×

Dynamics of a 3D autonomous quadratic system with an invariant algebraic surface. (English) Zbl 1331.34044

Summary: An invariant algebraic surface is calculated for a 3D autonomous quadratic system. Also, the dynamics near finite singularities and near infinite singularities on the invariant algebraic surface is analyzed. Furthermore, pitchfork bifurcation is analyzed using center manifold theorem and a first integral of this quadratic system for some special parameters is provided. Finally, the dynamics of this system at infinity using the Poincare compactification in \(R^3\) is investigated and the singularly degenerate heteroclinic cycles are presented by a first integral and verified by numerical simulations.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C29 Homoclinic and heteroclinic orbits for dynamical systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lorenz, E.N.: Deterministic nonperiodic flow[J]. J. Atmos. Sci. 20(2), 130C141 (1963) · Zbl 1417.37129 · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[2] Celikovsky, S., Vanecek, A.: Bilinear systems and chaos[J]. Kybernetika 30(4), 403-424 (1994) · Zbl 0823.93026
[3] Lü, J.H., Chen, G.R.: A new chaotic attractor coined[J]. Int. J. Bifurc. Chaos 12(3), 659-661 (2002) · Zbl 1063.34510 · doi:10.1142/S0218127402004620
[4] Tigan, Gh: Analysis of a dynamical system derived from the Lorenz system[J]. Sci. Bull. politeh. Univ. Timis. 50(64), 61-72 (2005) · Zbl 1107.37039
[5] Yang, Q.G., Chen, G.R.: A chaotic system with one saddle and two stable node-foci[J]. Int. J. Bifurc. Chaos 18(5), 1393-1414 (2008) · Zbl 1147.34306 · doi:10.1142/S0218127408021063
[6] Wang, Z.: Existence of attractor and control of a 3D differential system[J]. Nonlinear Dyn. 60(3), 369-373 (2010) · Zbl 1189.70103 · doi:10.1007/s11071-009-9601-1
[7] Wang, Z., Li, Y.X., Xi, X.J., Lv, L.: Heteoclinic orbit and backstepping control of a 3D chaotic system[J]. Acta Phys. Sin. 60(1), 010513 (2011)
[8] Liu, Y.J.: Dynamics of a new Lorenz-like chaotic system[J]. Nonlinear Anal. 11(4), 2563-2572 (2010) · Zbl 1202.34083
[9] Xi, X.J., Wang, Z., Sun, W.: Homoclinic orbits analysis of T chaotic system with periodic parametric perturbation[J]. Acta Phys. Sin. 62(13), 130507 (2013) · Zbl 1274.94024
[10] Jiang, B., Han, X.J., Bi, Q.J.: Hopf bifurcation analysis in the T system[J]. Nonlinear Anal. 11(4), 2563-2572 (2010) · Zbl 1202.34083
[11] Llibre, J., Zhang, X.: Invariat algebraic surfaces of the Lorenz system[J]. J. Math. Phys. 43(3), 1622-1645 (2002) · Zbl 1059.34005 · doi:10.1063/1.1435078
[12] Lü, T.H., Zhang, Z.: Darboux polynomials and algebraic integrability of the Chen system[J]. Int. J. Bifurc. Chaos 17(8), 2739-2748 (2007) · Zbl 1148.34001 · doi:10.1142/S0218127407018725
[13] Liu, Y.J., Yang, Q.G.: Dynamics of the Lü system on the invariant algebraic surface and at infinity[J]. Int. J. Bifurc. Chaos 21(9), 2559-2582 (2011) · Zbl 1248.34069 · doi:10.1142/S0218127411029938
[14] Llibre, J., Zhang, X.: Invariant algebraic surfaces of the Rikitake system[J]. J. Phys. A 33(42), 7613-7635 (2000) · Zbl 0967.34002 · doi:10.1088/0305-4470/33/42/310
[15] Zhang, X.: Integrals of motion of the Rabinovich system[J]. J. Phys. A 33(28), 5137-5155 (2000) · Zbl 0962.34034 · doi:10.1088/0305-4470/33/28/315
[16] Cao, J.L., Zhang, X.: Dynamics of the Lorenz system having an invariant algebraic surface[J]. J. Math. Phys. 48(8), 082702 (2007) · Zbl 1152.81361 · doi:10.1063/1.2767007
[17] Llibre, J., Messias, M., Silva, P.R.: Global dynamics in the Poincare ball of the Chen system having invariant algebraic surfaces[J]. Int. J. Bifurc. Chaos 22(6), 1250154 (2012) · Zbl 1270.34130 · doi:10.1142/S0218127412501544
[18] Wu, K.S., Zhang, X.: Darboux polynomials and rational first integrals of the generalized Lorenz systems[J]. Bull. Sci. Math. 163(3), 291-308 (2012) · Zbl 1246.34018 · doi:10.1016/j.bulsci.2011.11.005
[19] Wu, K.S., Zhang, X.: Global dynamics of the generalized Lorenz systems having invariant algebraic surfaces[J]. Physica D 244(1), 25-35 (2013) · Zbl 1338.37038 · doi:10.1016/j.physd.2012.10.011
[20] Llibre, J., Messias, M., Silva, P.R.: Global dynamics of the Lorenz system with invariant algebraic surfaces[J]. Int. J. Bifurc. Chaos 20(10), 3137-3155 (2010) · Zbl 1204.34069 · doi:10.1142/S0218127410027593
[21] Messias, M.: Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system[J]. J. Phys. A 42(11), 115101 (2009) · Zbl 1181.37019 · doi:10.1088/1751-8113/42/11/115101
[22] Ye, Y.Q.: Theory of Limit Cycles[M]. Shanghai Science and Technology Press, Shanghai (1982)
[23] Dumortier, F., Llibre, J., Artes, J.C.: Qualitative Theory of Planar Differential Systems[M]. Springer, Berlin (2006) · Zbl 1110.34002
[24] Zhang, Z.F., Ding, T.R., Huang, W.Z., Dong, Z.X.: Qualitative Theory of Differential Equations[M]. Science Press, Beijing (2003)
[25] Dumortier, F., Herssens, C.: Polynomial Lienard equations near infinity[J]. J. Differ. Equ. 153(1), 1-29 (1999) · Zbl 0928.34028 · doi:10.1006/jdeq.1998.3543
[26] Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields[M]. Springer, Berlin (2002) · Zbl 0515.34001
[27] Cima, A., Llibre, J.: Bounded polynomial vector fields[J]. Trans. Am. Math. Soc. 318(2), 557-579 (1990) · Zbl 0695.34028 · doi:10.1090/S0002-9947-1990-0998352-5
[28] Liu, Y.J.: Analysis of global dynamics in an unusual 3D chaotic system[J]. Nonlinear Dyn. 70(3), 2203-2212 (2012) · Zbl 1268.34092
[29] Messias, M., Gouveia, M.R.: Dynamics at infinity and other global dynamical aspects of Shimizu-Morioka equations[J]. Nonlinear Dyn. 69(1-2), 577-587 (2012) · Zbl 1256.37007 · doi:10.1007/s11071-011-0288-8
[30] Li, J.B., Zhao, X.H., Liu, Z.R.: Theory of generalized Hamilton system and its applications[M]. Science Press, Beijing (2007)
[31] Smith, P.: The multiple scales method, homoclinic bifurcation and Melnikov’s method for autonomous systems[J]. Int. J. Bifurc. Chaos 8(11), 2099-2105 (1998) · Zbl 0996.37066 · doi:10.1142/S021812749800173X
[32] Belhaq, M., Fiedler, B., Lakrad, F.: Homoclinic connections in strongly selfexcited nonlinear oscillators: the Melnikov function and the elliptic Lindstedt-Poincare method[J]. Nonlinear Dyn. 23(1), 67-86 (2000) · Zbl 0967.70019
[33] Chen, Y.Y., Chen, S.H.: Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by the hyperbolic perturbation method[J]. Nonlinear Dyn. 58(1-2), 417-429 (2009) · Zbl 1183.70045
[34] Belykh, V.N.: Bifurcations of separatrices of a saddle point of the Lorenz system[J]. Differ. Equ. 20(10), 1184-1191 (1984) · Zbl 0576.34027
[35] Tigan, G., Turaev, D.: Analytical search for homoclinic bifurcations in the Shimizu-Morioka model[J]. Physica D 240(12), 985-989 (2011) · Zbl 1218.34046 · doi:10.1016/j.physd.2011.02.013
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.