dimensional systems to the infinite dimensional case by utilizing results on embedding techniques for infinite dimensional systems. We show how to implement this approach for the analysis of delay differential equations and illustrate the feasibility of our implementation by computing invariant sets for three different delay differential equations.
Citation: Michael Dellnitz, Mirko Hessel-Von Molo, Adrian Ziessler. On the computation of attractors for delay differential equations. Journal of Computational Dynamics, 2016, 3 (1) : 93-112. doi: 10.3934/jcd.2016005
References:
[1]

A. Arneodo, P. H. Coullet, E. A. Spiegel and C. Tresser, Asymptotic chaos,, Physica D: Nonlinear Phenomena, 14 (1985), 327.  doi: 10.1016/0167-2789(85)90093-4.  Google Scholar

[2]

A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations,, Oxford University Press, (2013).   Google Scholar

[3]

C. Chicone, Inertial and slow manifolds for delay equations with small delays,, Journal of Differential Equations, 190 (2003), 364.  doi: 10.1016/S0022-0396(02)00148-1.  Google Scholar

[4]

P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations,, Springer-Verlag, (1989).  doi: 10.1007/978-1-4612-3506-4.  Google Scholar

[5]

J. D. Crawford and S. Omohundro, On the global structure of period doubling flows,, Physica D: Nonlinear Phenomena, 13 (1984), 161.  doi: 10.1016/0167-2789(84)90275-6.  Google Scholar

[6]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems,, in Ergodic Theory, (2001), 145.   Google Scholar

[7]

M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors,, Numerische Mathematik, 75 (1997), 293.  doi: 10.1007/s002110050240.  Google Scholar

[8]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491.  doi: 10.1137/S0036142996313002.  Google Scholar

[9]

M. Dellnitz, O. Junge, M. Lo, J. E. Marsden, K. Padberg, R. Preis, S. Ross and B. Thiere, Transport of Mars-crossing asteroids from the quasi-Hilda region,, Physical Review Letters, 94 (2005).  doi: 10.1103/PhysRevLett.94.231102.  Google Scholar

[10]

R. D. Driver, On Ryabov's asymptotic characterization of the solutions of quasi-linear differential equations with small delays,, SIAM Review, 10 (1968), 329.  doi: 10.1137/1010058.  Google Scholar

[11]

J. Dugundji, An extension of Tietze's theorem,, Pacific J. Math., 1 (1951), 353.  doi: 10.2140/pjm.1951.1.353.  Google Scholar

[12]

N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory,, Interscience Publishers, (1957).   Google Scholar

[13]

J. D. Farmer, Chaotic attractors of an infinite-dimensional dynamical system,, Physica D, 4 (1982), 366.  doi: 10.1016/0167-2789(82)90042-2.  Google Scholar

[14]

C. Foias, M. Jolly, I. Kevrekidis, G. Sell and E. Titi, On the computation of inertial manifolds,, Physics Letters A, 131 (1988), 433.  doi: 10.1016/0375-9601(88)90295-2.  Google Scholar

[15]

G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost invariant sets and cycles,, SIAM Journal on Scientific Computing, 24 (2003), 1839.  doi: 10.1137/S106482750238911X.  Google Scholar

[16]

G. Froyland, C. Horenkamp, V. Rossi, N. Santitissadeekorn and A. Sen Gupta, Three-dimensional characterization and tracking of an Agulhas ring,, Ocean Modelling, 52 (2012), 69.   Google Scholar

[17]

G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems,, Physica D: Nonlinear Phenomena, 239 (2010), 1527.  doi: 10.1016/j.physd.2010.03.009.  Google Scholar

[18]

C. W. Gear, T. J. Kaper, I. G. Kevrekidis and A. Zagaris, Projecting to a slow manifold: Singularly perturbed systems and legacy codes,, SIAM Journal on Applied Dynamical Systems, 4 (2005), 711.  doi: 10.1137/040608295.  Google Scholar

[19]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Applied mathematical sciences, (1993).  doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[20]

B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces,, Nonlinearity, 12 (1999), 1263.  doi: 10.1088/0951-7715/12/5/303.  Google Scholar

[21]

B. Krauskopf and H. Osinga, Two-dimensional global manifolds of vector fields,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 9 (1999), 768.  doi: 10.1063/1.166450.  Google Scholar

[22]

I. Kukavica and J. C. Robinson, Distinguishing smooth functions by a finite number of point values, and a version of the Takens embedding theorem,, Physica D: Nonlinear Phenomena, 196 (2004), 45.  doi: 10.1016/j.physd.2004.04.004.  Google Scholar

[23]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Science, 197 (1977), 287.  doi: 10.1126/science.267326.  Google Scholar

[24]

I. Mezić and A. Banaszuk, Comparison of systems with complex behavior,, Physica D: Nonlinear Phenomena, 197 (2004), 101.  doi: 10.1016/j.physd.2004.06.015.  Google Scholar

[25]

J. C. Robinson, A topological delay embedding theorem for infinite-dimensional dynamical systems,, Nonlinearity, 18 (2005), 2135.  doi: 10.1088/0951-7715/18/5/013.  Google Scholar

[26]

T. Sahai and A. Vladimirsky, Numerical methods for approximating invariant manifolds of delayed systems,, SIAM J. Applied Dynamical Systems, 8 (2009), 1116.  doi: 10.1137/080718772.  Google Scholar

[27]

T. Sauer, J. A. Yorke and M. Casdagli, Embedology,, J. Stat. Phys., 65 (1991), 579.  doi: 10.1007/BF01053745.  Google Scholar

[28]

C. Schütte, W. Huisinga and P. Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems,, in Ergodic Theory, (2001), 191.   Google Scholar

[29]

J. Stark, Delay embeddings for forced systems. I. Deterministic forcing,, Journal of Nonlinear Science, 9 (1999), 255.  doi: 10.1007/s003329900072.  Google Scholar

[30]

F. Takens, Detecting strange attractors in turbulence,, Springer Lecture Notes in Mathematics, 898 (1981), 366.   Google Scholar

[31]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997), 978.  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[32]

C. Vandekerckhove, I. Kevrekidis and D. Roose, An efficient Newton-Krylov implementation of the constrained runs scheme for initializing on a slow manifold,, Journal of Scientific Computing, 39 (2009), 167.  doi: 10.1007/s10915-008-9256-y.  Google Scholar

[33]

S. Willard, General Topology,, Addison-Wesley, (1970).   Google Scholar

show all references

References:
[1]

A. Arneodo, P. H. Coullet, E. A. Spiegel and C. Tresser, Asymptotic chaos,, Physica D: Nonlinear Phenomena, 14 (1985), 327.  doi: 10.1016/0167-2789(85)90093-4.  Google Scholar

[2]

A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations,, Oxford University Press, (2013).   Google Scholar

[3]

C. Chicone, Inertial and slow manifolds for delay equations with small delays,, Journal of Differential Equations, 190 (2003), 364.  doi: 10.1016/S0022-0396(02)00148-1.  Google Scholar

[4]

P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations,, Springer-Verlag, (1989).  doi: 10.1007/978-1-4612-3506-4.  Google Scholar

[5]

J. D. Crawford and S. Omohundro, On the global structure of period doubling flows,, Physica D: Nonlinear Phenomena, 13 (1984), 161.  doi: 10.1016/0167-2789(84)90275-6.  Google Scholar

[6]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems,, in Ergodic Theory, (2001), 145.   Google Scholar

[7]

M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors,, Numerische Mathematik, 75 (1997), 293.  doi: 10.1007/s002110050240.  Google Scholar

[8]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491.  doi: 10.1137/S0036142996313002.  Google Scholar

[9]

M. Dellnitz, O. Junge, M. Lo, J. E. Marsden, K. Padberg, R. Preis, S. Ross and B. Thiere, Transport of Mars-crossing asteroids from the quasi-Hilda region,, Physical Review Letters, 94 (2005).  doi: 10.1103/PhysRevLett.94.231102.  Google Scholar

[10]

R. D. Driver, On Ryabov's asymptotic characterization of the solutions of quasi-linear differential equations with small delays,, SIAM Review, 10 (1968), 329.  doi: 10.1137/1010058.  Google Scholar

[11]

J. Dugundji, An extension of Tietze's theorem,, Pacific J. Math., 1 (1951), 353.  doi: 10.2140/pjm.1951.1.353.  Google Scholar

[12]

N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory,, Interscience Publishers, (1957).   Google Scholar

[13]

J. D. Farmer, Chaotic attractors of an infinite-dimensional dynamical system,, Physica D, 4 (1982), 366.  doi: 10.1016/0167-2789(82)90042-2.  Google Scholar

[14]

C. Foias, M. Jolly, I. Kevrekidis, G. Sell and E. Titi, On the computation of inertial manifolds,, Physics Letters A, 131 (1988), 433.  doi: 10.1016/0375-9601(88)90295-2.  Google Scholar

[15]

G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost invariant sets and cycles,, SIAM Journal on Scientific Computing, 24 (2003), 1839.  doi: 10.1137/S106482750238911X.  Google Scholar

[16]

G. Froyland, C. Horenkamp, V. Rossi, N. Santitissadeekorn and A. Sen Gupta, Three-dimensional characterization and tracking of an Agulhas ring,, Ocean Modelling, 52 (2012), 69.   Google Scholar

[17]

G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems,, Physica D: Nonlinear Phenomena, 239 (2010), 1527.  doi: 10.1016/j.physd.2010.03.009.  Google Scholar

[18]

C. W. Gear, T. J. Kaper, I. G. Kevrekidis and A. Zagaris, Projecting to a slow manifold: Singularly perturbed systems and legacy codes,, SIAM Journal on Applied Dynamical Systems, 4 (2005), 711.  doi: 10.1137/040608295.  Google Scholar

[19]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Applied mathematical sciences, (1993).  doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[20]

B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces,, Nonlinearity, 12 (1999), 1263.  doi: 10.1088/0951-7715/12/5/303.  Google Scholar

[21]

B. Krauskopf and H. Osinga, Two-dimensional global manifolds of vector fields,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 9 (1999), 768.  doi: 10.1063/1.166450.  Google Scholar

[22]

I. Kukavica and J. C. Robinson, Distinguishing smooth functions by a finite number of point values, and a version of the Takens embedding theorem,, Physica D: Nonlinear Phenomena, 196 (2004), 45.  doi: 10.1016/j.physd.2004.04.004.  Google Scholar

[23]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Science, 197 (1977), 287.  doi: 10.1126/science.267326.  Google Scholar

[24]

I. Mezić and A. Banaszuk, Comparison of systems with complex behavior,, Physica D: Nonlinear Phenomena, 197 (2004), 101.  doi: 10.1016/j.physd.2004.06.015.  Google Scholar

[25]

J. C. Robinson, A topological delay embedding theorem for infinite-dimensional dynamical systems,, Nonlinearity, 18 (2005), 2135.  doi: 10.1088/0951-7715/18/5/013.  Google Scholar

[26]

T. Sahai and A. Vladimirsky, Numerical methods for approximating invariant manifolds of delayed systems,, SIAM J. Applied Dynamical Systems, 8 (2009), 1116.  doi: 10.1137/080718772.  Google Scholar

[27]

T. Sauer, J. A. Yorke and M. Casdagli, Embedology,, J. Stat. Phys., 65 (1991), 579.  doi: 10.1007/BF01053745.  Google Scholar

[28]

C. Schütte, W. Huisinga and P. Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems,, in Ergodic Theory, (2001), 191.   Google Scholar

[29]

J. Stark, Delay embeddings for forced systems. I. Deterministic forcing,, Journal of Nonlinear Science, 9 (1999), 255.  doi: 10.1007/s003329900072.  Google Scholar

[30]

F. Takens, Detecting strange attractors in turbulence,, Springer Lecture Notes in Mathematics, 898 (1981), 366.   Google Scholar

[31]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997), 978.  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[32]

C. Vandekerckhove, I. Kevrekidis and D. Roose, An efficient Newton-Krylov implementation of the constrained runs scheme for initializing on a slow manifold,, Journal of Scientific Computing, 39 (2009), 167.  doi: 10.1007/s10915-008-9256-y.  Google Scholar

[33]

S. Willard, General Topology,, Addison-Wesley, (1970).   Google Scholar

[1]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324

[2]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[3]

Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021004

[4]

Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390

[5]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[6]

John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044

[7]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042

[8]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[9]

Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020409

[10]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020399

[11]

Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028

[12]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[13]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[14]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[15]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[16]

The Editors. The 2019 Michael Brin Prize in Dynamical Systems. Journal of Modern Dynamics, 2020, 16: 349-350. doi: 10.3934/jmd.2020013

[17]

Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003

[18]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020105

[19]

Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391

[20]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

 Impact Factor: 

Metrics

  • PDF downloads (187)
  • HTML views (0)
  • Cited by (2)

[Back to Top]