Abstract
In this paper we study the pseudospectral approximation of delay differential equations formulated as abstract differential equations in the ∗-space. This formalism also allows us to define rigorously the abstract variation-of-constants formula, where the ∗-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 ∗-shift operator and of the ∗-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.
Original language | English |
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Pages (from-to) | 2575-2602 |
Number of pages | 28 |
Journal | Discrete and Continuous Dynamical Systems - Series B |
Volume | 13 |
Issue number | 9 |
DOIs | |
Publication status | Published - 1 Sept 2020 |
Funding
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. ∗ Corresponding author: Rossana Vermiglio.
Keywords
- Chebyshev differentiation matrix
- Collocation
- Delay differential equations
- Differential operators
- Pseudospectral discretization
- Shift semigroup
- Sun-star formalism