An explicit expression of the first Liapunov and period constants with applications

Journal of Mathematical Analysis and Applications

Volumn 211, Issue 1, 1997, Pages 190-212

  Gasull, Armengol ^a   Guillamon, Antoni ^b   Mañosa, Víctor ^b  

^a UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

^b UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

ALGEBRA; CALCULATIONS; COMPUTER APPLICATIONS; DIFFERENTIAL EQUATIONS; EIGENVALUES AND EIGENFUNCTIONS; FUNCTIONS; INTEGRATION; LYAPUNOV METHODS; PROBLEM SOLVING;

LIAPUNOV CONSTANTS; PURE IMAGINARY EIGENVALUES;

MATHEMATICAL TECHNIQUES;

EID: 0031172970     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmaa.1997.5455     Document Type: Article

References (18)

1. 1. M. A. M. Alwash and N. G. Lloyd, Non-Autonomous equations related to polynomial two-dimensional systems, Proc. Roy. Soc. Edinburgh Sect. A 105 (1987), 129-152.
2. 2. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, "Theory of Bifurcations of Dynamic Systems on a Plane," Wiley, New York/Toronto, 1973.
3. 3. A. Cima, A. Gasull, V. Mañosa, and F. Mañosas, Algebraic properties of the Liapunov and period constants, Rocky Mountain J. Math., in press.
4. 4. C. Chicone and M. Jacobs, Bifurcations of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989), 433-486.
5. 5. R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1 (1961), 445-466.
6. 6. W. W. Farr, C. Li, I. S. Labouriau, and W. F. Langford, Degenerate Hopf-bifurcation formulas and Hubert's 16th problem, SIAM J. Math. Anal. 20 (1989), 13-29.
7. 7. E. Gamero, "Computación simbólica y bifurcaciones de sistemas dinámicos," Ph.D. Thesis, Univ. de Sevilla.
8. 8. A. Gasull, A. Guillamon, C. Li, and Z. Zhifen, Study of perturbed Lotka-Volterra systems via Abelian integrals, J. Math. Anal. Appl. 198 (1996), 703-728.
9. 9. M. Golubitsky and W. Langford, Classification and unfoldings of degenerate Hopf bifurcations, J. Differential Equations 41 (1981), 375-415.
10. 10. A. Gasull and R. Prohens, Effective computation of the first Liapunov quantities for a planar differential equation, preprint, Univ. Autònoma de Barcelona.
11. 11. I. S. Gradshteyn, and I. M. Ryzhik, "Table of Integrals, Series and Products," Academic Press, New York, 1980.
12. 12. F. Göbber and K. D. Willamowski, Liapunov approach to multiple Hopf bifurcation, J. Math. Anal. Appl. 71 (1979), 333-350.
13. 13. I. D. Hsü and N. D. Kazarinoff, An applicable Hopf bifurcation formula and instability of small periodic solutions of the Field-Noyes model, J. Math. Anal. Appl. 55 (1976), 61-89.
14. 14. B. Hassard and Y. H. Wan, Bifurcation formulae derived from center manifold theory, J. Math. Anal. Appl. 63 (1978), 297-312.
15. 15. N. G. Lloyd, Limit cycles of polynomial systems: New directions in dynamical systems, in "London Math. Soc. Lecture Note Ser.," Vol. 127, pp. 192-234, Cambridge Univ. Press, Cambridge, UK, 1988.
16. 16. E. J. Ponce-Núñez and E. Gamero, Generating Hopf bifurcation formulae with MAPLE, Internat. Ser. Numer. Math. 97 (1991), 295-298.
17. 17. V. G. Romanovskii, Calculation of Lyapunov numbers in the case of two pure imaginary roots, Differential Equations 29 (1993), 782-784.
18. 18. M. Villarini, Regularity properties of the period function near a centre of a planar vector field, Nonlinear Anal. 19 (1992), 787-803.