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The Flow of ODEs: Formalization of Variational Equation and Poincaré Map

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Abstract

Formal analysis of ordinary differential equations (ODEs) and dynamical systems requires a solid formalization of the underlying theory. The formalization needs to be at the correct level of abstraction, in order to avoid drowning in tedious reasoning about technical details. The flow of an ODE, i.e., the solution depending on initial conditions, and a dedicated type of bounded linear functions yield suitable abstractions. The dedicated type integrates well with the type-class based analysis in Isabelle/HOL and we prove advanced properties of the flow, most notably, differentiable dependence on initial conditions via the variational equation. Moreover, we formalize the notion of first return or Poincaré map and prove its differentiability. We provide rigorous numerical algorithm to solve the variational equation and compute the Poincaré map.

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Notes

  1. This means that f does not depend on t. This restriction makes the presentation clearer. Many of our results are also formalized for non-autonomous ODEs. Moreover, for a large class of ODEs, a reduction to the autonomous case is possible.

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Acknowledgements

We would like to thank Professor Dr. Martin Brokate for supervising part of this work as an “interdisciplinary project”. Johannes Hölzl’s suggestions related to filters were very helpful. We would also like to thank the anonymous reviewers for all their suggestions and comments. Part of this work was supported by DFG RTG 1480 and DFG NI 491/16-1.

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  1. Institut für Informatik, Technische Universität München, Garching, Germany

    Fabian Immler & Christoph Traut

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  1. Fabian Immler
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  2. Christoph Traut
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Correspondence to Fabian Immler.

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Immler, F., Traut, C. The Flow of ODEs: Formalization of Variational Equation and Poincaré Map. J Autom Reasoning 62, 215–236 (2019). https://doi.org/10.1007/s10817-018-9449-5

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