Lyapunov quantities, limit cycles and strange behavior of trajectories in two-dimensional quadratic systems

Journal of Vibroengineering

Volumn 10, Issue 4, 2008, Pages 460-467

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

^a ST PETERSBURG STATE UNIVERSITY   (Russian Federation)

Author keywords

(Normal amplitude) large limit cycle; Lienard equation; Lyapunov quantity; Period constant; Poincare Lyapunov constant; Quadratic system; Small limit cycle; Two dimensional autonomous systems

Indexed keywords

EID: 58149248005     PISSN: 13928716     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Conference Paper

References (29)

1. Bautin N.N. Behavior of Dynamic Systems Near the Boundaries of the Stability Region, Gostekhizdat, Moscow-Leningrad, 1949 [in Russian]
2. Andronov A. A., Vitt A. A., Khaikin S. E. Theory of oscillators, Oxford, New York: Pergamon Press, 1966
3. Poincare H. Memoire sur les courbes definies par lesequations diffeentielles, J. de Mathemantiques Pures et Appliquees, (4) Vol. 1, pp. 167-244, 1885
4. Lyapunov A. M. Stability of Motion. New York and London: Academic Press, 1966 (1892)
5. Lloyd N. G. Limit cycles of polynomial systems - some recent developments. In New Direction in Dynamical Systems. Cambridge University Press, pp. 182-234, 1988
6. Li J. Hilbert's 16th problem and bifurcation of planar polynomial vector fields, Int. J. Bifurcation Chaos 13, pp. 47-106, 2003
7. Chavarriga J., Grau M. Some open problems related to 16th Hilbert problem, Sci. Ser. A Math. Sci. N.S., 9, pp. 1-26, 2003
8. Lynch S. Symbolic computation of Lyapunov quantities and the second part of Hilbert's sixteenth problem. Chapter in the book Differential Equations with Symbolic Computations, pp. 1-26, 2005
9. Gine J. On some problems in planar differential systems and Hilbert's 16th problem, Chaos, Solutions and Fractals, Vol. 31, pp. 1118-1134, 2007
10. Yu P., Chen G. Computation of focus values with applications, Nonlinear Dynamics, 51(3), 2008
11. Leonov G. A. Criterion for cycles existence in quadratic systems. Vestnik of St.Petersburg University, 3, pp. 31-41, 2007
12. Kuznetsov N. V., Leonov G. A. Computation of the first Lyapunov quantity for the second-order dynamical system. Third IFAC Workshop Periodic Control System, 2007.
13. Leonov G. A. Hilbert's 16th problem for quadratic system. New method based on a transformation to the Lienard equation. International journal of bifurcation and chaos, 18(3), pp. 877-884, 2008
14. Kuznetsov N.V., Leonov G.A. Computation of Lyapunov quantities. 6th EUROMECH Nonlinear Dynamics Conference (http://lib.physcon.ru/ ?item=1802), 2008
15. Leonov G. A., Kuznetsov N. V., Kudryashova E. V. Limit cycles of two dimensional systems. Calculations, proofs, experiments. Vestnik of St.Petersburg University, N3, pp.25-62, 2008
16. Dumortier F., Llibre J., Artes J. Qualitative Theory of Planar Differential Systems, Springer, 2006
17. Shilnikov L., Turaev D., Chua L. Methods of Qualitative Theory in Nonlinear Dynamics: Part 2, World Scientific, 2001
18. Christopher C., Li C. Limit Cycles of Differential Equations, Birkhauser, 2007
19. Perko L. Differential Equations and Dynamical Systems, Springer, 2001
20. Roussarie R. Bifurcation of Planar Vector Fields and Hilbert's Sixteenth Problem, Birkhauser, 1998
21. Gasull A., Guillamon A. and Manosa V. An explicit expression of the first Liapunov and period constants with applications. J. Math. Anal. Appl., 211, pp. 190-212, 1997
22. Leonov G. A. Family of transversal curves for two-dimensional dynamical systems. Vestnik St.Petersburg University, 1(4), pp. 48-78, 2006
23. Serebryakova N. Behavior of dynamical system with one degree of freedom near the points of boundaries of stability domain, where a safe boundary goes to the dangerous one [in Russian]. Izvestiya AN SSSR. Division of technical sciences, mechanics and engineering, 2, pp. 178-182, 1959
24. Lefschetz S. Differential Equations: Geometric Theory. Interscience Publishers. New York. 1957
25. Cesari L. Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. Springer. Berlin. 1959
26. Hartman P. Ordinary differential equation. John Willey & Sons. New York. 1964
27. Leonov G.A., Kuznetsov N.V. Time-Varying Linearization and Perron effects, International journal of bifurcation and chaos. Vol. 17, Iss. 4, pp. 1-29, 2007
28. Leonov G.A. Two-Dimensional Quadratic Systems as a Lienard Equation, Differential Equations and Dynamical Systems, Vol. 5. N3/4, 1997
29. Shi S.L. A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica 23. pp. 153-158, 1980