On the number of limit cycles in generalized Abel equations. (English) Zbl 1472.34065
Consider the generalized Abel differential equation
\[
\frac{{dx}}{{d\theta}} = A(\theta) x^p + B(\theta) x^q,\tag{1}
\]
where \( p,q \) are natural numbers satisfying \( p \neq q, p,q \ge 2\), \(A\) and \(B \) are trigonometric polynomials of degree \(n \ge 1\) and \(m \ge 1\), respectively. Let the number \(H_{p,q}(n,m) \) denote the maximum number of isolated periodic solutions (limit cycles) of (1). By means of the second order Melnikov function the authors prove a lower bound for \(H_{p,q}(n,m) \), which is better than known ones. Especially, they obtain for the classical Abel equation (i.e. \( p=3, q=2\)) the estimate \( H_{3,2}(n,m) \geq 2(n+m)-1 \).
Reviewer: Klaus R. Schneider (Berlin)
MSC:
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 37C60 | Nonautonomous smooth dynamical systems |
References:
| [1] | N. Alkoumi and P. J. Torres, On the number of limit cycles of a generalized Abel equation, Czechoslovak Math. J., 61 (2011), pp. 73-83, https://doi.org/10.1007/s10587-011-0018-x. · Zbl 1224.34097 |
| [2] | N. M. H. Alkoumi and P. J. Torres, Estimates on the number of limit cycles of a generalized Abel equation, Discrete Contin. Dyn. Syst., 31 (2011), pp. 25-34, https://doi.org/10.3934/dcds.2011.31.25. · Zbl 1372.34075 |
| [3] | A. Álvarez, J. L. Bravo, and M. Fernández, Limit cycles of Abel equations of the first kind, J. Math. Anal. Appl., 423 (2015), pp. 734-745, https://doi.org/10.1016/j.jmaa.2014.10.019. · Zbl 1326.34027 |
| [4] | M. J. Álvarez, J. L. Bravo, M. Fernández, and R. Prohens, Centers and limit cycles for a family of Abel equations, J. Math. Anal. Appl., 453 (2017), pp. 485-501, https://doi.org/10.1016/j.jmaa.2017.04.017. · Zbl 1369.34059 |
| [5] | M. J. Álvarez, J. L. Bravo, M. Fernández, and R. Prohens, Alien limit cycles in Abel equations, J. Math. Anal. Appl., 482 (2020), 123525, https://doi.org/10.1016/j.jmaa.2019.123525. · Zbl 1434.34035 |
| [6] | M. J. Álvarez, A. Gasull, and J. Yu, Lower bounds for the number of limit cycles of trigonometric Abel equations, J. Math. Anal. Appl., 342 (2008), pp. 682-693, https://doi.org/10.1016/j.jmaa.2007.12.016. · Zbl 1362.34068 |
| [7] | A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Ma\u\ier, Theory of Bifurcations of Dynamic Systems on a Plane, Halsted Press, New York, 1973. · Zbl 0282.34022 |
| [8] | V. I. Arnol’d, Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields, Funct. Anal. Appl., 11 (1977), pp. 85-92, https://doi.org/10.1007/BF01081886. · Zbl 0411.58013 |
| [9] | A. Buică, On the equivalence of the Melnikov functions method and the averaging method, Qual. Theory Dyn. Syst., 16 (2017), pp. 547-560, https://doi.org/10.1007/s12346-016-0216-x. · Zbl 1392.34052 |
| [10] | L. A. Cherkas, The number of limit cycles of a certain second order autonumous system, Differencial’nye Uravnenija, 12 (1976), pp. 944-946. · Zbl 0326.34036 |
| [11] | A. Cima, A. Gasull, V. Man͂osa, and F. Man͂osas, Algebraic properties of the Liapunov and period constants, Rocky Mountain J. Math., 27 (1997), pp. 471-501, https://doi.org/10.1216/rmjm/1181071923. · Zbl 0911.34025 |
| [12] | J. Devlin, N. G. Lloyd, and J. M. Pearson, Cubic systems and Abel equations, J. Differential Equations, 147 (1998), pp. 435-454, https://doi.org/10.1006/jdeq.1998.3420. · Zbl 0911.34020 |
| [13] | E. Fossas, J. M. Olm, and H. Sira-Ramírez, Iterative approximation of limit cycles for a class of Abel equations, Phys. D, 237 (2008), pp. 3159-3164, https://doi.org/10.1016/j.physd.2008.05.011. · Zbl 1160.34025 |
| [14] | A. Frabetti and D. Manchon, Five interpretations of Faà di Bruno’s formula, in Faà di Bruno Hopf Algebras, Dyson-Schwinger Equations, and Lie-Butcher Series, IRMA Lect. Math. Theor. Phys. 21, European Mathematical Society, Zürich, 2015, pp. 91-147. · Zbl 1355.16031 |
| [15] | A. Gasull, From Abel’s differential equations to Hilbert’s sixteenth problem, Butl. Soc. Catalana Mat., 28 (2013), pp. 123-146. · Zbl 1322.34052 |
| [16] | A. Gasull and A. Guillamon, Limit cycles for generalized Abel equations, Internat. J. Bifur. Chaos, 16 (2006), pp. 3737-3745, https://doi.org/10.1142/S0218127406017130. · Zbl 1140.34348 |
| [17] | A. Gasull, C. Li, and J. Torregrosa, A new Chebyshev family with applications to Abel equations, J. Differential Equations, 252 (2012), pp. 1635-1641, https://doi.org/10.1016/j.jde.2011.06.010. · Zbl 1233.41003 |
| [18] | A. Gasull and J. Llibre, Limit cycles for a class of Abel equations, SIAM J. Math. Anal., 21 (1990), pp. 1235-1244, https://doi.org/10.1137/0521068. · Zbl 0732.34025 |
| [19] | A. Gasull and J. Torregrosa, Some results on rigid systems, in EQUADIFF 2003, World Scientific, Hackensack, NJ, 2005, pp. 340-345, https://doi.org/10.1142/9789812702067_0052. · Zbl 1107.34310 |
| [20] | A. Gasull and Y. Zhao, On a family of polynomial differential equations having at most three limit cycles, Houston J. Math., 39 (2013), pp. 191-203. · Zbl 1283.34030 |
| [21] | J. Giné, L. F. S. Gouveia, and J. Torregrosa, Lower Bounds for the Local Cyclicity for Families of Centers, preprint, 2020. |
| [22] | L. F. Gouveia and J. Torregrosa, The Local Cyclicity Problem. Melnikov Method Using Lyapunov Constants, preprint, 2020. |
| [23] | T. Harko and M. K. Mak, Relativistic dissipative cosmological models and Abel differential equation, Comput. Math. Appl., 46 (2003), pp. 849-853, https://doi.org/10.1016/S0898-1221(03)90147-7. · Zbl 1100.83514 |
| [24] | T. Harko and M. K. Mak, Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach, Math. Biosci. Eng., 12 (2015), pp. 41-69, https://doi.org/10.3934/mbe.2015.12.41. · Zbl 1316.35070 |
| [25] | J. Huang and H. Liang, Estimate for the number of limit cycles of Abel equation via a geometric criterion on three curves, NoDEA Nonlinear Differential Equations Appl., 24 (2017), 47, https://doi.org/10.1007/s00030-017-0469-3. · Zbl 1382.34031 |
| [26] | Y. Ilyashenko, Centennial history of Hilbert’s \(16\) th problem, Bull. Amer. Math. Soc. (N.S.), 39 (2002), pp. 301-354. · Zbl 1004.34017 |
| [27] | W. P. Johnson, The curious history of Faà di Bruno’s formula, Amer. Math. Monthly, 109 (2002), pp. 217-234, https://doi.org/10.2307/2695352. · Zbl 1024.01010 |
| [28] | S. Karlin and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Pure Appl. Math. XV, Interscience, New York, 1966. · Zbl 0153.38902 |
| [29] | O. G. Khuda\u\iberenov and N. O. Khuda\u\iverenov, An estimate for the number of periodic solutions of the Abel equation, Differential Equations, 40 (2004), pp. 1209-1212, 1152, https://doi.org/10.1023/B:DIEQ.0000049841.77318.dc. · Zbl 1087.34517 |
| [30] | M. Krusemeyer, Why Does the Wronskian Work?, Amer. Math. Monthly, 95 (1988), pp. 46-49, https://doi.org/10.2307/2323448. · Zbl 0661.34009 |
| [31] | A. Lins Neto, On the number of solutions of the equation \(dx/dt=\sum^n_{j=0}\,a_j(t)x^j, 0\leq t\leq 1\), for which \(x(0)=x(1)\), Invent. Math., 59 (1980), pp. 67-76, https://doi.org/10.1007/BF01390315. · Zbl 0448.34012 |
| [32] | M.-L. Mazure, Chebyshev spaces and Bernstein bases, Constr. Approx., 22 (2005), pp. 347-363, https://doi.org/10.1007/s00365-004-0583-4. · Zbl 1116.41006 |
| [33] | V. A. Pliss, Nonlocal Problems of the Theory of Oscillations, Academic Press, New York, 1966. · Zbl 0151.12104 |
| [34] | G. Pólya and G. Szegö, Problems and Theorems in Analysis II, Classics Math., Springer, Berlin, 1998, https://doi.org/10.1007/978-3-642-61905-2_7. · Zbl 1024.00003 |
| [35] | T. J. Rivlin, Chebyshev Polynomials, From Approximation Theory to Algebra and Number Theory, 2nd ed., Pure Appl. Math., Wiley, New York, 1990. · Zbl 0734.41029 |
| [36] | R. Roussarie, Bifurcations of Planar Vector Fields and Hilbert’s Sixteenth Problem, Progr. Math. 164, Springer, Berlin, 1998. · Zbl 0898.58039 |
| [37] | S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), pp. 7-15, https://doi.org/10.1007/BF03025291. · Zbl 0947.01011 |
| [38] | A. V. Yurov, A. V. Yaparova, and V. A. Yurov, Application of the Abel equation of the 1st kind to inflation analysis of non-exactly solvable cosmological models, Gravit. Cosmol., 20 (2014), pp. 106-115, https://doi.org/10.1134/S0202289314020121. · Zbl 1338.83232 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
