Paper title: Integrability and a Limit Cycle Solver for A Generalization of Polynomial Liénard Differential Systems
DOI: https://doi.org/10.4316/JACSM.201802007
Published in: Issue 2, (Vol. 12) / 2018Download
Publishing date: 2018-10-15
Pages: 41-47
Author(s): FENENICHE Fatima, REZAOUI Med-Salem
Abstract. n this paper, we study the periodic orbits of the second- order differential equation. We find a way of study the integrability liénard systems in the plane. In this article such problem is formulated in the more general framework of Poincarée Bendixson structures, which include Hamiltonian systems as a particular case. We deal with the analyticity of the second integral of any (possibly degenerate) center of an analytic planar differential system. A concrete example as application is given
Keywords: Integrating Factors, Limit Cycles, Ordinary Differential Equations
References:1. A.A. Andronov, Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues, C. R. Acad. Sci. Paris 89 (1929) 559-561.
2. A. Agarwal and N. Ananthkrishnan, Bifurcation analysis for onset and ces-sation of surge in axial flow compressors, Internat. J. Turbo Jet Engines, 17-3 (2000), 207-217.
3. A. Buica, J. Giné, J. Llibre, Bifurcation of limit cycles from a polynomial degenerate center, Adv. Nonlinear Stud. 10 (2010) 597-609.
4. A. Liénard, Étude des oscillations entrenues, Rev. Gén. Électricité, 23 (1928),946-954.
5. A. L. Hodgkin and A. F. Huxley, A qualitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol.,117 (1952), 500-544; reproduced in Bull. Math. Biol., 52 (1990), 25-71.
6. Anco, S. C. Bluman, G. W. , Integrating factors and first integrals for ordinary differential equations , University of British Columbia 1997
7. B. van der Pol, On relaxation-oscillations , Phil. Mag. 2 (1926) 978-992.
8. C. C. Rocsoreanu, A. Georgeson, and N. Giurgiteanu, The Fitzhugh-Nagumo Model: Bifur-cation and Dynamics, Kluwer, Dordrecht, Netherlands, 2000.
9. C. Liu, G. Chen and J. Yang, On the hyperelliptic limit cycles of Lienard systems, Puplished 20 April 2012.
10. D. B. Owens, F. J. Capone, R. M. Hall, J. M. Brandon, and J. R. Cham-bers, Transonic free-to-roll analysis of abrupt wing stall on military aircraft, J. Aircraft , 41-3 (2004), 474-484
11. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer-Verlag, Berlin, 1996.
12. J. Bricmont, Introduction á la dynamique non linéaire, Unité de la physique théorique et mathématique académique 2009-2010.
13. J. le Bourlot, Introductions aux systémes dynamique dissipatifs , P6.P7.P11.observatoire de paris 2011-2012.
14. J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating 1214-nerve axons, Proc. IRL, 50 (1970), 2061-2070.
15. J.Quinet, J.Fazekas, Cours élémentaire de Mathématiques supérieures Tome 5 page 81-94 Dunod, Bordas, paris,1998, ISBN 2-04-010079-2.
16. J. Rayleigh, The Theory of Sound , Dover, New York, 1945.
17. H. Chen, Y. Li, Bifurcation and stability of periodic solutions of Duffing equations , Nonlinearity 21 (2008) 2485-2503.
18. H. Giacomini, S.Neukirch, Algebraic approximations to bifurcation curves of limit cycles for the lienard equation, Chao-dyn =9707002 v130 Jun 1997.
19. L. Forest, N. Glade, J. Demongeot, Lienard systems and potentiel Hamiltonian decomposition appliations in biology, Hal -00346975, version 1-13 Dec 2008.
20. L.Perko, Differential equation and dynamical systems , ISBN 0-387-95116-4 www.springer-ny.com.
21. O.Abu Arqub, Adaptation of reproducing Kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations, Neural Comput& Applic (2017)28:1591-1610.
22. O.Abu Arqub, Z. Abo-Hammour, S.Momani, Application of continuous Genetic algorithm for nonlinear system of second-order boundary value problems, Appl.Math.Inf.Sci.8,No.1, 235-248(2014).
23. O.Abu Arqub, Z. Abo-Hammour, Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm, Information Sciences 279 (2014) 396-415.
24. R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membranes, J. Biophys., 1182 (1961), 445-466.
25. V.I.ARNOLD, Méthodes mathématiques de la mécanique classique. Mir, Moscou 1976
26. W. Xu, C. Li, Limit cycles of some polynomial Liénard systems, J. Math. Anal. Appl. 389 (2012) 367-378.

Back to the journal content
Creative Commons License
This article is licensed under a
Creative Commons Attribution-ShareAlike 4.0 International License.
Home | Editorial Board | Author info | Archive | Contact
Copyright JACSM 2007-2018