Center-focus problem for discontinuous planar differential equations

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 13, Issue 7, 2003, Pages 1755-1765

  Gasull, Armengol ^a   Torregrosa, Joan ^a  

^a UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

Author keywords

Center problem; Discontinuous ordinary differential equation; Limit cycle; Lyapunov constant; Nonsmooth system

Indexed keywords

LYAPUNOV METHODS; MATHEMATICAL MODELS; ORDINARY DIFFERENTIAL EQUATIONS;

POLAR COORDINATES;

BIFURCATION (MATHEMATICS);

EID: 0141454809     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127403007618     Document Type: Conference Paper

References (23)

1. Andronov, A. A., Leontovich, E. A., Gordon, I. I. & Maǐer, A. G. [1973] Theory of Bifurcations of Dynamic Systems on a Plane (Halsted Press, A division of John Wiley & Sons, NY-Toronto, Ont), translated from Russian.
2. Andronov, A. A., Vitt, A. A. & Khaǐkin, S. È. [1987] Theory of Oscillators (Dover Publications Inc., NY), translated from Russian by Immirzi, F. Reprint of the 1966 translation.
3. Bautin, N. N. [1954] "On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type," Amer. Math. Soc. Transi. 100.
4. Cartan, E. [1970] Differential Forms (Houghton Mifflin Co., Boston, MA).
5. Coll, B., Prohens, R. & Gasull, A. [1999] "The center problem for discontinuous Liénard differential equation," Int. J. Bifurcation and Chaos 9, 1751-1761.
6. Coll, B., Gasull, A. & Prohens, R. [2000] "Center-focus and isochronous center problems for discontinuous differential equations," Discr. Contin. Dyn. Syst. 6, 609-624.
7. Coll, B., Gasull, A. & Prohens, R. [2001] "Degenerate Hopf bifurcation in discontinuous planar systems," J. Math. Anal. Appl. 253, 671-690.
8. Filippov, A. F. [1988] Differential Equation with Discontinuous Righthand Sides (Kluwer Academic Pub., Netherlands).
9. Françoise, J. P. [1996] "Successive derivatives of a first return map, application to the study of quadratic vector fields," Ergod. Th. Dyn. Syst. 16, 87-96.
10. Françoise, J. P. [1997] "The first derivative of the period function of a plane vector field," Publ. Matemat. 41, 127-134.
11. Françoise, J. P. [1998] "The successive derivatives of the period function of a plane vector field," J. Diff. Eqs. 146, 320-335.
12. Gasull, A. & Torregrosa, J. [2001] "A relation between small amplitude limit cycles and big limit cycles," Rocky Mountain J. Math. 31, 1277-1303.
13. Iliev, I. D. [1998] "On second order bifurcations of limit cycles," J. London Math. Soc. 58, 353-366.
14. Iliev, I. D. & Perko, L. M. [1999] "Higher order bifurcations of limit cycles," J. Diff. Eqs. 154, 339-363.
15. Kunze, M. [2000] Non-smooth Dynamical Systems (Springer-Verlag, Berlin).
16. Lunkevič, V. A. [1968] "The problem of the center for differential equations with discontinuous right-hand sides," Diff. Eqs. 4, 837-844; Translation [1968] Diff. Uravn. 4, 1625-1639.
17. Lunkevič, V. A. [1968] "The problem of the center for differential equations with discontinuous right-hand sides," Diff. Eqs. 4, 837-844; Translation [1968] Diff. Uravn. 4, 1625-1639.
18. Lunkevič, V. A. & Sibirskiľ, K. S. [1982] "Integrals of a general quadratic differential system in cases of the center," Diff. Eqs. 18, 563-568; Translation [1982] Diff. Uravn. 18, 786-792.
19. Lunkevič, V. A. & Sibirskiľ, K. S. [1982] "Integrals of a general quadratic differential system in cases of the center," Diff. Eqs. 18, 563-568; Translation [1982] Diff. Uravn. 18, 786-792.
20. Pleškan, 1.1. & Sibirskiľ, K. S. [1973] "On the problem of the center for systems with discontinuous right-hand sides," Diff. Eqs. 9, 1396-1402; Translation [1973] Diff. Uravn. 9, 1817-1825.
21. Pleškan, 1.1. & Sibirskiľ, K. S. [1973] "On the problem of the center for systems with discontinuous right-hand sides," Diff. Eqs. 9, 1396-1402; Translation [1973] Diff. Uravn. 9, 1817-1825.
22. Poggiale, J. C. [1994] "Applications des variétés invariantes à la modélisation de l'hétérogénéité en dynamique des populations," Ph.D. thesis, Université de Bourgogne.
23. Roussarie, R. [1998] Bifurcation of Planar Vector Fields and Hilbert's Sixteenth Problem (Birkhäuser Verlag, Basel).