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| We will present a rigorous integrator for Delay Differential Equations of the form $\dot{x} = f(x(t-1), x(t))$ and we will show how to implement rigorous Poincare Map for such systems. The integrator, together with topological arguments, can be used in rigorous computer assisted proofs of existence of various dynamical phenomena, such as stable/unstable stationary points, periodic solutions, connecting orbits, etc. We will show some preliminary results which can be obtained for scalar equations when $f(x(t-1), x(t)) = -K \cdot sin(x(t-1))$ and $f(x(t-1), x(t)) = -x(t) + g(x(t-1))$, where $g$ is polynomial of degree $5$. Our method can be used for any r.h.s. sufficiently smooth and may be simply extended to any dimension and any number of discrete delays. |
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