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# Using automatic differentiation to compute periodic orbits of delay differential equations

• * Corresponding author: Joan Gimeno

This work has been supported by the Spanish grants PGC2018-100699-B-I00 (MCIU/AEI/FEDER, UE) and the Catalan grant 2017 SGR 1374. The project leading to this application has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 734557

• In this paper we focus on the computation of periodic solutions of Delay Differential Equations (DDEs) with constant delays. The method is based on defining a Poincaré section in a suitable functional space and looking for a fixed point of the flow in this section. This is done by applying a Newton method on a suitable discretisation of the section. To avoid computing and storing large matrices we use a GMRES method to solve the linear system because in this case GMRES converges very fast due to the compactness of the flow of the DDE. The derivatives of the Poincaré map are obtained in a simple way, by applying Automatic Differentiation to the numerical integration. The stability of the periodic orbit is also obtained in a very efficient way by means of Arnoldi methods. The examples considered include temporal and spatial Poincaré sections.

Mathematics Subject Classification: Primary: 37C27; Secondary: 65F10 65Q20.

 Citation: • • Figure 3.1.  Space Poincaré section $\sigma(x)$ starting with initial symbols $\boldsymbol{u}'$. After $t_S$ units of time the solution is crossing the section again with directional derivatives $\boldsymbol{v}'$ but it may not be in the section, so a projection to it can be done with the tangent vector $\frac{d}{dt} x_{t_S}$ and the normal to the section $\frac{d}{dx}s(x_{t_S})$

Figure 4.1.  Continuation of periodic orbits of (4.3) with respect to parameters; (a) has $\varepsilon = 10^{-4}$ and $\tau = 1$, (b) has $\alpha = 1.5$ and $\tau = 1$, and (c) has $\alpha = 1.5$ and $\varepsilon = 10^{-4}$. Black colour means stable

Figure 4.2.  Periodic orbit of the equation (4.1). In the left hand side, the orbit is displayed with the initial condition in $-1\leq t \leq 0$ and final lag-segment once the second has been crossed two times. The phase space of the periodic orbit is shown in the right hand side

Figure 4.3.  Continuation of periodic orbits with respect to $\alpha$ starting at the periodic orbit computed for $\alpha = 1.57$, $\varepsilon = 10^{-4}$ and $\tau = 1$ of Equation (4.1). The $y$-axis represents the $\infty$-norm of the initial condition of each of the periodic orbits. Black colour means stable

Figure 4.4.  Same continuation as in Figure 4.3, left. Left plot: positions vs. time. Right plot: derivatives vs. positions

Figure 4.5.  Same continuation as in Figure 4.3, left, now showing the evolution of the spectral radius. We have added a horizontal straight line at 1 to visualise the changes of stability

Figure 4.6.  Periodic orbit of (4.4) with parameters $\lambda_1 = \lambda_2 = 2.5$, $\lambda_3 = 0.25$, $\tau_1 = 1.65$, $\tau_2 = 0.35$ and $\tau_3 = 1$. The final period is almost $3.5894$ units of time

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