Research output: Contribution to journal › Article › peer-review
Cycles of two-dimensional systems : Computer calculations, proofs, and experiments. / Leonov, G. A.; Kuznetsov, N. V.; Kudryashova, E. V.
In: Vestnik St. Petersburg University: Mathematics, Vol. 41, No. 3, 01.09.2008, p. 216-250.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - Cycles of two-dimensional systems
T2 - Computer calculations, proofs, and experiments
AU - Leonov, G. A.
AU - Kuznetsov, N. V.
AU - Kudryashova, E. V.
PY - 2008/9/1
Y1 - 2008/9/1
N2 - One of the central problems in studying small cycles in the neighborhood of equilibrium involves computation of Lyapunov's quantities. While Lyapunov's first and second quantities were computed in the general form in the 1940s-1950s, Lyapunov's third quantity was calculated only for certain special cases. In the present work, we present general formulas for calculation of Lyapunov's third quantity. Together with the classical Lyapunov method for calculation of Lyapunov's quantities, which is based on passing to the polar coordinates, we suggest a method developed for the Euclidian coordinates and for the time domain. The calculation of Lyapunov's quantities by two different analytic methods involving modern software tools for symbolic computing enables us to justify the formulas obtained for Lyapunov's third quantity. For quadratic systems in which Lyapunov's first and second quantities vanish, while the third one does not, large cycles were calculated. In the calculations, the quadratic system was reduced to the Liénard equation, which was used to evaluate the domain of parameters corresponding to the existence of four cycles (three "small" cycles and a "large" one). This domain extends the region of parameters obtained by S. L. Shi in 1980 for a quadratic system with four limit cycles.
AB - One of the central problems in studying small cycles in the neighborhood of equilibrium involves computation of Lyapunov's quantities. While Lyapunov's first and second quantities were computed in the general form in the 1940s-1950s, Lyapunov's third quantity was calculated only for certain special cases. In the present work, we present general formulas for calculation of Lyapunov's third quantity. Together with the classical Lyapunov method for calculation of Lyapunov's quantities, which is based on passing to the polar coordinates, we suggest a method developed for the Euclidian coordinates and for the time domain. The calculation of Lyapunov's quantities by two different analytic methods involving modern software tools for symbolic computing enables us to justify the formulas obtained for Lyapunov's third quantity. For quadratic systems in which Lyapunov's first and second quantities vanish, while the third one does not, large cycles were calculated. In the calculations, the quadratic system was reduced to the Liénard equation, which was used to evaluate the domain of parameters corresponding to the existence of four cycles (three "small" cycles and a "large" one). This domain extends the region of parameters obtained by S. L. Shi in 1980 for a quadratic system with four limit cycles.
UR - http://www.scopus.com/inward/record.url?scp=77951910410&partnerID=8YFLogxK
U2 - 10.3103/S1063454108030047
DO - 10.3103/S1063454108030047
M3 - Article
AN - SCOPUS:77951910410
VL - 41
SP - 216
EP - 250
JO - Vestnik St. Petersburg University: Mathematics
JF - Vestnik St. Petersburg University: Mathematics
SN - 1063-4541
IS - 3
ER -
ID: 52008331