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September  2020, 13(9): 2575-2602. doi: 10.3934/dcdss.2020196

## Pseudospectral discretization of delay differential equations in sun-star formulation: Results and conjectures

 1 Department of Mathematics, Utrecht University, P.O. Box 80010, 3508 TA Utrecht, The Netherlands 2 Department of Mathematics and Statistics, University of Helsinki, P.O. 68 (Pietari Kalmin katu 5), FI-00014 Helsinki, Finland 3 Department of Mathematics and Statistics, York University, 4700 Keele St, Toronto, ON M3J 1P3, Canada 4 Department of Mathematics, Computer Science and Physics, University of Udine, via delle Scienze 206, I-33100 Udine, Italy

* Corresponding author: Rossana Vermiglio

Received  January 2019 Revised  May 2019 Published  December 2019

Fund Project: The research of the second author was supported by Domast (Doctoral Programme in Mathematics and Statistics, University of Helsinki), and by the Centre of Excellence in Analysis and Dynamics Research, Academy of Finland. F.S. and R.V. are members of the INdAM Research group GNCS, and of CDLab (Computational Dynamics Laboratory), Department of Mathematics, Computer Science and Physics, University of Udine

In this paper we study the pseudospectral approximation of delay differential equations formulated as abstract differential equations in the $\odot*$-space. This formalism also allows us to define rigorously the abstract variation-of-constants formula, where the $\odot*$-shift operator plays a fundamental role. By applying the pseudospectral discretization technique we derive a system of ordinary differential equations, whose dynamics can be efficiently analyzed by existing bifurcation tools. To better understand to what extent the resulting finite-dimensional system "mimics" the dynamics of the original infinite-dimensional one, we study the pseudospectral approximations of the $\odot*$-shift operator and of the $\odot*$-generator in the supremum norm, which is the natural choice for delay differential equations, when the discretization parameter increases. In this context there are still open questions. We collect the most relevant results from the literature and we present some conjectures, supported by various numerical experiments, to illustrate the behavior w.r.t. the discretization parameter and to indicate the direction of ongoing and future research.

Citation: Odo Diekmann, Francesca Scarabel, Rossana Vermiglio. Pseudospectral discretization of delay differential equations in sun-star formulation: Results and conjectures. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2575-2602. doi: 10.3934/dcdss.2020196
##### References:
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##### References:
 [1] A. H. Al-Mohy and N. J. Higham, A new scaling and squaring algorithm for the matrix exponential, SIAM J. Matrix Anal. Appl., 31 (2009), 970-989.  doi: 10.1137/09074721X.  Google Scholar [2] A. Andò, D. Breda, L. Davide, S. Maset, F. Scarabel and R. Vermiglio, 15 years or so of pseudospectral collocation methods for stability and bifurcation of delay equations, in Advances on Delays and Dynamics, Springer, New York, 2019. Google Scholar [3] C. T. H. Baker, The Numerical Treatment of Integral Equations, Monographs on Numerical Analysis, Clarendon Press, Oxford, 1977.   Google Scholar [4] C. Baker, Numerical analysis of Volterra functional and integral equations, in The State of the Art in Numerical Analysis, Math. Appl. Conf. Ser. New Ser., 63, Oxford Univ. Press, New York, 1997.  Google Scholar [5] H. T. Banks, J. A. Burns and E. M. Cliff, Spline-based approximation methods for control and identification of hereditary systems, in International Symposium on Systems Optimization and Analysis, Lecture Notes in Control and Information Sci., 14, Springer, Berlin-New York, 1979, 314–320. doi: 10.1007/BFb0002662.  Google Scholar [6] A. Batkai and S. Piazzera, Semigroup for Delay Equations, Research Notes in Mathematics, 10, A K Peters, Ltd., Wellesley, MA, 2005.  Google Scholar [7] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, 2003.   Google Scholar [8] D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel and R. Vermiglio, Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Syst., 15 (2016), 1-23.  doi: 10.1137/15M1040931.  Google Scholar [9] D. Breda, O. Diekmann, D. Liessi and F. 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Left: plot of $\sigma(D_M)$ for different $M$, together with the line $\mathrm{Re} \lambda = \log \epsilon \approx -36.0437$ connected with the instability phenomenon studied in [63]. Right: plot of $\frac{-\alpha(D_M)}{\log(M)}$, versus $M$
Error $\| \psi_M - \psi\|$ for the resolvent operators applied to $\beta = 0$, $\varphi = 1$, versus $M$. Left: $\lambda = 1$; right: $\lambda = 10$. Note the spectral accuracy and that the convergence is slower when $\lambda$ has larger modulus
Left: plot of $\log \|{\rm e}^{D_M t}\|$ for $t\in [0, 2]$ with different values of $M$. Notice divergence in $[0, 1]$ for increasing $M$. Right: plot of $\max_{t\in I} \frac{\log \|{\rm e}^{D_M t}\|}{\log(M)}$ versus $M,$ for $I$ specified in the legend. Notice that the convergence gains one order at every time interval
$\|{\rm e}^{D_M t}\|$ computed with different routines: built-in Matlab function $text{expm}$ and $\text{norm(., inf)}$ versus maximum over $n = 100$ random initial vectors of the solution of the ODE system. Left: $\|{\rm e}^{D_M t}\|$ versus time. Right: $\max_{t \in [0, 2]} \|{\rm e}^{D_M t}\|$ versus $M$. The dotted line is the reference line $\log(M)$. The fact that the effective norm of the exponential matrix diverges, while the norm computed by selecting a random set of vectors is uniformly bounded, suggests that the "bad" behavior of the norm is due to a small set of vectors
Left: plot of the function $\sum\limits_{i = 1}^M |\ell_i(\theta)|$ for $M = 8$. Right: plot of $\|{\rm e}^{D_M t}\|$ (blue) and $C_M(t)$ in (45) (red) versus time for $M = 8$
Left: plot of $\psi(\theta) = H(\theta)$ (black) and its interpolating polynomial for $M = 8$ (red). Right: $\log \log$ plot of the error $\|{\rm e}^{D_M t}\psi(\Theta_M)-R_M \mathcal{H}_{0}(t)\psi\|$ for $t = 0.5, 1.5, 2.5$ versus $M$. Note the uniform bound for $t = 0.5$ and the convergence for $t>1$. The dotted lines are the reference lines $M^{-k}$, $k = 1, \dots, 4$
Same as Figure 6 for $\psi(\theta) = 0.5-|\theta+0.5|$. Note that $\psi \in Y$ and $\psi^{(1)}$ has bounded variation. The convergence rate for $t = 0.25, 0.5$ is $O(M^{-1})$ (right)
Same as Figure 6 for $\psi(\theta) = {\rm e}^{\theta^2}-1$. Note that the interpolating polynomial is indistinguishable from the function (left) and that $\psi^{(2)}$ has bounded variation. The convergence rate for $t = 0.5$ is $O(M^{-2})$ (right)
Same as Figure 6 for $\psi(\theta) = -\theta^3.$ Note that the interpolating polynomial is indistinguishable from the function (left) and that $\psi^{(3)}$ has bounded variation. The convergence rate for $t = 0.5$ is $O(M^{-3})$ (right)
Same as Figure 6 for $\psi(\theta) = {\rm e}^{-1/\theta^2}.$ Note that the interpolating polynomial is indistinguishable from the function (left) and the spectral convergence for $t = 0.5, 1.5, 2.5$ (right)
Same as Figure 6 for $\psi(\theta) = \sin\frac{1}{\theta}$, if $\theta<0$, $\psi(0) = 0$. Note there is no convergence for $t = 0.5$
Same as Figure 6 for $\psi(\theta) = \sin(6 \theta)+sign(\sin(\theta+{\rm e}^{2\theta}))$ [60,pag.10]. Note there is no convergence for $t = 0.5$
Left: plot of the function $z$ (saw function) Right: error $\|\int_0^t e^{(t-s)D_M} z(s) {\rm d} s- w(t)(\Theta_M) \|$ for $t = 0.5, 1, 1.5$ versus $M$. The dotted line is the reference line $\log(M)/M$
Same as Figures 13 for the function $z$ built so that it is continuously differentiable with jumps in the second derivative at $t = 0.5,1,1.5$
$\|D_M\|$ and estimation of the order $\log_2\frac{\|D_{2M}\|}{\|D_M\|}$ varying $M.$
 $M$ $\|D_M\|$ order $\|D_M\|/M^2$ 4 31 1.9375 8 127 2.0345 1.9844 16 511 2.0085 1.9961 32 2047 2.0021 1.9990 64 8191 2.0005 1.9998 128 32767 2.0001 1.9999 256 131071 2.0000 2.0000
 $M$ $\|D_M\|$ order $\|D_M\|/M^2$ 4 31 1.9375 8 127 2.0345 1.9844 16 511 2.0085 1.9961 32 2047 2.0021 1.9990 64 8191 2.0005 1.9998 128 32767 2.0001 1.9999 256 131071 2.0000 2.0000
$\mu(D_M)$ and estimation of the order $\log_2\frac{\mu(D_{2M})}{\mu(D_M)}$ varying $M.$
 $M$ $\mu(D_M)$ order $\mu(D_M)/M^2$ 4 6.6569e+00 4.1605e-01 8 3.8921e+01 2.5476 6.0814e-01 16 1.6698e+02 2.1011 6.5227e-01 32 6.7900e+02 2.0237 6.6308e-01 64 2.7270e+03 2.0058 6.6577e-01 128 1.0919e+04 2.0015 6.6644e-01 256 4.3687e+04 2.0004 6.6661e-01
 $M$ $\mu(D_M)$ order $\mu(D_M)/M^2$ 4 6.6569e+00 4.1605e-01 8 3.8921e+01 2.5476 6.0814e-01 16 1.6698e+02 2.1011 6.5227e-01 32 6.7900e+02 2.0237 6.6308e-01 64 2.7270e+03 2.0058 6.6577e-01 128 1.0919e+04 2.0015 6.6644e-01 256 4.3687e+04 2.0004 6.6661e-01
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