| Abstract: |
| We study the following class of Delay Differential Equations (DDEs):
\begin{align*}
x`(t) &= f(x(t), x(t-\tau)) & t \ge 0 \
x(t) &= \psi(t) & t \in [-\tau, 0],
\end{align*}
with $\tau > 0$ fixed, $x \in \mathbb{R}^d$, $\psi \in \mathcal{C}^0([-\tau, 0], \mathbb{R}^d) =: \mathcal{C}$.
While significant work has been done on the dynamics of such systems--most
notably regarding chaos in the Mackey-Glass equation--the infinite-dimensional
nature of DDEs presents some challenges for carrying out the rigorous computations
and obtaining estimates good enough for computer-assisted proofs remains a challenge.
This talk will present a recent attempt to reduce the number of unknown factors
affecting the quality of rigorous numerics by using a pseudospectral approximation,
which reduces the DDE to a relatively small system of
ODEs while preserving numerically observed
dynamical features. Due to the low-dimensionality of
the resulting approximation and its great theoretical accuracy, the computations
are less demanding and can be done using known tools, such as CAPD,
to efficiently verify some dynamical phenomena that closely mirror those of the full system.
We present a proof of existence of symbolic dynamics in the ODE approximating DDE
and we discuss lessons learned that might be used to guide a similar
proof for the full DDE system.
The author acknowledge the support of Polish National Science Center (NCN) grant no. 2023/49/B/ST6/02801. |
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