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A general framework for validated continuation of periodic orbits in systems of polynomial ODEs

The authors are partially supported by NWO-VICI grant 639033109

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  • In this paper a parametrized Newton-Kantorovich approach is applied to continuation of periodic orbits in arbitrary polynomial vector fields. This allows us to rigorously validate numerically computed branches of periodic solutions. We derive the estimates in full generality and present sample continuation proofs obtained using an implementation in Matlab. The presented approach is applicable to any polynomial vector field of any order and dimension. A variety of examples is presented to illustrate the efficacy of the method.

    Mathematics Subject Classification: Primary: 37C27, 34A12; Secondary: 65G40.


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  • Figure 1.  Representation of the matrix introduced in Equation (25)

    Figure 2.  The shape of the operator $ Q $, as defined in Lemma 5.1. The vertical columns represent the vectors $ y^{[k]} $, where $ k $ denotes the column index

    Figure 3.  The boundaries of the validation interval $ (r_{\min},r_{\max}) $ versus the number of modes $ K $ for various values of $ \nu $

    Figure 4.  On the right: the minimal number of modes $ K $ necessary for the validation of a periodic orbit of (52) depending on $ \mu $. The stepsize in $ \mu $ is not chosen uniformly, because the problem is more sensitive to changes in $ \mu $ for $ \mu $ bigger, therefore requesting a steeper change in the number of modes. One the left: several of the validated periodic orbits

    Figure 5.  The computation time versus the number of modes $ K $. The reference line has a slope corresponding to $ K^3 $

    Figure 6.  The computation time versus the dimension $ N $ of the system (53). The reference lines have slopes corresponding to quadratic and cubic dependence on $ N $

    Figure 7.  The computation time versus the order $ D $ of the system (54). The reference lines have slopes corresponding to quadratic and cubic dependence on $ D $

    Figure 8.  Validation of a periodic solution for the Lorenz system (55) with the classical parameter values $ \sigma = 10 $, $ \beta = 8/3 $ and $ \rho = 28 $. The solution has a period close to 25. This solution was validated with $ \nu = 1+10^{-6} $ and $ K = 800 $

    Figure 9.  Step size versus $ \mu $. Some oscillation is noticeable in the step size. This is due to the fact that the stepsize is decreased when the number of modes is changed, see Section 9. During this validation, the number of modes has been increased automatically from 4 to about 400

    Figure 10.  Validated continuation for the Lorenz system (55) from $ \rho = 28 $ to $ \rho = 14.8 $. In the left graph, the lower orbit still has two distinct swirls in the left lobe, but they are too close to each other to be seen in this graph

    Figure 11.  Validated continuation for $ 10 $ coupled Lorenz systems (77) from $ \epsilon = 0.14205 $ to $ \epsilon = 0.14252 $. In the left graph we have depicted in blue the orbit of the first three component, in green the orbit of the last three component, with the biggest validated $ \epsilon $, in the right graph we depicted the values of $ \epsilon $, where we can notice the adaptation of the stepsize

    Figure 12.  Continuation through the fold of the Rychkov system (78). The validation started at the top, where a small initial stepsize was chosen. One can see how the stepsize was automatically adjusted along the validated curve. As can be seen from the values along the horizontal axis, we only depict a validated continuation for parameter values very close to saddle-node

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