×

REDUCE and the bifurcation of limit cycles. (English) Zbl 0702.68072

Summary: A technique is described which has been used extensively to investigate the bifurcation of limit cycles in polynomial differential systems. Its implementation requires a Computer Algebra System, in the case REDUCE. Concentration is on the computational aspects of the work, and a brief resume is given of some of the results which have been obtained.

MSC:

68W30 Symbolic computation and algebraic computation
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations

Software:

REDUCE; ALGOL 68
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alwash, M. A.M.; Lloyd, N. G., Non-autonomous equations related to polynomial two-dimensional systems, Proc. Roy. Soc. Edinburgh Sect. A, 105, 129-152 (1987) · Zbl 0618.34026
[2] Basarab-Horwath, P.; Lloyd, N. G., Co-existing fine foci and bifurcating limit cycles, Nieuw Arch. Wisk., 6, 4, 295-302 (1988) · Zbl 0676.34019
[3] Bautin, N. N., On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or centre type, Amer. Math. Soc. Transl., 100 (1954) · Zbl 0059.08201
[4] Blows, T. R.; Lloyd, N. G., The number of limit cycles of certain polynomial differential equations, Proc. Roy. Soc. Edinburgh, Sect. A, 98, 215-239 (1984) · Zbl 0603.34020
[5] Blows, T. R.; Lloyd, N. G., The number of small-amplitude limit cycles of Liénard equations, Math. Proc. Cambridge Philos. Soc., 95, 359-366 (1984) · Zbl 0532.34022
[6] Lansun, Chen; Mingshu, Wang, The relative position and number of limit cycles of the quadratic differential system, Acta Math. Sinica, 22, 751-758 (1979) · Zbl 0433.34022
[7] Ecalle, J.; Martinet, J.; Moussu, R.; Ramis, J.-P., Non-accumulation des cycles-limites (I), C.R. Acad. Sci. Paris Sér. I Math., 304, 375-377 (1987) · Zbl 0615.58011
[8] Ecalle, J.; Martinet, J.; Moussu, R.; Ramis, J.-P., Non-accumulation des cycles-limites (II), C.R. Acad. Sci. Paris Sér. I Math., 304, 431-434 (1987) · Zbl 0615.58011
[9] Guckenheimer, J.; Rand, R.; Schlomiuk, D., Degenerate homoclinic cycles in perturbations of quadratic Hamiltonian systems, Nonlinearity, 2, 405-418 (1989) · Zbl 0677.58038
[10] Joyal, P.; Rousseau, C., Saddle quantities and applications, J. Differential Equations, 78, 374-399 (1989) · Zbl 0684.34033
[11] Landis, E. M.; Petrovskii, I. G., On the number of limit cycles of the equation d \(y\)/d \(x= P(x,y)/Q(x,y)\) where \(P\) and \(Q\) are polynomials of the second degree, Amer. Math. Soc. Transl., 16, 2, 177-221 (1958) · Zbl 0080.07502
[12] Landis, E. M.; Petrovskii, I. G., On the number of limit cycles of the equation d \(y\)/d \(x= P(x,y)/Q(x,y)\) where \(P\) and \(Q\) are polynomials, Amer. Math. Soc. Transl., 14, 2, 181-200 (1960) · Zbl 0094.06304
[13] Landis, E. M.; Petrovskii, I. G., Corrections to the articles “On the number of limit cycles of the equation d \(y\)/d \(x= P(x,y)/Q(x,y)\) where \(P\) and \(Q\) are polynomials ofthe second degree” and “On the number of limit cycles of the equation d \(y\)/d \(x= P(x,y)/Q(x,y) \), where \(P\) and \(Q\) are polynomials”, Mat. Sb. N.S., 48, 90, 253-255 (1959)
[14] Li, C. F.; Rousseau, C., A system with three limit cycles bifurcating from a homoclinic loop, J. Differential Equations, 79, 132-167 (1989)
[15] Lloyd, N. G., Limit cycles of polynomial systems, (Bedford, T.; Swift, J., New Directions in Dynamical Systems. New Directions in Dynamical Systems, L.M.S. Lecture Notes, 127 (1988), Cambridge University Press: Cambridge University Press Cambridge), 192-234 · Zbl 0587.34026
[16] Lloyd, N. G., The number of limit cycles of polynomial systems in the plane, Bull. Inst. Math. Appl., 24, 161-165 (1988) · Zbl 0679.34026
[17] Lloyd, N. G.; Blows, T. R.; Kalenge, M. C., Some cubic systems with several limit cycles, Nonlinearity, 1, 653-669 (1988) · Zbl 0691.34024
[18] Lloyd, N. G.; Lynch, S., Small amplitude limit cycles of certain Liénard systems, Proc. Roy. Soc. London, Ser. A, 418, 199-208 (1988) · Zbl 0657.34030
[19] Long, F. W.; Danicic, I., Algebraic manipulation of polynomials in several indeterminates, (Proc. Conf. on Applications of Algol 68. Proc. Conf. on Applications of Algol 68, University of East Anglia, March (1976)), 112-115
[20] Lynch, S., Small-amplitude limit cycles of Liénard systems (1989), University College Wales: University College Wales Aberystwyth, Preprint · Zbl 0728.34026
[21] Nemytskii, V. V.; Stepanov, V. V., (Qualitative Theory of Differential Equations (1960), Princeton University Press: Princeton University Press Princeton, NJ) · Zbl 0089.29502
[22] Songling, Shi, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica Ser. A, 23, 153-158 (1980) · Zbl 0431.34024
[23] Dongming, Wang, The applications of characteristic sets and Gröbner bases to problems concerning Liapunov constants, (Technical Report 88-49.0 (1988), RISC-LINZ)
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.