Conditions for polynomial Liénard centers. (English) Zbl 1353.34040
The main result is the following theorem:
Theorem. Let \(V\) be the center variety of the Liénard system \[ \dot x= y,\quad y=-x-b_2 x^2- b_3 x^3- b_4 x^4- b_5 x^5- y(a_1x+ a_2 x^2+ a_3 x^3+ a_4 x^4+ a_5 x^5). \] Then \[ V= \bigcup^3_{i=1} V(J_i), \] where \[ \begin{aligned} J_1 &=\langle a_2- a_1b_2, 3a_4- 5a_3 b_2, 3a_5- 2a_3 b^2_2, 3b_4- 5b_2 b_3, 3b_5- 2b^2_2 b_3\rangle,\\ J_2 &=\langle a_2- a_1 b_2, a_3- a_1 b_3, a_4- a_1 b_4, a_5- a_1 b_5\rangle,\\ J_3 &=\langle a_2,a_4, b_2, b_4\rangle.\end{aligned} \]
Theorem. Let \(V\) be the center variety of the Liénard system \[ \dot x= y,\quad y=-x-b_2 x^2- b_3 x^3- b_4 x^4- b_5 x^5- y(a_1x+ a_2 x^2+ a_3 x^3+ a_4 x^4+ a_5 x^5). \] Then \[ V= \bigcup^3_{i=1} V(J_i), \] where \[ \begin{aligned} J_1 &=\langle a_2- a_1b_2, 3a_4- 5a_3 b_2, 3a_5- 2a_3 b^2_2, 3b_4- 5b_2 b_3, 3b_5- 2b^2_2 b_3\rangle,\\ J_2 &=\langle a_2- a_1 b_2, a_3- a_1 b_3, a_4- a_1 b_4, a_5- a_1 b_5\rangle,\\ J_3 &=\langle a_2,a_4, b_2, b_4\rangle.\end{aligned} \]
Reviewer: A. P. Sadovskii (Minsk)
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 68W30 | Symbolic computation and algebraic computation |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 34C25 | Periodic solutions to ordinary differential equations |
Software:
SINGULARReferences:
| [1] | Becker T, Weispfenning V. Gröbner Bases: A Computational Approach to Commutative Algebra. New York: Springer, 1993 · Zbl 0772.13010 · doi:10.1007/978-1-4612-0913-3 |
| [2] | Blows T R, Lloyd N G. The number of small-amplitude limit cycles of Liénard equations. Math Proc Cambridge, 1984, 95: 359-366 · Zbl 0532.34022 · doi:10.1017/S0305004100061636 |
| [3] | Buchberger B. Ein algorithmus zum auffinden der basiselemente des restklassenringes nach einem nulldimensional polynomideal. PhD thesis. Austria: Universität Innsbruck, 1965 · Zbl 1245.13020 |
| [4] | Cherkas L A. Conditions for a Liénard equation to have a center. Differ Equ, 1977, 12: 201-206 · Zbl 0353.34031 |
| [5] | Christopher C. An algebraic approach to the classification of centers in polynomial Liénard systems. J Math Anal Appl, 1999, 229: 319-329 · Zbl 0921.34033 · doi:10.1006/jmaa.1998.6175 |
| [6] | Cox D, Little J, O’Shea D. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd ed. New York: Springer, 2007 · Zbl 1118.13001 · doi:10.1007/978-0-387-35651-8 |
| [7] | De Maesschalck P, Dumortier F. Bifurcations of multiple relaxation oscillations in polynomial Liénard equations. Proc Amer Math Soc, 2011, 139: 2073-2085 · Zbl 1230.37061 · doi:10.1090/S0002-9939-2010-10610-X |
| [8] | Dumortier F, Li C. Quadratic Liénard equations with quadratic damping. J Differential Equations, 1997, 139: 41-59 · Zbl 0881.34046 · doi:10.1006/jdeq.1997.3291 |
| [9] | Filippov A F. Sufficient conditions for existence of stable limit cycles of second order eqution (in Russian). Mat Sb, 1952, 30: 171-180 · Zbl 0046.31705 |
| [10] | Gasull A, Giacomini H. A new criterion for controlling the number of limit cycles of some generalized Li nard equations. J Differential Equations, 2002, 185: 54-73 · Zbl 1032.34028 · doi:10.1006/jdeq.2002.4172 |
| [11] | Gianni P, Trager B, Zacharias G. Gröbner bases and primary decomposition of polynomials. J Symbolic Comput, 1988, 6: 146-167 · Zbl 0667.13008 · doi:10.1016/S0747-7171(88)80040-3 |
| [12] | Greuel G-M, Pfister G. A Singular Introduction to Commutative Algebra, 2nd ed. Berlin: Springer, 2007 · Zbl 1133.13001 |
| [13] | Guckenheimer, J.; Holmes, P., Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (1983) · Zbl 0515.34001 |
| [14] | Hale J K. Ordinary Differential Equations, 2nd ed. Malabar: Krieger Publishing, 1980 · Zbl 0433.34003 |
| [15] | He Z, Zhang W. Subharmonic bifurcations in a perturbed nonlinear oscillation. Nonlinear Anal, 2005, 61: 1057-1091 · Zbl 1083.34033 · doi:10.1016/j.na.2005.01.095 |
| [16] | Hirsch, M. W.; Smale, S.; Devaney, R. L., Differential Equations, Dynamical Systems, and an Introduction to Chaos (2004) · Zbl 1135.37002 |
| [17] | Li C, Llibre J. Uniqueness of limit cycles for Li enard differential equations of degree four. J Differential Equations, 2012, 252: 3142-3162 · Zbl 1248.34035 · doi:10.1016/j.jde.2011.11.002 |
| [18] | Liénard A. Etude des oscillations entretenues. Rev Gen Electr, 1928, 23: 901-912; 946-954 |
| [19] | Llibre J, Mereu A C, Teixeira M A. Limit cycles of the generalized polynomial Liénard differential equations. Math Proc Cambridge, 2010, 148: 363-383 · Zbl 1198.34051 · doi:10.1017/S0305004109990193 |
| [20] | Mawhin J. Global Results for the Forced Pendulum Equation. Dordrecht: Elsevier, 2004 · Zbl 1091.34019 |
| [21] | Mcharg E A. A differential equation. J London Math Soc, 1947, 22: 83-85 · Zbl 0029.21302 · doi:10.1112/jlms/s1-22.2.83 |
| [22] | Opial Z. Sur un théorème de A. Filippoff. Ann Polon Math, 1958, 5: 67-75 · Zbl 0085.07402 |
| [23] | Reissig R, Sansone G, Conti R. Qualitative Theorie Nichtlinearer Differentialgleichungen. Rome: Cremonese, 1963 · Zbl 0114.04302 |
| [24] | Romanovski V G, Shafer D S. The Center and Cyclicity Problems: A Computational Algebra Approach. Boston: Birkhäuser, 2009 · Zbl 1192.34003 |
| [25] | Sansone G, Conti R. Nonlinear Differential Equations. New York: Macmillan, 1964 · Zbl 0128.08403 |
| [26] | Soudack A C, Barkham P G D. On the transient solution of the unforced duffing equation with large damping. Int J Control, 1971, 13: 767-769 · Zbl 0217.28301 · doi:10.1080/00207177108931981 |
| [27] | Sugie J, Hara T. Classification of global phase portraits of a system of Liénard type. J Math Anal Appl, 1995, 193: 264-281 · Zbl 0840.34020 · doi:10.1006/jmaa.1995.1234 |
| [28] | Wendel J G. On a van der Pol equation with odd coefficients. J London Math Soc, 1949, 24: 65-67 · Zbl 0031.39704 · doi:10.1112/jlms/s1-24.1.65 |
| [29] | Ye Y, Lo C. Theory of Limit Cycles, 2nd ed. Providence, RI: Amer Math Soc, 1986 · Zbl 0588.34022 |
| [30] | Yu S, Zhang J. On the center of the Liénard equation. J Differential Equations, 1993, 102: 53-61 · Zbl 0781.34022 · doi:10.1006/jdeq.1993.1021 |
| [31] | Zhang Z, Ding T, Huang W, et al. Qualitative Theory of Differential Equations. Providence, RI: Amer Math Soc, 1992 · Zbl 0779.34001 |
| [32] | Zhou Y, Wang X. On the conditions of a center of the Liénard equation. J Math Anal Appl, 1993, 100: 43-59 · Zbl 0809.34046 · doi:10.1006/jmaa.1993.1381 |
| [33] | Zou L, Chen X, Zhang W. Local bifurcations of critial periods for cubic Liénard equations with cubic damping. J Comput Appl Math, 2008, 222: 404-410 · Zbl 1163.34349 · doi:10.1016/j.cam.2007.11.005 |
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