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 -