Geodesic
Computer algebra methods in the study of nonlinear differential systems
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 6, pp. 1027-1040 Cet article a éte moissonné depuis la source Math-Net.Ru

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Some issues concerning computer algebra methods as applied to the qualitative analysis of differential equations with first integrals are discussed. The problems of finding stationary sets and analyzing their stability and bifurcations are considered. Special attention is given to algorithms for finding and analyzing peculiar stationary sets. It is shown that computer algebra tools, combined with qualitative analysis methods for differential equations, make it possible not only to enhance the computational efficiency of classical algorithms, but also to implement new approaches to the solution of well-known problems and, in this way, to obtain new results. 
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V. D. Irtegov; T. N. Titorenko. Computer algebra methods in the study of nonlinear differential systems. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 6, pp. 1027-1040. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_6_a14/

[1] Borisov A. V., Mamaev I. S., Sokolov V. V., “Novyi integriruemyi sluchai na $so (4)$”, Dokl. AN, 381:5 (2001), 614–615 | MR

[2] Morozov V. P., “Topologiya sloenii Liuvillya sluchaev integriruemosti Steklova i Sokolova”, Matem. sb., 195:3 (2004), 69–114 | DOI | MR

[3] Ryabov P. E., “Bifurkatsii pervykh integralov v sluchae Sokolova”, Teoreticheskaya i matem. fiz., 134:2 (2003), 207–226 | DOI | MR | Zbl

[4] Lyapunov A. M., “O postoyannykh vintovykh dvizheniyakh tela v zhidkosti”, Sobr. soch., v. 1, AN SSSR, 1954 | MR

[5] Irtegov V. D., Titorenko T. N., “Ob invariantnykh mnogoobraziyakh sistem s pervymi integralami”, PMM, 73:4 (2009), 531–537 | MR

[6] Koks D., Littl Dzh., O'Shi D., Idealy, mnogoobraziya i algoritmy, Mir, M., 2000

[7] Zharkov A. Yu., Blinkov Yu. A., “Involution approach to solving systems of algebraic equations”, Proc. of Internat. IMACS Symposium on Symbolic Computation: New Trends and Developments (1993), 11–16

[8] Gerdt V. P., “Gröbner bases and involutive methods for algebraic and differential equations”, Math. Comput. Modelling, 25:8–9 (1997), 75–90 | DOI | MR | Zbl

[9] Wang D., “An elimination method for polynomial systems”, J. Symbolic Comput., 16:2 (1993), 83–114 | DOI | MR | Zbl

[10] Wu Wen-Tsun, “Basic principles of mechanical theorem proving in elementary geometries”, J. Automated Reasoning, 1986, no. 2, 221–252 | DOI | MR

[11] Kapur D., “Automated geometric reasoning. Dixon resultants, Gröbner bases, and characteristic sets”, Automated Deduction in Geometry, Proc. of Conf., LNCS, 1360, Springer, Berlin, 1997, 1–36

[12] Weispfenning V., “Comprehensive Gröbner bases”, J. Symbolic Comput., 14:1 (1992), 636–667 | DOI | MR

[13] Montes A., “A new algorithm for discussing Gröbner bases with parameters”, J. Symbolic Computation, 33:2 (2002), 183–208 | DOI | MR | Zbl

[14] Chen C., Maza M., “Comprehensive triangular decomposition”, Computer Algebra in Scientific Computing, Proc. of Conf., LNCS, 4770, Springer, Berlin, 2007, 73–101 | Zbl

[15] Devenport Dzh., Sire I., Turne E., Kompyuternaya algebra. Sistemy i algoritmy algebraicheskikh vychislenii, Mir, M., 1991 | MR

[16] Collins G. E., “Quantifier elimination for real closed fields by cylindrical algebraic decomposition”, Automata Theory and Formal Languages, Proc. of Conf., LNCS, 33, Springer, Berlin, 1975, 134–183 | MR

[17] Gantmakher F. R., Teoriya matrits, Fizmatlit, M., 2010

[18] Banschikov A. V., Burlakova L. A., Irtegov V. D., Titorenko T. N., Programmnyi kompleks dlya vydeleniya i issledovaniya ustoichivosti statsionarnykh mnozhestv, Svidetelstvo o gos. registratsii programm dlya EVM No 2011615235, FGU-FIPS, 2011