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December  2020, 25(12): 4853-4867. doi: 10.3934/dcdsb.2020130

## 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

* Corresponding author: Joan Gimeno

Received  October 2019 Revised  January 2020 Published  December 2020 Early access  April 2020

Fund Project: This work has been supported by the Spanish grants PGC2018-100699-B-I00 (MCIU/AEI/FEDER, UE) and the Catalan grant 2017 SGR 1374. The project leading to this application has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 734557

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.

Citation: Joan Gimeno, Àngel Jorba. Using automatic differentiation to compute periodic orbits of delay differential equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4853-4867. doi: 10.3934/dcdsb.2020130
##### 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.
Space Poincaré section $\sigma(x)$ starting with initial symbols $\boldsymbol{u}'$. After $t_S$ units of time the solution is crossing the section again with directional derivatives $\boldsymbol{v}'$ but it may not be in the section, so a projection to it can be done with the tangent vector $\frac{d}{dt} x_{t_S}$ and the normal to the section $\frac{d}{dx}s(x_{t_S})$
Continuation of periodic orbits of (4.3) with respect to parameters; (a) has $\varepsilon = 10^{-4}$ and $\tau = 1$, (b) has $\alpha = 1.5$ and $\tau = 1$, and (c) has $\alpha = 1.5$ and $\varepsilon = 10^{-4}$. Black colour means stable
Periodic orbit of the equation (4.1). In the left hand side, the orbit is displayed with the initial condition in $-1\leq t \leq 0$ and final lag-segment once the second has been crossed two times. The phase space of the periodic orbit is shown in the right hand side
Continuation of periodic orbits with respect to $\alpha$ starting at the periodic orbit computed for $\alpha = 1.57$, $\varepsilon = 10^{-4}$ and $\tau = 1$ of Equation (4.1). The $y$-axis represents the $\infty$-norm of the initial condition of each of the periodic orbits. Black colour means stable
Same continuation as in Figure 4.3, left. Left plot: positions vs. time. Right plot: derivatives vs. positions
Same continuation as in Figure 4.3, left, now showing the evolution of the spectral radius. We have added a horizontal straight line at 1 to visualise the changes of stability
Periodic orbit of (4.4) with parameters $\lambda_1 = \lambda_2 = 2.5$, $\lambda_3 = 0.25$, $\tau_1 = 1.65$, $\tau_2 = 0.35$ and $\tau_3 = 1$. The final period is almost $3.5894$ units of time
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