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

  • * Corresponding author: Rossana Vermiglio

    * Corresponding author: Rossana Vermiglio

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

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  • 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.

    Mathematics Subject Classification: Primary: 65F35, 65L03, 65L07, 65M70, 65P99, 65Q20, 37M20X, 34K07, 34K20, 41A10; Secondary: 34K08, 34K28, 37K45.


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  • Figure 1.  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 $

    Figure 2.  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

    Figure 3.  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

    Figure 4.  $ \|{\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

    Figure 5.  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 $

    Figure 6.  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 $

    Figure 7.  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)

    Figure 8.  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)

    Figure 9.  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)

    Figure 10.  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)

    Figure 11.  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 $

    Figure 12.  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 $

    Figure 13.  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 $

    Figure 14.  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 $

    Table 1.  $ \|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
     | Show Table
    DownLoad: CSV

    Table 2.  $ \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
     | Show Table
    DownLoad: CSV
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