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April  2022, 9(2): 103-121. doi: 10.3934/jcd.2022004

Piecewise discretization of monodromy operators of delay equations on adapted meshes

 CDLab – Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics, University of Udine, via delle Scienze 206, 33100 Udine UD, Italy

* Corresponding author: Davide Liessi

Received  March 2021 Revised  March 2022 Published  April 2022 Early access  April 2022

Periodic solutions of delay equations are usually approximated as continuous piecewise polynomials on meshes adapted to the solutions' profile. In practical computations this affects the regularity of the (coefficients of the) linearized system and, in turn, the effectiveness of assessing local stability by approximating the Floquet multipliers. To overcome this problem when computing multipliers by collocation, the discretization grid should include the piecewise adapted mesh of the computed periodic solution. By introducing a piecewise version of existing pseudospectral techniques, we explain why and show experimentally that this choice is essential in presence of either strong mesh adaptation or nontrivial multipliers whose eigenfunctions' profile is unrelated to that of the periodic solution.

Citation: Dimitri Breda, Davide Liessi, Rossana Vermiglio. Piecewise discretization of monodromy operators of delay equations on adapted meshes. Journal of Computational Dynamics, 2022, 9 (2) : 103-121. doi: 10.3934/jcd.2022004
References:
 [1] A. Andò, Collocation Methods for Complex Delay Problems of Structured Populations, PhD thesis, University of Udine, 2020. Available from: http://cdlab.uniud.it/theses/Ando2020.pdf. [2] A. Andò, Convergence of collocation methods for solving periodic boundary value problems for renewal equations defined through finite-dimensional boundary conditions, Comput. Math. Methods, 3 (2021), Paper No. e1190, 12 pp. doi: 10.1002/cmm4.1190. [3] A. Andò and D. Breda, Convergence analysis of collocation methods for computing periodic solutions of retarded functional differential equations, SIAM J. Numer. Anal., 58 (2020), 3010–3039. Full-length version at arxiv: 2008.07604 [math.NA]. doi: 10.1137/19M1295015. [4] A. Andò and D. Breda, Piecewise orthogonal collocation for computing periodic solutions of renewal equations, submitted. [5] U. M. Ascher, R. M. M. Mattheij and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice Hall, Englewood Cliffs, NJ, 1988. [6] G. Bader, Solving boundary value problems for functional differential equations by collation, in Numerical Boundary Value ODEs (eds. U. M. Ascher and R. D. Russell), Progr. Sci. Comput., Birkhäuser, Boston, 5 (1985), 227–243. [7] F. Borgioli, D. Hajdu, T. Insperger, G. Stépán and W. Michiels, Pseudospectral method for assessing stability robustness for linear time-periodic delayed dynamical systems, Internat. J. Numer. Methods Engrg., 121 (2020), 3505-3528.  doi: 10.1002/nme.6368. [8] D. Breda, O. Diekmann, D. Liessi and F. Scarabel, Numerical bifurcation analysis of a class of nonlinear renewal equations, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), Paper No. 65, 24 pp. doi: 10.14232/ejqtde.2016.1.65. [9] D. Breda and D. Liessi, Approximation of eigenvalues of evolution operators for linear renewal equations, SIAM J. Numer. Anal., 56 (2018), 1456-1481.  doi: 10.1137/17M1140534. [10] D. Breda and D. Liessi, Approximation of eigenvalues of evolution operators for linear coupled renewal and retarded functional differential equations, Ric. Mat., 69 (2020), 457-481.  doi: 10.1007/s11587-020-00513-9. [11] D. Breda and D. Liessi, Floquet theory and stability of periodic solutions of renewal equations, J. Dynam. Differential Equations, 33 (2021), 457-481.  doi: 10.1007/s10884-020-09826-7. [12] D. Breda, D. Liessi and R. Vermiglio, A practical guide to piecewise pseudospectral collocation for Floquet multipliers of delay equations in MATLAB, submitted. [13] D. Breda, S. Maset and R. Vermiglio, Approximation of eigenvalues of evolution operators for linear retarded functional differential equations, SIAM J. Numer. Anal., 50 (2012), 1456-1483.  doi: 10.1137/100815505. [14] D. Breda, S. Maset and R. Vermiglio, Stability of Linear Delay Differential Equations: A Numerical Approach with MATLAB, SpringerBriefs Control Autom. Robot., Springer, New York, 2015. doi: 10.1007/978-1-4939-2107-2. [15] A. M. Castelfranco and H. W. Stech, Periodic solutions in a model of recurrent neural feedback, SIAM J. Appl. Math., 47 (1987), 573-588.  doi: 10.1137/0147039. [16] O. Diekmann, P. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2008), 1023-1069.  doi: 10.1137/060659211. [17] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional-, Complex- and Nonlinear Analysis, no. 110 in Appl. Math. Sci., Springer, New York, 1995. doi: 10.1007/978-1-4612-4206-2. [18] O. Diekmann and S. M. Verduyn Lunel, Twin semigroups and delay equations, J. Differential Equations, 286 (2021), 332-410.  doi: 10.1016/j.jde.2021.02.052. [19] E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations, in Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems (eds. B. Krauskopf, H. M. Osinga and J. Galán-Vioque), no. 116 in Underst. Complex Syst., Springer, Dordrecht, 2007, 1–49. doi: 10.1007/978-1-4020-6356-5. [20] K. Engelborghs, T. Luzyanina, K. J. in 't Hout and D. Roose, Collocation methods for the computation of periodic solutions of delay differential equations, SIAM J. Sci. Comput., 22 (2001), 1593-1609.  doi: 10.1137/S1064827599363381. [21] K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Softw., 28 (2002), 1-21.  doi: 10.1145/513001.513002. [22] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, no. 99 in Appl. Math. Sci., Springer, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [23] D. Liessi, Pseudospectral Methods for the Stability of Periodic Solutions of Delay Models, PhD thesis, University of Udine, 2018. Available from: http://www.liessi.it/mathematics/phdthesis. [24] T. Luzyanina and K. Engelborghs, Computing Floquet multipliers for functional differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 2977-2989.  doi: 10.1142/S0218127402006291. [25] R. E. Plant, A FitzHugh differential-difference equation modeling recurrent neural feedback, SIAM J. Appl. Math., 40 (1981), 150-162.  doi: 10.1137/0140012. [26] H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4th edition, Prentice Hall, 2010. [27] J. Sieber, K. Engelborghs, T. Luzyanina, G. Samaey and D. Roose, DDE-BIFTOOLmanual: Bifurcation analysis of delay differential equations, arXiv: 1406.7144 [math.DS]. [28] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts Appl. Math., Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [29] L. N. Trefethen, Spectral Methods in MATLAB, Software Environ. Tools, Society for Industrial and Applied Mathematics, Philadelphia, 2000. doi: 10.1137/1.9780898719598. [30] S. Yanchuk, S. Ruschel, J. Sieber and M. Wolfrum, Temporal dissipative solitons in time-delay feedback systems, Phys. Rev. Lett., 123 (2019), 053901, 6pp. doi: 10.1103/PhysRevLett.123.053901.

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References:
 [1] A. Andò, Collocation Methods for Complex Delay Problems of Structured Populations, PhD thesis, University of Udine, 2020. Available from: http://cdlab.uniud.it/theses/Ando2020.pdf. [2] A. Andò, Convergence of collocation methods for solving periodic boundary value problems for renewal equations defined through finite-dimensional boundary conditions, Comput. Math. Methods, 3 (2021), Paper No. e1190, 12 pp. doi: 10.1002/cmm4.1190. [3] A. Andò and D. Breda, Convergence analysis of collocation methods for computing periodic solutions of retarded functional differential equations, SIAM J. Numer. Anal., 58 (2020), 3010–3039. Full-length version at arxiv: 2008.07604 [math.NA]. doi: 10.1137/19M1295015. [4] A. Andò and D. Breda, Piecewise orthogonal collocation for computing periodic solutions of renewal equations, submitted. [5] U. M. Ascher, R. M. M. Mattheij and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice Hall, Englewood Cliffs, NJ, 1988. [6] G. Bader, Solving boundary value problems for functional differential equations by collation, in Numerical Boundary Value ODEs (eds. U. M. Ascher and R. D. Russell), Progr. Sci. Comput., Birkhäuser, Boston, 5 (1985), 227–243. [7] F. Borgioli, D. Hajdu, T. Insperger, G. Stépán and W. Michiels, Pseudospectral method for assessing stability robustness for linear time-periodic delayed dynamical systems, Internat. J. Numer. Methods Engrg., 121 (2020), 3505-3528.  doi: 10.1002/nme.6368. [8] D. Breda, O. Diekmann, D. Liessi and F. Scarabel, Numerical bifurcation analysis of a class of nonlinear renewal equations, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), Paper No. 65, 24 pp. doi: 10.14232/ejqtde.2016.1.65. [9] D. Breda and D. Liessi, Approximation of eigenvalues of evolution operators for linear renewal equations, SIAM J. Numer. Anal., 56 (2018), 1456-1481.  doi: 10.1137/17M1140534. [10] D. Breda and D. Liessi, Approximation of eigenvalues of evolution operators for linear coupled renewal and retarded functional differential equations, Ric. Mat., 69 (2020), 457-481.  doi: 10.1007/s11587-020-00513-9. [11] D. Breda and D. Liessi, Floquet theory and stability of periodic solutions of renewal equations, J. Dynam. Differential Equations, 33 (2021), 457-481.  doi: 10.1007/s10884-020-09826-7. [12] D. Breda, D. Liessi and R. Vermiglio, A practical guide to piecewise pseudospectral collocation for Floquet multipliers of delay equations in MATLAB, submitted. [13] D. Breda, S. Maset and R. Vermiglio, Approximation of eigenvalues of evolution operators for linear retarded functional differential equations, SIAM J. Numer. Anal., 50 (2012), 1456-1483.  doi: 10.1137/100815505. [14] D. Breda, S. Maset and R. Vermiglio, Stability of Linear Delay Differential Equations: A Numerical Approach with MATLAB, SpringerBriefs Control Autom. Robot., Springer, New York, 2015. doi: 10.1007/978-1-4939-2107-2. [15] A. M. Castelfranco and H. W. Stech, Periodic solutions in a model of recurrent neural feedback, SIAM J. Appl. Math., 47 (1987), 573-588.  doi: 10.1137/0147039. [16] O. Diekmann, P. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2008), 1023-1069.  doi: 10.1137/060659211. [17] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional-, Complex- and Nonlinear Analysis, no. 110 in Appl. Math. Sci., Springer, New York, 1995. doi: 10.1007/978-1-4612-4206-2. [18] O. Diekmann and S. M. Verduyn Lunel, Twin semigroups and delay equations, J. Differential Equations, 286 (2021), 332-410.  doi: 10.1016/j.jde.2021.02.052. [19] E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations, in Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems (eds. B. Krauskopf, H. M. Osinga and J. Galán-Vioque), no. 116 in Underst. Complex Syst., Springer, Dordrecht, 2007, 1–49. doi: 10.1007/978-1-4020-6356-5. [20] K. Engelborghs, T. Luzyanina, K. J. in 't Hout and D. Roose, Collocation methods for the computation of periodic solutions of delay differential equations, SIAM J. Sci. Comput., 22 (2001), 1593-1609.  doi: 10.1137/S1064827599363381. [21] K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Softw., 28 (2002), 1-21.  doi: 10.1145/513001.513002. [22] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, no. 99 in Appl. Math. Sci., Springer, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [23] D. Liessi, Pseudospectral Methods for the Stability of Periodic Solutions of Delay Models, PhD thesis, University of Udine, 2018. Available from: http://www.liessi.it/mathematics/phdthesis. [24] T. Luzyanina and K. Engelborghs, Computing Floquet multipliers for functional differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 2977-2989.  doi: 10.1142/S0218127402006291. [25] R. E. Plant, A FitzHugh differential-difference equation modeling recurrent neural feedback, SIAM J. Appl. Math., 40 (1981), 150-162.  doi: 10.1137/0140012. [26] H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4th edition, Prentice Hall, 2010. [27] J. Sieber, K. Engelborghs, T. Luzyanina, G. Samaey and D. Roose, DDE-BIFTOOLmanual: Bifurcation analysis of delay differential equations, arXiv: 1406.7144 [math.DS]. [28] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts Appl. Math., Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [29] L. N. Trefethen, Spectral Methods in MATLAB, Software Environ. Tools, Society for Industrial and Applied Mathematics, Philadelphia, 2000. doi: 10.1137/1.9780898719598. [30] S. Yanchuk, S. Ruschel, J. Sieber and M. Wolfrum, Temporal dissipative solitons in time-delay feedback systems, Phys. Rev. Lett., 123 (2019), 053901, 6pp. doi: 10.1103/PhysRevLett.123.053901.
Periodic solutions (rescaled to period $1$) of (11) computed by DDE-BIFTOOL with $L_{\rm{DB}} = 30$ and $m_{\rm{DB}} = 6$
Absolute errors of $\mathtt{eigTMN}$, varying $M = N$, on the trivial (A–D) and dominant nontrivial (E–H) multipliers of (11) linearized around the solutions computed by DDE-BIFTOOL. The gray horizontal lines show DDE-BIFTOOL's errors on the same multipliers. The reference values for the nontrivial multipliers are computed by DDE-BIFTOOL with $L_{\rm{DB}} = 60$ and $m_{\rm{DB}} = 10$. $\mathtt{eigTMN}$'s errors eventually decay with infinite order with barriers comparable to DDE-BIFTOOL's errors, except for the nontrivial multiplier with $r = 3$ (H), which seems to need even higher $M=N$. For the trivial multiplier, as $r$ increases (D), the errors initially rise exponentially (exceeding $10^7$ for $r = 3$), complicating the choice of $M = N$
Partitions of $[0, \omega]$ for the periodic solutions of Figure 1, showing mesh adaptation as performed by DDE-BIFTOOL. The vertical lines show the uniform partition
Example collocation grid with $\omega > \tau$, $L=4$ and $M = 3$. Ticks mark $t_{i}$ and $t_{i} - \omega$, the cross marks $-\tau$, dots mark the grid points
Absolute errors of $\mathtt{eigTMNpw}$ on the dominant multiplier of (12), whose coefficient is not differentiable at $1$. The reference value is computed by $\mathtt{eigTMNpw}$ with $L_{\rm{pw}} = 2$ and $M_{\rm{pw}} = 120$. When the mesh includes $1$ ($L_{\rm{pw}} = 2$) the convergence order is infinite, otherwise ($L_{\rm{pw}} = 1$) it is finite (precisely $2$)
Absolute errors of $\mathtt{eigTMNpw}$ on the trivial (left) and on the dominant nontrivial (right) multipliers of (15) with $\gamma = 4$ in (14). The reference value for the nontrivial multiplier is computed with $L_{\rm{pw}} = 40$ and $M_{\rm{pw}} = 15$. The convergence order is infinite for the SEM (top) and finite for the FEM (bottom, precisely $4$, $6$, $6$ and $8$ on the left and $4$, $6$, $6$ and $10$ on the right)
Periodic solution of (16) with $a = 0.7$, $b = 0.8$, $\eta = -2$, $r = 0.08$ and $\tau = 25$ ($\omega \approx 50.7326$, $v_{0} \approx -1.1994$), computed by DDE-BIFTOOL with $L_{\rm{DB}} = 30$ and $m_{\rm{DB}} = 5$
Partition of $[0, \omega]$ for the solution of Figure 7. The vertical lines show the uniform partition
Absolute errors of $\mathtt{eigTMN}$ for varying $M = N$ on the trivial (top) and dominant nontrivial (bottom) multipliers of (16) linearized around the solutions computed by DDE-BIFTOOL (parameters as in Figure 7), compared to the errors on the same multipliers of $\mathtt{eigTMNpw}$ with $L_{\rm{pw}} = L_{\rm{DB}}$ and $M_{\rm{pw}} = m_{\rm{DB}}$ and, for reference, of DDE-BIFTOOL (horizontal lines). The reference value for the nontrivial multiplier is computed by DDE-BIFTOOL with $L_{\rm{DB}} = 60$ and $m_{\rm{DB}} = 10$. While $\mathtt{eigTMNpw}$'s errors are comparable with those of DDE-BIFTOOL, $\mathtt{eigTMN}$'s ones either require very large $M = N$ to reach the desired magnitude (trivial multiplier) or exhibit an alternating behavior which makes choosing an appropriate $M = N$ difficult (nontrivial multiplier)
Value of the ratio $\rho$ for the solutions of (16) computed by DDE-BIFTOOL with $L_{\rm{DB}} = 30$ and $m_{\rm{DB}} = 5$ for varying $\tau$ (other parameters as in Figure 7)
Absolute errors of $\mathtt{eigTMN}$ and $\mathtt{eigTMNpw}$, compared to those of DDE-BIFTOOL, on the trivial (left) and dominant nontrivial (right) multipliers of (16) linearized around the solutions computed by DDE-BIFTOOL with $L_{\rm{DB}} = 30$, $m_{\rm{DB}} = 5$ and varying $\tau$ (other parameters as in Figure 7). In all cases $L_{\rm{pw}} = L_{\rm{DB}}$ and $M_{\rm{pw}} = m_{\rm{DB}}$ are used for $\mathtt{eigTMNpw}$, while the errors of $\mathtt{eigTMN}$ are the mean error for $M = N \in \{140, \dots, 160\}$. The reference value for the nontrivial multiplier is computed by DDE-BIFTOOL with $L_{\rm{DB}} = 60$ and $m_{\rm{DB}} = 10$. $\mathtt{eigTMNpw}$'s errors behave similarly to those of DDE-BIFTOOL, while $\mathtt{eigTMN}$'s ones gradually increase
Eigenfunctions relevant to the multiplier $\mu$ of (16) linearized around the solution computed by DDE-BIFTOOL with $L_{\rm{DB}} = 240$ and $m_{\rm{DB}} = 10$ (parameters as in Figure 7)
Absolute errors on the trivial multiplier (left) and on the nontrivial multiplier $\mu \approx 0.0612 \pm 0.0594\mathrm{i}$ (right) of (16) linearized around the solutions computed by DDE-BIFTOOL with $m_{\rm{DB}} = 5$ and varying $L_{\rm{DB}}$ (parameters as in Figure 7). For $\mathtt{eigTMNpw}$ $[0, \omega]$ is partitioned using the solution's mesh, a refinement of the latter (resulting in $129 \leq L_{\rm{pw}} \leq 153$), and a uniform mesh with $L_{\rm{pw}} = 153$ ($M_{\rm{pw}} = m_{\rm{DB}}$ in all cases). DDE-BIFTOOL uses the solution's mesh. The reference value for the nontrivial multiplier is computed by DDE-BIFTOOL with $L_{\rm{DB}} = 240$ and $m_{\rm{DB}} = 10$. If the eigenfunction has oscillations unrelated to the solution's mesh (right), using the latter seems to prevent the convergence, while a denser partition allows to reach fairly small error barriers, smaller if it is actually a refinement. For the trivial multiplier (left), the solution's mesh is well adapted also to the eigenfunction: the error vanishes similarly with both the original and the finer partition, while with the uniform one it reaches a barrier larger than in the other case
Absolute errors of $\mathtt{eigTMNpw}$ on the trivial and dominant nontrivial multipliers of (17) linearized around the solution of (16) computed by DDE-BIFTOOL with $L_{\rm{DB}} = 30$ and $m_{\rm{DB}} = 5$ (parameters as in Figure 7). $\mathtt{eigTMNpw}$ is used in a nonpiecewise fashion for varying $M_{\rm{pw}}$ and in a piecewise fashion with $L_{\rm{pw}} = L_{\rm{DB}}$ and $M_{\rm{pw}} = m_{\rm{DB}}$. The reference value for the nontrivial multiplier is computed by DDE-BIFTOOL for (16) with $L_{\rm{DB}} = 60$ and $m_{\rm{DB}} = 10$. $\mathtt{eigTMNpw}$'s errors are compared to those of DDE-BIFTOOL for (16): in the piecewise case the former are even more accurate than the latter
Absolute errors of $\mathtt{eigTMNpw}$ and DDE-BIFTOOL on the trivial and dominant nontrivial multipliers of (11) linearized around the solutions computed by DDE-BIFTOOL with $L_{\rm{DB}} = 30$ and $m_{\rm{DB}} = 6$. $\mathtt{eigTMNpw}$ uses $L_{\rm{pw}} = L_{\rm{DB}}$ and $M_{\rm{pw}} = m_{\rm{DB}}$. The reference values for the nontrivial multipliers are computed by DDE-BIFTOOL with $L_{\rm{DB}} = 60$ and $m_{\rm{DB}} = 10$. The errors are comparable in magnitude
 trivial multiplier dominant nontrivial multiplier r DDE-BIFTOOL $\mathtt{eigTMNpw}$ DDE-BIFTOOL $\mathtt{eigTMNpw}$ 1.6 7.270×10-12 9.353×10-13 7.889×10-13 8.943×10-12 2.3 4.463×10-10 2.444×10-10 1.243×10-13 6.597×10-12 3 1.577×10-4 3.443×10-4 5.082×10-16 3.960×10-15
 trivial multiplier dominant nontrivial multiplier r DDE-BIFTOOL $\mathtt{eigTMNpw}$ DDE-BIFTOOL $\mathtt{eigTMNpw}$ 1.6 7.270×10-12 9.353×10-13 7.889×10-13 8.943×10-12 2.3 4.463×10-10 2.444×10-10 1.243×10-13 6.597×10-12 3 1.577×10-4 3.443×10-4 5.082×10-16 3.960×10-15
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