American Institute of Mathematical Sciences

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

On the computation of attractors for delay differential equations

Michael Dellnitz 1, , Mirko Hessel-Von Molo 2, and  Adrian Ziessler 2,

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.
Keywords: set oriented numerical methods., delay differential equations, Attractors, invariant manifolds, infinite dimensional dynamical systems, embedding theory.
Mathematics Subject Classification: 34K19, 34K23, 34K28, 37L25, 37L30, 65P9.
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), 327-347. doi: 10.1016/0167-2789(85)90093-4.

[2]

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

[3]

C. Chicone, Inertial and slow manifolds for delay equations with small delays, Journal of Differential Equations, 190 (2003), 364-406, URL http://www.sciencedirect.com/science/article/pii/S0022039602001481. doi: 10.1016/S0022-0396(02)00148-1.

[4]

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

[5]

J. D. Crawford and S. Omohundro, On the global structure of period doubling flows, Physica D: Nonlinear Phenomena, 13 (1984), 161-180, URL http://www.sciencedirect.com/science/article/pii/0167278984902756. doi: 10.1016/0167-2789(84)90275-6.

[6]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer-Verlag, Berlin, 2001, 145-174, 805-807.

[7]

M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numerische Mathematik, 75 (1997), 293-317. doi: 10.1007/s002110050240.

[8]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM Journal on Numerical Analysis, 36 (1999), 491-515, URL http://epubs.siam.org/sinum/resource/1/sjnaam/v36/i2/p491_s1. doi: 10.1137/S0036142996313002.

[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), 231102, 4pp. doi: 10.1103/PhysRevLett.94.231102.

[10]

R. D. Driver, On Ryabov's asymptotic characterization of the solutions of quasi-linear differential equations with small delays, SIAM Review, 10 (1968), 329-341. doi: 10.1137/1010058.

[11]

J. Dugundji, An extension of Tietze's theorem, Pacific J. Math., 1 (1951), 353-367, URL http://projecteuclid.org/euclid.pjm/1103052106. doi: 10.2140/pjm.1951.1.353.

[12]

N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, Interscience Publishers, Inc., 1957.

[13]

J. D. Farmer, Chaotic attractors of an infinite-dimensional dynamical system, Physica D, 4 (1982), 366-393. doi: 10.1016/0167-2789(82)90042-2.

[14]

C. Foias, M. Jolly, I. Kevrekidis, G. Sell and E. Titi, On the computation of inertial manifolds, Physics Letters A, 131 (1988), 433-436. doi: 10.1016/0375-9601(88)90295-2.

[15]

G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost invariant sets and cycles, SIAM Journal on Scientific Computing, 24 (2003), 1839-1863. doi: 10.1137/S106482750238911X.

[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), 69-75.

[17]

G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Physica D: Nonlinear Phenomena, 239 (2010), 1527-1541. doi: 10.1016/j.physd.2010.03.009.

[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), 711-732. doi: 10.1137/040608295.

[19]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied mathematical sciences, Springer-Verlag, New York, Berlin, Heidelberg, 1993. doi: 10.1007/978-1-4612-4342-7.

[20]

B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity, 12 (1999), 1263-1275, URL http://stacks.iop.org/0951-7715/12/i=5/a=303. doi: 10.1088/0951-7715/12/5/303.

[21]

B. Krauskopf and H. Osinga, Two-dimensional global manifolds of vector fields, Chaos: An Interdisciplinary Journal of Nonlinear Science, 9 (1999), 768-774. doi: 10.1063/1.166450.

[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), 45-66. doi: 10.1016/j.physd.2004.04.004.

[23]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.

[24]

I. Mezić and A. Banaszuk, Comparison of systems with complex behavior, Physica D: Nonlinear Phenomena, 197 (2004), 101-133, URL http://www.sciencedirect.com/science/article/pii/S0167278904002507. doi: 10.1016/j.physd.2004.06.015.

[25]

J. C. Robinson, A topological delay embedding theorem for infinite-dimensional dynamical systems, Nonlinearity, 18 (2005), 2135-2143. doi: 10.1088/0951-7715/18/5/013.

[26]

T. Sahai and A. Vladimirsky, Numerical methods for approximating invariant manifolds of delayed systems, SIAM J. Applied Dynamical Systems, 8 (2009), 1116-1135. doi: 10.1137/080718772.

[27]

T. Sauer, J. A. Yorke and M. Casdagli, Embedology, J. Stat. Phys., 65 (1991), 579-616. doi: 10.1007/BF01053745.

[28]

C. Schütte, W. Huisinga and P. Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer-Verlag, Berlin, 2001, 191-223.

[29]

J. Stark, Delay embeddings for forced systems. I. Deterministic forcing, Journal of Nonlinear Science, 9 (1999), 255-332. doi: 10.1007/s003329900072.

[30]

F. Takens, Detecting strange attractors in turbulence, Springer Lecture Notes in Mathematics, 898 (1981), 366-381.

[31]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997, URL http://dx.doi.org/10.1007/978-1-4612-0645-3_9. doi: 10.1007/978-1-4612-0645-3.

[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), 167-188. doi: 10.1007/s10915-008-9256-y.

[33]

S. Willard, General Topology, Addison-Wesley, Reading, Mass., 1970.

show all references

References:
[1]

A. Arneodo, P. H. Coullet, E. A. Spiegel and C. Tresser, Asymptotic chaos, Physica D: Nonlinear Phenomena, 14 (1985), 327-347. doi: 10.1016/0167-2789(85)90093-4.

[2]

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

[3]

C. Chicone, Inertial and slow manifolds for delay equations with small delays, Journal of Differential Equations, 190 (2003), 364-406, URL http://www.sciencedirect.com/science/article/pii/S0022039602001481. doi: 10.1016/S0022-0396(02)00148-1.

[4]

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

[5]

J. D. Crawford and S. Omohundro, On the global structure of period doubling flows, Physica D: Nonlinear Phenomena, 13 (1984), 161-180, URL http://www.sciencedirect.com/science/article/pii/0167278984902756. doi: 10.1016/0167-2789(84)90275-6.

[6]

M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer-Verlag, Berlin, 2001, 145-174, 805-807.

[7]

M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numerische Mathematik, 75 (1997), 293-317. doi: 10.1007/s002110050240.

[8]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM Journal on Numerical Analysis, 36 (1999), 491-515, URL http://epubs.siam.org/sinum/resource/1/sjnaam/v36/i2/p491_s1. doi: 10.1137/S0036142996313002.

[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), 231102, 4pp. doi: 10.1103/PhysRevLett.94.231102.

[10]

R. D. Driver, On Ryabov's asymptotic characterization of the solutions of quasi-linear differential equations with small delays, SIAM Review, 10 (1968), 329-341. doi: 10.1137/1010058.

[11]

J. Dugundji, An extension of Tietze's theorem, Pacific J. Math., 1 (1951), 353-367, URL http://projecteuclid.org/euclid.pjm/1103052106. doi: 10.2140/pjm.1951.1.353.

[12]

N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, Interscience Publishers, Inc., 1957.

[13]

J. D. Farmer, Chaotic attractors of an infinite-dimensional dynamical system, Physica D, 4 (1982), 366-393. doi: 10.1016/0167-2789(82)90042-2.

[14]

C. Foias, M. Jolly, I. Kevrekidis, G. Sell and E. Titi, On the computation of inertial manifolds, Physics Letters A, 131 (1988), 433-436. doi: 10.1016/0375-9601(88)90295-2.

[15]

G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost invariant sets and cycles, SIAM Journal on Scientific Computing, 24 (2003), 1839-1863. doi: 10.1137/S106482750238911X.

[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), 69-75.

[17]

G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Physica D: Nonlinear Phenomena, 239 (2010), 1527-1541. doi: 10.1016/j.physd.2010.03.009.

[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), 711-732. doi: 10.1137/040608295.

[19]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied mathematical sciences, Springer-Verlag, New York, Berlin, Heidelberg, 1993. doi: 10.1007/978-1-4612-4342-7.

[20]

B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity, 12 (1999), 1263-1275, URL http://stacks.iop.org/0951-7715/12/i=5/a=303. doi: 10.1088/0951-7715/12/5/303.

[21]

B. Krauskopf and H. Osinga, Two-dimensional global manifolds of vector fields, Chaos: An Interdisciplinary Journal of Nonlinear Science, 9 (1999), 768-774. doi: 10.1063/1.166450.

[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), 45-66. doi: 10.1016/j.physd.2004.04.004.

[23]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.

[24]

I. Mezić and A. Banaszuk, Comparison of systems with complex behavior, Physica D: Nonlinear Phenomena, 197 (2004), 101-133, URL http://www.sciencedirect.com/science/article/pii/S0167278904002507. doi: 10.1016/j.physd.2004.06.015.

[25]

J. C. Robinson, A topological delay embedding theorem for infinite-dimensional dynamical systems, Nonlinearity, 18 (2005), 2135-2143. doi: 10.1088/0951-7715/18/5/013.

[26]

T. Sahai and A. Vladimirsky, Numerical methods for approximating invariant manifolds of delayed systems, SIAM J. Applied Dynamical Systems, 8 (2009), 1116-1135. doi: 10.1137/080718772.

[27]

T. Sauer, J. A. Yorke and M. Casdagli, Embedology, J. Stat. Phys., 65 (1991), 579-616. doi: 10.1007/BF01053745.

[28]

C. Schütte, W. Huisinga and P. Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer-Verlag, Berlin, 2001, 191-223.

[29]

J. Stark, Delay embeddings for forced systems. I. Deterministic forcing, Journal of Nonlinear Science, 9 (1999), 255-332. doi: 10.1007/s003329900072.

[30]

F. Takens, Detecting strange attractors in turbulence, Springer Lecture Notes in Mathematics, 898 (1981), 366-381.

[31]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997, URL http://dx.doi.org/10.1007/978-1-4612-0645-3_9. doi: 10.1007/978-1-4612-0645-3.

[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), 167-188. doi: 10.1007/s10915-008-9256-y.

[33]

S. Willard, General Topology, Addison-Wesley, Reading, Mass., 1970.

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