Hopf bifurcation and the centers on center manifold for a class of three-dimensional circuit system.(English)Zbl 1453.34069
The authors study the three-dimensional Chua’s circuit system with \(f(x)=c x^2\) and \(f(x) =c x^3\). They compute the order of all fine foci on the local center manifolds for both cases. The tools utilized here are calculations of normal forms and Darboux methods.
Reviewer: Hao Wu (Nanjing)
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 94C60 | Circuits in qualitative investigation and simulation of models |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 34C45 | Invariant manifolds for ordinary differential equations |
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Full Text:DOI
References:
| [1] | Li, J.; Liu, Y., New results on the study of Zq‐equivariant planar polynomial vector fields, Qual Theory Dyn Syst, 9, 167-219, 2010 ·Zbl 1213.34059 |
| [2] | Li, C.; Liu, C.; Yang, J., A cubic system with thirteen limit cycles, J, Differ Equ, 246, 3609-3619, 2009 ·Zbl 1176.34037 |
| [3] | Li, J., Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Int J Bifurc Chaos, 13, 47-106, 2003 ·Zbl 1063.34026 |
| [4] | Romanovski, VG; Shafer, DS, The center and cyclicity problems: a computational algebra approach, 2009, Springer Science & Business Media: Birkhäuser Boston, Inc.: Boston, MA ·Zbl 1192.34003 |
| [5] | Guo, L.; Yu, P.; Chen, Y., Twelve limit cycles in 3D quadratic vector fields with Z_3 symmetry, Int J Bifurc Chaos, 28, 2018 ·Zbl 1404.34033 |
| [6] | Dulac, H., D’etermination et int’egration d’une certaine classe d equations diff’erentielles ayant pour point singulier un centre, Bull Sci Math, 32, 230-252, 1908 ·JFM 39.0374.01 |
| [7] | Yu, P.; Tian, Y., Twelve limit cycles around a singular point in a planar cubic‐degree polynomial system, Commun Nonlinear Sci Numer Simulat, 19, 2690-2705, 2014 ·Zbl 1510.37083 |
| [8] | Wang, Q.; Liu, Y.; Chen, H., Hopf bifurcation for a class of three‐dimensional nonlinear dynamic systems, Bull Sci Math, 134, 786-798, 2010 ·Zbl 1204.37051 |
| [9] | Yu, P.; Han, M., Ten limit cycles around a center‐type singular point in a 3‐d quadratic system with quadratic perturbation, Appl Math Lett, 44, 17-20, 2015 ·Zbl 1336.34051 |
| [10] | Gyllenberg, M.; Yan, P., Four limit cycles for a 3D competitive Lotka‐Volterra system with a heteroclinic cycle, Comput Math Appl, 58, 649-669, 2009 ·Zbl 1189.34080 |
| [11] | Wang, Q.; Wu, H.; Li, B., Limit cycles and singular point quantities for a 3D Lotka‐Volterra system, Appl Math Comput, 217, 8856-8859, 2011 ·Zbl 1220.34051 |
| [12] | Liu, L.; Aybar, OO; Romanovski, VG; Zhang, W., Identifying weak foci and centers in Maxwell‐Bloch system, J Math, 430, 549-571, 2015 ·Zbl 1322.34050 |
| [13] | García, IA; Maza, S.; Shafer, DS, Cyclicity of polynomial nondegenerate centers on center manifolds, J Differ Equations, 265, 5767-5808, 2018 ·Zbl 1434.37030 |
| [14] | García, IA; Maza, S.; Shafer, DS, Center cyclicity of Lorenz, Chen and Lü systems, Nonlinear Anal, 188, 362-376, 2019 ·Zbl 1428.37046 |
| [15] | Matsumoto, T.; Chua, LO; Komuro, M., The double scroll, IEEE Trans Circuits Syst, 32, 797-818, 1985 ·Zbl 0578.94023 |
| [16] | Chua, LO; Komuro, M.; Matsumoto, T., The double scroll family, IEEE Trans Circuits Syst, 33, 1072-1118, 2003 ·Zbl 0634.58015 |
| [17] | Messias, M.; Braga, DC; Mello, LF, Degenerate Hopf bifurcations in Chua’s system, Int J Bifurc Chaos, 19, 497-515, 2009 ·Zbl 1170.34333 |
| [18] | Llibre, J.; Makhlouf, A.; Badi, S., 3‐dimensional Hopf bifurcation via averaging theory of second order, Discrete Cont Dyn Syst, 25, 1287-1295, 2009 ·Zbl 1186.37059 |
| [19] | Barreira, L.; Llibre, J.; Valls, C., Bifurcation of limit cycles from a 4‐dimensional center in ℝ^m in resonance 1: N, J Math Anal Appl, 389, 754-768, 2012 ·Zbl 1242.34066 |
| [20] | Buica, A.; Garca, I.; Maza, S., Existence of inverse Jacobi multipliers around Hopf points in ℝ^3: emphasis on the center problem, J Differential Equations, 252, 6324-6336, 2012 ·Zbl 1252.37040 |
| [21] | Buica, A.; Garca, I.; Maza, S., Multiple Hopf bifurcation in ℝ^3 and inverse Jacobi multipliers, J Differ Equations, 256, 310-325, 2014 ·Zbl 1346.37050 |
| [22] | Tian, Y.; Yu, P., An explicit recursive formula for computing the normal form and center manifold of n‐dimensional differential systems associated with Hopf bifurcation, Int J Bifurcation Chaos, 23, 2013 ·Zbl 1272.34004 |
| [23] | Edneral, VF; Mahdi, A.; Romanovskic, VG; Shafer, DS, The center problem on a center manifold in ℝ^3, Nonlinear Anal Real World Appl, 75, 2614-2622, 2012 ·Zbl 1259.34021 |
| [24] | Liu, Y.; Li, J., Theory of values of singular point in complex autonomous differential system, Sci China Ser A, 33, 10-24, 1990 ·Zbl 0686.34027 |
| [25] | Liu, Y., Theory of center‐focus for a class of higher‐degree critical points and infinite points, Sci China Series A: Math, 44, 365-377, 2001 ·Zbl 1012.34027 |
| [26] | Liu, Y.; Huang, H., A cubic system with twelve small amplitude limit cycles, Bulletin Des Sciences Mathematiques, 129, 83-98, 2005 ·Zbl 1086.34030 |
| [27] | Wang, Q.; Liu, Y.; Du, C., Small limit cycles bifurcating from fine focus points in quartic order z_3‐equivariant vector fields, J Math Anal Appl, 337, 524-536, 2008 ·Zbl 1134.34020 |
| [28] | Huang, W.; Liu, Y.; Zhu, F., The center‐focus problem of a class of polynomial differential systems with degenerate critical points, Int J Non‐Linear Sci Numer Simulat, 10, 1167-1179, 2009 |
| [29] | Huang, W.; Chen, A.; Xu, Q., Bifurcation of limit cycles and isochronous centers for a quartic system, Int J Bifurc Chaos, 10, 6637-6643, 2013 |
| [30] | Wang, Q.; Huang, W.; Wu, H., Bifurcation of limit cycles for 3D Lotka‐Volterra competitive systems, Acta Appl Math, 114, 207-218, 2011 ·Zbl 1223.34073 |
| [31] | Wang, Q.; Huang, W.; Feng, J., Multiple limit cycles and centers on center manifolds for Lorenz system, Appl Math Comput, 238, 281-288, 2014 ·Zbl 1334.37089 |
| [32] | Du, C.; Liu, Y.; Huang, W., A class of three‐dimensional quadratic systems with ten limit cycles, Int J Bifurc Chaos, 26, 2016 ·Zbl 1347.34049 |
| [33] | Carr, J., Applications of Centre Manifold Theory, Appl. Math. Sci. vol. 35, 1981, New York: Springer ·Zbl 0464.58001 |
| [34] | Wang, Q.; Huang, W., The equivalence between singular point quantities and Liapunov constants on center manifold, Advances in Difference Equations, 2012, 78, 2012 ·Zbl 1294.34045 |
| [35] | Llibre, J.; Zhang, X., On the Darboux integrability of polynomial differential systems, Qual Theory Dyn Syst, 2012, 129-144, 2012 ·Zbl 1266.34002 |
| [36] | Romanovski, VG; Xia, Y.; Zhang, X., Varieties of local integrability of analytic differential systems and their applications. J, Differential Equations, 257, 3079-3101, 2014 ·Zbl 1305.34022 |
| [37] | Mahdi, A.; Pessoa, C.; Shafer, DS, Centers on center manifolds in the L \(\ddot{\operatorname{u}}\) system, Phys Lett A, 375, 3509-3511, 2011 ·Zbl 1252.70046 |
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