Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T19:36:29.865Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Limit Cycles of Linear Differential Systems with Homogeneous Nonlinearities

Published online by Cambridge University Press:  20 November 2018

Jaume Llibre  and
Xiang Zhang
Jaume Llibre
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain e-mail: [email protected]
Xiang Zhang
Affiliation:
Department of Mathematics, MOE–LSC, Shanghai Jiao tong University, Shanghai, 200240, P. R. China e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the class of polynomial differential systems of the form $\dot{x} =\,\lambda x\,-\,y\,+\,{{P}_{n}}\left( x,\,y \right)$ , $\dot{y} =\,x\,+\,\lambda y\,+\,{{Q}_{n}}\left( x,\,y \right)$ where ${{P}_{n}}$ and ${{Q}_{n}}$ are homogeneous polynomials of degree $n$ . For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence.

Keywords

34C3534D30polynomial differential systemlimit cyclesdifferential equations on the cylinder
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 4 , 01 December 2015 , pp. 818 - 823
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Andronov, A. A., Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues. C.R. Acad. Sci. Paris 89(1929), 559–561.Google Scholar
[2] Carbonell, M. and Llibre, J., Limit cycles of a class of polynomial systems. Proc. Roy. Soc. Edinburgh Sect. A 109(1988), no. 1-2, 187–199. http://dx.doi.Org/10.1017/S0308210500026755 Google Scholar
[3] Carbonell, M. and Llibre, J., Hopf bifurcation, averaging methods and Liapunov quantities for polynomial systems with homogeneous nonlinearities. In: European Conference on Iteration Theory (Caldes de Malavella, 1987), World Sci. Publ., Teaneck, NJ, 1989, pp. 145–160.Google Scholar
[4] Cherkas, L. A., Number of limit cycles of an autonomous second-order system. Differential Equations 5(1976), 666–668.Google Scholar
[5] Coll, T., Gasull, A., and Prohens, R., Differential equations defined by the sum of two quasi-homogeneous vector fields. Canad. J. Math. 49(1997), 212–231. http://dx.doi.Org/10.4153/CJM-1997-011-0 Google Scholar
[6] Gasull, A. and Llibre, J., Limit cycles for a class of Abel equations. SIAM J. Math. Anal. 21(1990), 1235–1244. http://dx.doi.Org/10.1137/0521068 Google Scholar
[7] Liénard, A., Étude des oscillations entretenues. Rev. Générale de l'Electricité 23(1928), 901–912.Google Scholar
[8] Lins Neto, A., On the number of solutions of the equation dxjdt =E“=o aj(t)xi, 0 t < I, for which x(0) = %(1). Invent. Math. 59(1980), no. 1, 67-76. http://dx.doi.Org/10.1007/BF01390315 Google Scholar
[9] Liouville, R., Sur une équation différentielle du premier ordre. Acta Math. 27(1903), 55–78. http://dx.doi.Org/10.1007/BF02421296 Google Scholar
[10] Llibre, J. and Zhang, X., Non-existence, existence and uniqueness of limit cycles for quadratic polynomial differential systems. Proc. Roy. Soc. Edinburgh Sect. A, to appear.Google Scholar
[11] Lloyd, N. G., A note on the number of limit cycles in certain two-dimensional systems. J. London Math. Soc. 20(1979), 277–286. http://dx.doi.Org/10.1112/jlms/s2-20.2.2 77 Google Scholar
[12] Pliss, V. A., Non-local problems of the theory of oscillations. Academic Press, New York, 1966.Google Scholar
[13] Poincaré, H., Mémoire sur les courbes définies par une équation différentielle I, II. J. Math. Pures Appl. 7(1881), 375–422; 8(1882), 251-296; Sur les courbes définies pas les équations différentielles III, IV. 1(1885), 167-244; 2(1886), 155-217.Google Scholar
[14] van der Pol, , On relaxation-oscillations. Phil. Mag. 2(1926), 978-992.Google Scholar