Algebraic approach to the classification of centers in trigonometric Cherkas systems
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Abstract:
We give a complete algebraic characterization of the non-degenerated centers for planar trigonometric Cherkas systems. The main tools used in our proof are the classical results of Cherkas on planar analytic Liénard systems, the characterization of some subfields of the quotient field of the ring of trigonometric polynomials, and results due to Rosenlicht concerning algebraic solutions to transcendental equations. The results obtained are reminiscent of the ones for planar polynomial Cherkas systems, but the proofs are different.References
- A. F. Andreev, A. P. Sadovskiĭ, A. P. Sadovskiĭ, and V. A. Tsikalyuk, The center-focus problem for a system with homogeneous nonlinearities in the case of zero eigenvalues of the linear pact, Differ. Uravn. 39 (2003), no. 2, 147–153, 285 (Russian, with Russian summary); English transl., Differ. Equ. 39 (2003), no. 2, 155–164. MR 2134005, DOI 10.1023/A:1025192613518
- V. N. Belykh, N. F. Pedersen, and O. H. Soerensen, Shunted-Josephson-junction model. I. The autonomous case, Phys. Rev. B 16 (1977), 4853–4859.
- L. A. Čerkas, Conditions for a center for a certain Liénard equation, Differencial′nye Uravnenija 12 (1976), no. 2, 292–298, 379 (Russian). MR 0404755
- Colin Christopher, An algebraic approach to the classification of centers in polynomial Liénard systems, J. Math. Anal. Appl. 229 (1999), no. 1, 319–329. MR 1664344, DOI 10.1006/jmaa.1998.6175
- C. J. Christopher, N. G. Lloyd, and J. M. Pearson, On Cherkas’s method for centre conditions, Nonlinear World 2 (1995), no. 4, 459–469. MR 1360871
- Colin Christopher and Dana Schlomiuk, On general algebraic mechanisms for producing centers in polynomial differential systems, J. Fixed Point Theory Appl. 3 (2008), no. 2, 331–351. MR 2434452, DOI 10.1007/s11784-008-0077-2
- Anna Cima, Armengol Gasull, and Francesc Mañosas, A simple solution of some composition conjectures for Abel equations, J. Math. Anal. Appl. 398 (2013), no. 2, 477–486. MR 2990073, DOI 10.1016/j.jmaa.2012.09.006
- Anna Cima, Armengol Gasull, and Francesc Mañosas, An explicit bound of the number of vanishing double moments forcing composition, J. Differential Equations 255 (2013), no. 3, 339–350. MR 3053469, DOI 10.1016/j.jde.2013.04.009
- A. Gasull, A. Geyer, and F. Mañosas, On the number of limit cycles for perturbed pendulum equations, J. Differential Equations 261 (2016), no. 3, 2141–2167. MR 3501844, DOI 10.1016/j.jde.2016.04.025
- Armengol Gasull, Jaume Giné, and Claudia Valls, Center problem for trigonometric Liénard systems, J. Differential Equations 263 (2017), no. 7, 3928–3942. MR 3670041, DOI 10.1016/j.jde.2017.05.008
- Jaume Giné, Maite Grau, and Jaume Llibre, Universal centres and composition conditions, Proc. Lond. Math. Soc. (3) 106 (2013), no. 3, 481–507. MR 3048548, DOI 10.1112/plms/pds050
- Kenji Inoue, Perturbed motion of a simple pendulum, J. Phys. Soc. Japan 57 (1988), no. 4, 1226–1237. MR 949109, DOI 10.1143/JPSJ.57.1226
- A. D. Morozov, Limit cycles and chaos in equations of pendulum type, Prikl. Mat. Mekh. 53 (1989), no. 5, 721–730 (Russian); English transl., J. Appl. Math. Mech. 53 (1989), no. 5, 565–572 (1990). MR 1040438, DOI 10.1016/0021-8928(89)90101-9
- Albert D. Morozov, Quasi-conservative systems, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 30, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. Cycles, resonances and chaos. MR 1832323, DOI 10.1142/9789812796318
- Maxwell Rosenlicht, On the explicit solvability of certain transcendental equations, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 15–22. MR 258808
- J. A. Sanders, Report on the driven Josephson equation, Dyn. Syst. Chaos, Lecture Notes in Physics, vol. 179, Springer, Berlin-Heidelberg, 1983, pp. 297–298.
- Jan A. Sanders and Richard Cushman, Limit cycles in the Josephson equation, SIAM J. Math. Anal. 17 (1986), no. 3, 495–511. MR 838238, DOI 10.1137/0517039
- B. L. van der Waerden, Algebra. Teil I, Heidelberger Taschenbücher, Band 12, Springer-Verlag, Berlin-New York, 1966 (German). Siebte Auflage. MR 0263581
- Henryk Żołądek, The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal. 4 (1994), no. 1, 79–136. MR 1321810, DOI 10.12775/TMNA.1994.024
Additional Information
- Claudia Valls
- Affiliation: Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal
- MR Author ID: 636500
- Email: cvalls@math.ist.utl.pt
- Received by editor(s): October 24, 2017
- Received by editor(s) in revised form: April 13, 2018
- Published electronically: March 15, 2019
- Additional Notes: The author was partially supported by FCT/Portugal through UID/MAT/04459/2013.
- Communicated by: Wenxian Shen
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 2863-2875
- MSC (2010): Primary 34C25; Secondary 37C10, 37C27
- DOI: https://doi.org/10.1090/proc/14285
- MathSciNet review: 3973890