
-
Previous Article
A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games
- DCDS-B Home
- This Issue
-
Next Article
A continuous-time stochastic model of cell motion in the presence of a chemoattractant
Using automatic differentiation to compute periodic orbits of delay differential equations
Departament de Matemàtiques i Informàtica, Barcelona Graduate School of Mathematics (BGSMath), Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain |
In this paper we focus on the computation of periodic solutions of Delay Differential Equations (DDEs) with constant delays. The method is based on defining a Poincaré section in a suitable functional space and looking for a fixed point of the flow in this section. This is done by applying a Newton method on a suitable discretisation of the section. To avoid computing and storing large matrices we use a GMRES method to solve the linear system because in this case GMRES converges very fast due to the compactness of the flow of the DDE. The derivatives of the Poincaré map are obtained in a simple way, by applying Automatic Differentiation to the numerical integration. The stability of the periodic orbit is also obtained in a very efficient way by means of Arnoldi methods. The examples considered include temporal and spatial Poincaré sections.
References:
[1] |
Z.-Z. Bai,
Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems, Appl. Math. Comput., 109 (2000), 273-285.
doi: 10.1016/S0096-3003(99)00027-2. |
[2] |
R. Baltensperger, J.-P. Berrut and B. Noël,
Exponential convergence of a linear rational interpolant between transformed Chebyshev points, Math. Comp., 68 (199), 1109-1120.
doi: 10.1090/S0025-5718-99-01070-4. |
[3] |
A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical
Mathematics and Scientific Computation, Oxford University Press, Oxford, 2013. |
[4] |
J. Duintjer Tebbens and G. Meurant,
Any Ritz value behavior is possible for Arnoldi and for GMRES, SIAM J. Matrix Anal. Appl., 33 (2012), 958-978.
doi: 10.1137/110843666. |
[5] |
J. Gimeno, À. Jorba, M. Jorba-Cuscó, N. Miguel and M. Zou, Numerical integration of high order variational equations of ODEs, Preprint, (2020). |
[6] |
A. Griewank and G. F. Corliss, editors, Automatic Differentiation of Algorithms: Theory, Implementation, and Application, SIAM, Philadelphia, Penn., 1991. |
[7] |
A. Griewank, Evaluating Derivatives: Principles and techniques of algorithmic differentiation, Frontiers in Applied Mathematics, 19. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. |
[8] |
J. Hale, Theory of Functional Differential Equations, Second edition, Applied Mathematical Sciences, Vol. 3. Springer-Verlag, New York-Heidelberg, 1977. |
[9] |
E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations. I. {N}onstiff Problems, Springer Series in Computational Mathematics, 8. Springer-Verlag, Berlin, 1987.
doi: 10.1007/978-3-662-12607-3. |
[10] |
Y. Hino, S. Murakami and T. Naito., Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics, 1473. Springer-Verlag, Berlin, 1991.
doi: 10.1007/BFb0084432. |
[11] |
À. Jorba and M. R. Zou,
A software package for the numerical integration of ODEs by means of high-order Taylor methods, Exp. Math., 14 (2005), 99-117.
doi: 10.1080/10586458.2005.10128904. |
[12] |
G. Kiss and J.-P. Lessard,
Computational fixed-point theory for differential delay equations with multiple time lags, J. Differential Equations, 252 (2012), 3093-3115.
doi: 10.1016/j.jde.2011.11.020. |
[13] |
R. B. Lehoucq, D. C. Sorensen and C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, Software, Environments, and Tools, 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998.
doi: 10.1137/1.9780898719628. |
[14] |
T. Luzyanina, K. Engelborghs, K. Lust and D. Roose,
Computation, continuation and bifurcation analysis of periodic solutions of delay differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 7 (1997), 2547-2560.
doi: 10.1142/S0218127497001709. |
[15] |
U. Naumann, The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation, Software, Environments, and Tools, 24. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012. |
[16] |
R. D. Nussbaum, Differential-delay equations with two time lags, Mem. Amer. Math. Soc., 16 (1978).
doi: 10.1090/memo/0205. |
[17] |
Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, second edition, 2003.
doi: 10.1137/1.9780898718003. |
[18] |
J. Sieber, K. Engelborghs, T. Luzyanina, G. Samaey and D. Roose, DDE-BIFTOOL v. 3.0 Manual - Bifurcation analysis of delay differential equations, 125 (2015), 265–275, arXiv: 1406.7144. |
[19] |
C. Simó, On the analytical and numerical approximation of invariant manifolds, Modern methods in celestial mechanics, Ed. Frontières, (1990), 285–300. |
[20] |
A. L. Skubachevskii and H.-O. Walther,
On the Floquet multipliers of periodic solutions to non-linear functional differential equations, J. Dynam. Differential Equations, 18 (2006), 257-355.
doi: 10.1007/s10884-006-9006-5. |
[21] |
D. C. Sorensen, Implicitly restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations, Parallel Numerical Algorithms (Hampton, VA, 1994), ICASE/LaRC Interdiscip.
Ser. Sci. Eng., Kluwer Acad. Publ., Dordrecht, 4 (1997), 119–165.
doi: 10.1007/978-94-011-5412-3_5. |
[22] |
R. Szczelina and P. Zgliczyński,
Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation, Found. Comput. Math., 18 (2018), 1299-1332.
doi: 10.1007/s10208-017-9369-5. |
show all references
References:
[1] |
Z.-Z. Bai,
Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems, Appl. Math. Comput., 109 (2000), 273-285.
doi: 10.1016/S0096-3003(99)00027-2. |
[2] |
R. Baltensperger, J.-P. Berrut and B. Noël,
Exponential convergence of a linear rational interpolant between transformed Chebyshev points, Math. Comp., 68 (199), 1109-1120.
doi: 10.1090/S0025-5718-99-01070-4. |
[3] |
A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical
Mathematics and Scientific Computation, Oxford University Press, Oxford, 2013. |
[4] |
J. Duintjer Tebbens and G. Meurant,
Any Ritz value behavior is possible for Arnoldi and for GMRES, SIAM J. Matrix Anal. Appl., 33 (2012), 958-978.
doi: 10.1137/110843666. |
[5] |
J. Gimeno, À. Jorba, M. Jorba-Cuscó, N. Miguel and M. Zou, Numerical integration of high order variational equations of ODEs, Preprint, (2020). |
[6] |
A. Griewank and G. F. Corliss, editors, Automatic Differentiation of Algorithms: Theory, Implementation, and Application, SIAM, Philadelphia, Penn., 1991. |
[7] |
A. Griewank, Evaluating Derivatives: Principles and techniques of algorithmic differentiation, Frontiers in Applied Mathematics, 19. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. |
[8] |
J. Hale, Theory of Functional Differential Equations, Second edition, Applied Mathematical Sciences, Vol. 3. Springer-Verlag, New York-Heidelberg, 1977. |
[9] |
E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations. I. {N}onstiff Problems, Springer Series in Computational Mathematics, 8. Springer-Verlag, Berlin, 1987.
doi: 10.1007/978-3-662-12607-3. |
[10] |
Y. Hino, S. Murakami and T. Naito., Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics, 1473. Springer-Verlag, Berlin, 1991.
doi: 10.1007/BFb0084432. |
[11] |
À. Jorba and M. R. Zou,
A software package for the numerical integration of ODEs by means of high-order Taylor methods, Exp. Math., 14 (2005), 99-117.
doi: 10.1080/10586458.2005.10128904. |
[12] |
G. Kiss and J.-P. Lessard,
Computational fixed-point theory for differential delay equations with multiple time lags, J. Differential Equations, 252 (2012), 3093-3115.
doi: 10.1016/j.jde.2011.11.020. |
[13] |
R. B. Lehoucq, D. C. Sorensen and C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, Software, Environments, and Tools, 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998.
doi: 10.1137/1.9780898719628. |
[14] |
T. Luzyanina, K. Engelborghs, K. Lust and D. Roose,
Computation, continuation and bifurcation analysis of periodic solutions of delay differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 7 (1997), 2547-2560.
doi: 10.1142/S0218127497001709. |
[15] |
U. Naumann, The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation, Software, Environments, and Tools, 24. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012. |
[16] |
R. D. Nussbaum, Differential-delay equations with two time lags, Mem. Amer. Math. Soc., 16 (1978).
doi: 10.1090/memo/0205. |
[17] |
Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, second edition, 2003.
doi: 10.1137/1.9780898718003. |
[18] |
J. Sieber, K. Engelborghs, T. Luzyanina, G. Samaey and D. Roose, DDE-BIFTOOL v. 3.0 Manual - Bifurcation analysis of delay differential equations, 125 (2015), 265–275, arXiv: 1406.7144. |
[19] |
C. Simó, On the analytical and numerical approximation of invariant manifolds, Modern methods in celestial mechanics, Ed. Frontières, (1990), 285–300. |
[20] |
A. L. Skubachevskii and H.-O. Walther,
On the Floquet multipliers of periodic solutions to non-linear functional differential equations, J. Dynam. Differential Equations, 18 (2006), 257-355.
doi: 10.1007/s10884-006-9006-5. |
[21] |
D. C. Sorensen, Implicitly restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations, Parallel Numerical Algorithms (Hampton, VA, 1994), ICASE/LaRC Interdiscip.
Ser. Sci. Eng., Kluwer Acad. Publ., Dordrecht, 4 (1997), 119–165.
doi: 10.1007/978-94-011-5412-3_5. |
[22] |
R. Szczelina and P. Zgliczyński,
Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation, Found. Comput. Math., 18 (2018), 1299-1332.
doi: 10.1007/s10208-017-9369-5. |







[1] |
Ferenc A. Bartha, Hans Z. Munthe-Kaas. Computing of B-series by automatic differentiation. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 903-914. doi: 10.3934/dcds.2014.34.903 |
[2] |
Tom Maertens, Joris Walraevens, Herwig Bruneel. Controlling delay differentiation with priority jumps: Analytical study. Numerical Algebra, Control and Optimization, 2011, 1 (4) : 657-673. doi: 10.3934/naco.2011.1.657 |
[3] |
Bun Theang Ong, Masao Fukushima. Global optimization via differential evolution with automatic termination. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 57-67. doi: 10.3934/naco.2012.2.57 |
[4] |
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 |
[5] |
Hermann Brunner, Stefano Maset. Time transformations for delay differential equations. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 751-775. doi: 10.3934/dcds.2009.25.751 |
[6] |
Klaudiusz Wójcik, Piotr Zgliczyński. Topological horseshoes and delay differential equations. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 827-852. doi: 10.3934/dcds.2005.12.827 |
[7] |
Serhiy Yanchuk, Leonhard Lücken, Matthias Wolfrum, Alexander Mielke. Spectrum and amplitude equations for scalar delay-differential equations with large delay. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 537-553. doi: 10.3934/dcds.2015.35.537 |
[8] |
Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057 |
[9] |
Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215 |
[10] |
Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633 |
[11] |
Sana Netchaoui, Mohamed Ali Hammami, Tomás Caraballo. Pullback exponential attractors for differential equations with delay. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1345-1358. doi: 10.3934/dcdss.2020367 |
[12] |
Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361 |
[13] |
Eduardo Liz, Manuel Pinto, Gonzalo Robledo, Sergei Trofimchuk, Victor Tkachenko. Wright type delay differential equations with negative Schwarzian. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 309-321. doi: 10.3934/dcds.2003.9.309 |
[14] |
Igor Chueshov, Michael Scheutzow. Invariance and monotonicity for stochastic delay differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1533-1554. doi: 10.3934/dcdsb.2013.18.1533 |
[15] |
C. M. Groothedde, J. D. Mireles James. Parameterization method for unstable manifolds of delay differential equations. Journal of Computational Dynamics, 2017, 4 (1&2) : 21-70. doi: 10.3934/jcd.2017002 |
[16] |
Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577 |
[17] |
Zhengxin Zhou. On the Poincaré mapping and periodic solutions of nonautonomous differential systems. Communications on Pure and Applied Analysis, 2007, 6 (2) : 541-547. doi: 10.3934/cpaa.2007.6.541 |
[18] |
Benjamin Seibold, Rodolfo R. Rosales, Jean-Christophe Nave. Jet schemes for advection problems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1229-1259. doi: 10.3934/dcdsb.2012.17.1229 |
[19] |
Mickaël D. Chekroun, Michael Ghil, Honghu Liu, Shouhong Wang. Low-dimensional Galerkin approximations of nonlinear delay differential equations. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4133-4177. doi: 10.3934/dcds.2016.36.4133 |
[20] |
Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]