Abstract
We discuss how the geometric theory of differential equations can be used for the numerical integration and visualisation of implicit ordinary differential equations, in particular around singularities of the equation. The Vessiot theory automatically transforms an implicit differential equation into a vector field distribution on a manifold and thus reduces its analysis to standard problems in dynamical systems theory like the integration of a vector field and the determination of invariant manifolds. For the visualisation of low-dimensional situations we adapt the streamlines algorithm of Jobard and Lefer to 2.5 and 3 dimensions. A concrete implementation in Matlab is discussed and some concrete examples are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
It is well-known that the fibration \(\pi ^{q}_{q-1}:J_{q}\pi \rightarrow J_{q-1}\pi \) defines an affine bundle. A differential equation \(\mathcal {R}_{q}\subset J_{q}\pi \) is quasi-linear, if it is an affine subbundle.
Note that, strictly speaking, we are dealing here with a vector field on a two-dimensional manifold. If we had a nice parametrisation of the manifold, we could express the vector field X in these parameters and would obtain a \(2\times 2\) Jacobian. As it is in general difficult to find such parametrisations, we use instead the three coordinates of the ambient space \(J_{1}\pi \). Consequently, we obtain a too large Jacobian and must see which two eigenvalues are the right ones. This is easily decided by checking whether the corresponding (generalised) eigenvectors are tangential to \(\mathcal {R}_{1}\).
In [5] it is shown that for these parameter values there are infinitely many solutions reaching the “tip” \(\rho \).
References
Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren der mathematischen Wissenschaften 250, 2nd edn. Springer, New York (1988)
Bächler, T., Gerdt, V.P., Lange-Hegermann, M., Robertz, D.: Algorithmic Thomas decomposition of algebraic and differential systems. J. Symb. Comput. 47, 1233–1266 (2012)
Beyn, W.J., Kleß, W.: Numerical Taylor expansion of invariant manifolds in large dynamical systems. Numer. Math. 80, 1–38 (1998)
Braun, E.: Numerische Analyse und Visualisierung von voll-impliziten gewöhnlichen Differentialgleichungen. Master thesis, Institut für Mathematik, Universität Kassel (2017)
Brunovský, P., Černý, A., Winkler, M.: A singular differential equation stemming from an optimal control problem in financial economics. Appl. Math. Opt. 68, 255–274 (2013)
Chen, C.K., Yan, S., Yu, H., Max, N., Ma, K.L.: An illustrative visualization framework for 3D vector fields. Comput. Graph. Forum 30, 1941–1951 (2011)
Dumortier, F., Llibre, J., Artés, J.C.: Qualitative Theory of Planar Differential Systems. Universitext. Springer, Berlin (2006)
Eirola, T., von Pfaler, J.: Taylor expansion for invariant manifolds. Numer. Math. 80, 1–38 (1998)
Fesser, D.: On Vessiot’s Theory of Partial Differential Equations. PhD thesis, Fachbereich Mathematik, Universität Kassel (2008)
Fesser, D., Seiler, W.M.: Existence and construction of Vessiot connections. SIGMA 5, 092 (2009)
Jobard, B., Lefer, W.: Creating evenly-spaced streamlines of arbitrary density. In: Lefer, W., Grave, M. (eds.) Visualization in Scientific Computing, Eurographics, pp. 43–55. Springer, Berlin (1997)
Kant, U., Seiler, W.M.: Singularities in the geometric theory of differential equations. In: Feng, W., Feng, Z., Grasselli, M., Lu, X., Siegmund, S., Voigt, J. (eds.) Dynamical Systems, Differential Equations and Applications (Proceedings of 8th AIMS Conference, Dresden 2010), vol. 2, pp. 784–793. AIMS (2012)
Lychagin, V.V.: Homogeneous geometric structures and homogeneous differential equations. In: Lychagin, V.V. (ed.) The Interplay Between Differential Geometry and Differential Equations, Amer. Math. Soc. Transl. 167, pp. 143–164. Amer. Math. Soc., Providence (1995)
Marchesin, S., Chen, C.K., Ho, C., Ma, K.L.: View-dependent streamlines for 3D vector fields. IEEE Trans. Vis. Comput. Graph. 16, 1578–1586 (2010)
Pommaret, J.F.: Systems of Partial Differential Equations and Lie Pseudogroups. Gordon & Breach, London (1978)
Saunders, D.J.: The Geometry of Jet Bundles. London Mathematical Society Lecture Notes Series 142. Cambridge University Press, Cambridge (1989)
Seiler, W.M.: Involution—The Formal Theory of Differential Equations and its Applications in Computer Algebra. Algorithms and Computation in Mathematics 24. Springer, Berlin (2010)
Seiler, W.M.: Singularities of implicit differential equations and static bifurcations. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing—CASC 2013. Lecture Notes in Computer Science 8136, pp. 355–368. Springer, Berlin (2013)
Seiler, W.M., Seiß, M.: Singular initial value problems for quasi-linear ordinary differential equations (2018, in preparation)
Sit, W.Y.: An algorithm for solving parametric linear systems. J. Symb. Comput. 13, 353–394 (1992)
Tuomela, J.: On singular points of quasilinear differential and differential–algebraic equations. BIT 37, 968–977 (1997)
Tuomela, J.: On the resolution of singularities of ordinary differential equations. Numer. Algorithms 19, 247–259 (1998)
Vessiot, E.: Sur une théorie nouvelle des problèmes généraux d’intégration. Bull. Soc. Math. Fr. 52, 336–395 (1924)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was partially performed as part of the European H2020-FETOPEN-2016-2017-CSA project \(SC^{2}\) (712689) and partially supported by the bilateral project ANR-17-CE40-0036 and DFG-391322026 SYMBIONT.
Rights and permissions
About this article
Cite this article
Braun, E., Seiler, W.M. & Seiß, M. On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations. Math.Comput.Sci. 14, 281–293 (2020). https://doi.org/10.1007/s11786-019-00423-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11786-019-00423-6