Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation

Abstract

We present an algorithm for the rigorous integration of Delay Differential Equations (DDEs) of the form $x'(t)=f(x(t-τ),x(t))$. As an application, we give a computer assisted proof of the existence of two attracting periodic orbits (before and after the first period-doubling bifurcation) in the Mackey-Glass equation.

Figures (5)

• Figure 1: A graphical presentation of the integrator scheme. We set $n=2$ and $p=4$. A $(p,n)$-representation is depicted as dots at grid points and rectangles stretching on the whole intervals between consecutive grid points. The dot is used to stress the fact that the corresponding coefficient represents the value at a given grid point. Rectangles are used to stress the fact that remainders are bounds for derivative over whole intervals. Below the time line we have an initial $(p,n)$-representation. Above the time line we see a representation after one step of size $h = \frac{1}{p}$. Black solid dots and grey rectangles represent the values we do not need to compute - this is the shift part. The forward part, i.e. the elements to be computed, are presented as empty dots and an empty rectangle. The doubly bordered dot represents the exact value of the solution at the time $t = h = \frac{1}{p}$ (in practical rigorous computations it is an interval bound on the value). The doubly bordered empty rectangle is an enclosure for the $n+1$-st derivative on the interval $\left[0, h\right]$.
• Figure 2: An illustration of the wrapping effect problem for a classical, idealized mathematical pendulum ODE $\ddot{x} = -x$. The picture shows a set of solutions in a phase space $(x, \dot{x})$. The grey boxes present points of initial box moved by the flow. The colored boxes present the wrapping effect occurring at each step when we want to enclose the moving points in a product of intervals in the basic coordinate system. For example, the blue square on the left encloses the image of the first iteration. Its image is then presented with blue rhombus which is enclosed again by an orange square. Then the process goes on.
• Figure 3: Left: an illustration of the problem with big difference in transition time $t_S$ for poorly chosen section $S$. If $t_S \in q \cdot {\tau \over p} + [\varepsilon_1, \varepsilon_2]$ and $|\varepsilon_1 - \varepsilon_2|$ is large, then $I_{[\varepsilon_1, \varepsilon_2]}$ produces estimates on solutions distant from the section (blue rectangles) so the interval enclosure $W$ of all solutions tend to be very large (green rectangle). Right: if the section is chosen carefully, then all the solution obtained from $I_{[\varepsilon_1, \varepsilon_2]}$ are close to the section, so the set $W$ is small.
• Figure 4: Top: approximate function $\hat{x}$ (blue) and estimates on the value of the true solution obtained from computer-assisted proof (red). Bottom: solution plotted as parametric curve $r(t) = (\hat{x}(t), \hat{x}(t-\tau))$.
• Figure 5: Top: approximate function $\hat{x}$ (blue) and estimates on the value of the true solution obtained from computer-assisted proof (red). Bottom: solution plotted as parametric curve $r(t) = (\hat{x}(t), \hat{x}(t-\tau))$.

Theorems & Definitions (28)

• Lemma 1: Continuous (local) semiflow
• Lemma 2: Smoothing property
• Remark 3
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 4
• Definition 5
• Lemma 5