Symbolic computations of first integrals for polynomial vector fields. (English) Zbl 1485.34006
In the paper, the authors generalize the extactic curve introduced by J. Pereira (in fact, proposed first by Mikhail Nikolaevich Lagutinskii (1871–1915)) to the Darbouxian, Liouvillian and Riccati integrability. Using this approach, the authors present new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral for polynomial planar vector fields with bounded degree. Probabilistic and deterministic algorithms are presented where it is also analyzed the arithmetic complexity of the probabilistic one. The algorithms presented improves previous ones. The algorithms have been implemented in Maple and are available on the authors’ websites. In the last section of the paper some examples are presented showing the efficiency of the algorithms. The algorithms presented generalize to the Darbouxian, Liouvillian and Riccati cases the algorithm proposed in [A. Bostan et al., Math. Comput. 85, No. 299, 1393–1425 (2016; Zbl 1335.34002)] for computing rational first integrals. It is important to remark that the method presented avoids the computation of Darboux polynomials and then does not need a recombination step of the associated cofactors.
Reviewer: Jaume Giné (Lleida)
MSC:
| 34A05 | Explicit solutions, first integrals of ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 68W30 | Symbolic computation and algebraic computation |
| 68W40 | Analysis of algorithms |
Citations:
Zbl 1335.34002References:
| [1] | Avellar, J.; Duarte, LGS; da Mota, LACP, PSsolver: A Maple implementation to solve first order ordinary differential equations with Liouvillian solutions, Computer Physics Communications, 183, 10, 2313-2314 (2012) · doi:10.1016/j.cpc.2012.04.012 |
| [2] | Avellar, J.; Duarte, LGS; Duarte, SES; da Mota, LACP, Integrating first-order differential equations with Liouvillian solutions via quadratures: a semi-algorithmic method, Journal of Computational and Applied Mathematics, 182, 2, 327-332 (2005) · Zbl 1071.65095 · doi:10.1016/j.cam.2004.12.014 |
| [3] | Beckermann, B.; Labahn, G., A uniform approach for the fast computation of matrix-type Padé approximants, SIAM J. Matrix Anal. Appl., 15, 3, 804-823 (1994) · Zbl 0805.65008 · doi:10.1137/S0895479892230031 |
| [4] | D. Bini and V. Pan. Polynomial and matrix computations. Vol. 1. Progress in Theoretical Computer Science. Birkhäuser Boston Inc., Boston, MA, 1994. Fundamental algorithms. · Zbl 0809.65012 |
| [5] | Bostan, A.; Chèze, G.; Cluzeau, T.; Weil, J-A, Efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields, Math. Comp., 85, 299, 1393-1425 (2016) · Zbl 1335.34002 · doi:10.1090/mcom/3007 |
| [6] | A. Bostan, F. Chyzak, F. Ollivier, B. Salvy, É. Schost, and A. Sedoglavic. Fast computation of power series solutions of systems of differential equations. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1012-1021, New York, 2007. ACM. · Zbl 1302.65180 |
| [7] | A. Bostan, G. Lecerf, E. Schost, E. Salvy, and B. Wiebelt. Complexity issues in bivariate polynomial factorization. In Proceedings of ISSAC’04, pages 42-49. ACM Press, 2004. · Zbl 1134.68595 |
| [8] | M. Bronstein. Symbolic integration. I, volume 1 of Algorithms and Computation in Mathematics. Springer-Verlag, Berlin, second edition, 2005. Transcendental functions, With a foreword by B. F. Caviness. |
| [9] | Casale, G., Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières, Annales de l’Institut Fourier, 56, 3, 735-779 (2006) · Zbl 1155.32020 · doi:10.5802/aif.2198 |
| [10] | Chèze, G., Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time, J. Complexity, 27, 2, 246-262 (2011) · Zbl 1215.65040 · doi:10.1016/j.jco.2010.10.004 |
| [11] | G. Chèze. Décomposition et intégrales premières rationnelles: algorithmes et complexité. Habilitation thesis, Université Paul Sabatier, Toulouse 3, 2014. |
| [12] | Chèze, G., Bounding the number of remarkable values via Jouanolou’s theorem, Journal of Differential Equations, 258, 10, 3535-3545 (2015) · Zbl 1343.34036 · doi:10.1016/j.jde.2015.01.027 |
| [13] | C. Christopher. Liouvillian first integrals of second order polynomial differential equations. Electron. J. Differential Equations, pages No. 49, 7 pp. (electronic), 1999. · Zbl 0939.34002 |
| [14] | Christopher, C.; Llibre, J.; Pereira, J., Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math., 229, 1, 63-117 (2007) · Zbl 1160.34003 · doi:10.2140/pjm.2007.229.63 |
| [15] | A. Clairaut. Recherches générales sur le calcul intégral. Mémoires de l’Académie Royale des Sciences, pages 425-436, 1739. |
| [16] | G. Darboux. Mémoire sur les équations diff’érentielles du premier ordre et du premier degré. Bull. Sci. Math., 32:60-96, 123-144, 151-200, 1878. · JFM 10.0214.01 |
| [17] | Duarte, LGS; Duarte, SES; da Mota, LACP, Analysing the structure of the integrating factors for first-order ordinary differential equations with Liouvillian functions in the solution, J. Phys. A, 35, 4, 1001-1006 (2002) · Zbl 1002.34002 · doi:10.1088/0305-4470/35/4/312 |
| [18] | Duarte, LGS; Duarte, SES; da Mota, LACP, A method to tackle first-order ordinary differential equations with Liouvillian functions in the solution, J. Phys. A, 35, 17, 3899-3910 (2002) · Zbl 1040.34006 · doi:10.1088/0305-4470/35/17/306 |
| [19] | F. Dumortier, J. Llibre, and J. Artés. Qualitative theory of planar differential systems. Universitext. Springer-Verlag, Berlin, 2006. · Zbl 1110.34002 |
| [20] | Ferragut, A.; Giacomini, H., A new algorithm for finding rational first integrals of polynomial vector fields, Qual. Theory Dyn. Syst., 9, 1-2, 89-99 (2010) · Zbl 1216.34001 · doi:10.1007/s12346-010-0021-x |
| [21] | Gabrielov, A., Multiplicities of zeroes of polynomials on trajectories of polynomial vector fields and bounds on degree of nonholonomy, Math. Res. Lett., 2, 4, 437-451 (1995) · Zbl 0845.32003 · doi:10.4310/MRL.1995.v2.n4.a5 |
| [22] | von zur Gathen, J.; Gerhard, J., Modern computer algebra (1999), New York: Cambridge University Press, New York · Zbl 0936.11069 |
| [23] | A. Goriely. Integrability and nonintegrability of dynamical systems, volume 19 of Advanced Series in Nonlinear Dynamics. World Scientific Publishing Co. Inc., River Edge, NJ, 2001. · Zbl 1002.34001 |
| [24] | C.-P. Jeannerod, V. Neiger, E. Schost, and G. Villard. Computing minimal interpolation bases. Journal of Symbolic Computation, 83(Supplement C):272 - 314, 2017. Special issue on the conference ISSAC 2015: Symbolic computation and computer algebra. · Zbl 1375.65013 |
| [25] | J.-P. Jouanolou. Équations de Pfaff algébriques, volume 708 of Lecture Notes in Mathematics. Springer, Berlin, 1979. · Zbl 0477.58002 |
| [26] | D. Lay. Linear Algebra and Its Applications (5th Edition). Pearson, 2015. |
| [27] | Lecerf, G., Sharp precision in Hensel lifting for bivariate polynomial factorization, Math. Comp., 75, 254, 921-933 (2006) · Zbl 1125.12003 · doi:10.1090/S0025-5718-06-01810-2 |
| [28] | Man, Y-K, Computing closed form solutions of first order odes using the Prelle-Singer procedure, Journal of Symbolic Computation, 16, 5, 423-443 (1993) · Zbl 0793.34002 · doi:10.1006/jsco.1993.1057 |
| [29] | Man, Y-K; MacCallum, M., A rational approach to the Prelle-Singer algorithm, J. Symbolic Comput., 24, 1, 31-43 (1997) · Zbl 0922.12007 · doi:10.1006/jsco.1997.0111 |
| [30] | Pereira, J., Vector fields, invariant varieties and linear systems, Ann. Inst. Fourier (Grenoble), 51, 5, 1385-1405 (2001) · Zbl 1107.37038 · doi:10.5802/aif.1858 |
| [31] | É. Picard. Sur les intégrales doubles de fonctions rationnelles dont tous les résidus sont nuls. Bulletin des sciences mathématiques, série 2, 26, 1902. · JFM 33.0435.02 |
| [32] | Prelle, M.; Singer, M., Elementary first integrals of differential equations, Trans. Amer. Math. Soc., 279, 1, 215-229 (1983) · Zbl 0527.12016 · doi:10.1090/S0002-9947-1983-0704611-X |
| [33] | J.-J. Risler. A bound for the degree of nonholonomy in the plane. Theoret. Comput. Sci., 157(1):129-136, 1996. Algorithmic complexity of algebraic and geometric models (Creteil, 1994). · Zbl 0871.93024 |
| [34] | Ruppert, W., Reduzibilität Ebener Kurven, J. Reine Angew. Math., 369, 167-191 (1986) · Zbl 0584.14012 |
| [35] | A. Schinzel. Polynomials with special regard to reducibility, volume 77 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2000. With an appendix by Umberto Zannier. · Zbl 0956.12001 |
| [36] | Schlomiuk, D., Elementary first integrals and algebraic invariant curves of differential equations, Exposition. Math., 11, 5, 433-454 (1993) · Zbl 0791.34004 |
| [37] | Singer, M., Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc., 333, 2, 673-688 (1992) · Zbl 0756.12006 · doi:10.1090/S0002-9947-1992-1062869-X |
| [38] | X. Zhang. Integrability of Dynamical Systems: Algebra and Analysis. Developments in Mathematics. Springer Singapore, 2017. · Zbl 1373.37053 |
| [39] | Zhou, W.; Labahn, G., Efficient algorithms for order basis computation, J. Symbolic Comput., 47, 7, 793-819 (2012) · Zbl 1258.65046 · doi:10.1016/j.jsc.2011.12.009 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
