This paper presents a collocation method with an iterative linear system solver to compute periodic solutions of a system of autonomous delay differential equations (DDEs). We show that the linearized collocation system is equivalent to a discretization of the linearized periodic boundary value problem (BVP). This linear BVP is solved using the Newton-Picard single shooting method ([Int. J. Bifurcation Chaos, 7 (1997), pp. 2547– 2560]). The Newton-Picard method combines a direct method in the subspace of the weakly stable and unstable modes with an iterative solver in the orthogonal complement. As a side effect, we also obtain good estimates for the dominant Floquet multipliers. We have implemented the method in the DDE-BIFTOOL environment to test our algorithm.
5 Figures and Tables
Table 1: The values of the mesh characteristics , , and .
Figure 2: The orbits of the periodic solutions A.I (left) and A.II (right) with (dashed line) and (dash-dotted line).
Table 3: CPU time in seconds for DDE-BIFTOOL (“D.-B.”) and the Newton-Picard collocation method (“NP-c”) for the computation of a periodic solution (“P”) and the computation of both a periodic solution and the Floquet multipliers (“P+F”).
Figure 5: For periodic solutions A.I (left) and A.II (right): ( , dotted line), ( , dotted line), ( , dashed line) and an estimation of the higher-order terms ( ) vs. the Newton-Picard step.
Figure 6: For periodic solution B: ( , dotted line), ( , dotted line), ( , dashed line) and an estimation of the higher-order terms ( ) vs. the Newton-Picard step.
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