Cycles of two-dimensional systems: Computer calculations, proofs, and experiments

Vestnik St. Petersburg University: Mathematics

Volumn 41, Issue 3, 2008, Pages 216-250

  Leonov, G A ^a   Kuznetsov, N V ^a   Kudryashova, E V ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77951910410     PISSN: 10634541     EISSN: 19347855     Source Type: Journal    
DOI: 10.3103/S1063454108030047     Document Type: Article

References (31)

1. N. N. Bautin, "Differential Equation with a Limit Cycle," Zh. Tekh. Fiz. 9(7), 601-611 (1939). On the Number of Limit Cycles which Appear with the Variation of Coefficients from an Equilibrium Position of Focus or Center Type (1954).
2. N. N. Bautin, "On the Number of Limit Cycles which Appear with the Variation of Coefficients from an Equilibrium Position of Focus or Center Type," Mat. Sb. (N. S.) 30(72) (1), 181-196 (1952) [AMS Transl. 5 (1) 396-413 (1962)].
3. A. A. Andronov and E. A. Leontovich, "The Generation of Limit Cycles from a Fine (Multiple) Focus or Center, or from a Fine (Multiple) Limit Cycle," Mat. Sb. (N. S.) 40(82) (2), 179-224 (1956).
4. A. A. Andronov, A. A. Vitt, and S. E., Khaikin, Theory of Oscillators (Nauka, Moscow, 1981; Pergamon, Oxford, 1966).
5. L. A. Cherkass, "The Number of Limit Cycles of a Second Order Autonomous System," Diff. Eq. 5, 666-668 (1976).
6. L. M. Perko, "Global Families of Limit Cycles of Planar Analytic Systems," Trans. Amer. Math. Soc. 322, 627-656 (1990).
7. T. R. Blows and L. M. Perko, "Bifurcation of Limit Cycles from Centers," Preprint Univ. N. Arizona, 1990.
8. A. M. Lyapunov, The General Problem of the Stability of Motion (Kharkov, 1892).
9. N. N. Bautin and E. A. Leontovich, Methods and Ways of Quality Analysis of Dynamic Systems in the Plane (Nauka, Moscow, 1976).
10. J. Marsden and M. McCracken, Hopf Bifurcation and its Applications (Springer, New York, 1976).
11. N. G. Lloyd and J. Pearson, "Reduce and the Bifurcation of Limit Cycles," J. of Symb. Comp. 9(2), 215-224 (1990).
12. S. Lynch, "Symbolic Computation of Lyapunov Quantities and the Second Part of Hilbert's Sixteenth Problem", in Differential Equations with Symbolic Computations, Dongming Wang, Zhiming Zheng Eds., Trends in Mathematics, pp. 1-26 (Birkhäuser, 2005).
13. P. Yu, "Computation of Normal Forms via a Perturbation Technique," J. Sound Vibr. 211, 19-38 (1998).
14. N. N. Serebryakova, "On the Behavior of a Dynamical System with One Degree of Freedom Near the Points of the Boundary of the Domain of Stability where the Safe Boundary Becomes Dangerous," Izv. AN SSSR, Mech. Mach., No. 2, 178-182 (1959).
15. N. G. Lloyd and J. Pearson, "Five Limit Cycles for a Simple Cubic System," Publ. Matem. 41, 199-208 (1997).
16. P. Yu. and M. Han, "Twelve Limit Cycles in a Cubic Case of the 16th Hilbert Problem," Int. J. Bifurcation Chaos, 15(7), 2191-2205 (2005).
17. G. A. Leonov, "The Family of Transversal Curves of Two-Dimensional Dynamic Systems,", Vestn. St. Peterb. Univ., Ser. 1, no. 4, 48-78 (2006).
18. N. V. Kuznetsov and G. A. Leonov, "Computation of the First Lyapunov Quantity for the Second-Order Dynamical System", Third IFAC Workshop Periodic Control System, 2007.
19. G. A. Leonov, "A Criterion for Existence of Cycles in Quadratic Systems," Vestn. St. Peterb. Univ., Ser. 1, no. 3, 172-181 (2007).
20. G. A. Leonov, "Hilbert's 16th Problem for Quadratic System. New Method Based on a Transformation to the Lienard Equation," Int. J. Bifurcation Chaos 18(3), 877-884 (2008).
21. S. Lefschetz, Differential Equations: Geometric Theory (Interscience Publishers, New York, 1957).
22. L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations (Springer, Berlin, 1959).
23. P. Hartman, Ordinary Differential Equations (Willey, New York, 1964).
24. G. A. Leonov and N. V. Kuznetsov, "Time-Varying Linearization and Perron Effects," Int. J. Bifurcation Chaos, 17(4), 1-29 (2007).
25. G. A. Leonov, Strange Attractors and Classical Stability Theory (St. Peterb. Gos. Univ., St. Petersburg, 2008).
26. V. A. Zorich, Mathematical Analysis. Parts I, II (Nauka, Moscow, 1984).
27. J. K. Hale and H. Kocak, Dynamics and Bifurcations (Springer, New York, 1991).
28. W. A. Albarakati, N. G. Lloyd, and J. M. Pearson, "Transformation to Lienard Form," Electron. J. Diff. Eq., no. 76, 1-11 (2000).
29. G. A. Leonov, "The Problem of Estimation of the Number of Cycles for Two-Dimensional Quadratic Systems from the Point of View of Nonlinear Mechanics," Ukr. Math. J. 50(1), 55-65 (1998).
30. V. I. Arnol'd, "Experimental Mathematics," (Fazis, Moscow, 2005).
31. S. L. Shi, "A Concrete Example of the Existence of Four Limit Cycles for Plane Quadratic Systems," Sci. Sinica. 23, 153-158 (1980).