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Dynamical analysis of the generalized Sprott C system with only two stable equilibria. (English) Zbl 1252.93067
Summary: A generalized Sprott C system with only two stable equilibria is investigated by detailed theoretical analysis as well as dynamic simulation, including some basic dynamical properties, Lyapunov exponent spectra, fractal dimension, bifurcations, and routes to chaos. In the parameter space where the equilibria of the system are both asymptotically stable, chaotic attractors coexist with period attractors and stable equilibria. Moreover, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated. Periodic solutions and chaotic attractors can be found when these cycles disappear.

MSC:
93C15 Control/observation systems governed by ordinary differential equations
37N35 Dynamical systems in control
34H10 Chaos control for problems involving ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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[1] Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963) · Zbl 1417.37129 · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[2] Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976) · Zbl 1371.37062 · doi:10.1016/0375-9601(76)90101-8
[3] Sprott, J.C.: Some simple chaotic flows. Phys. Rev. E 50, 647–650 (1994) · doi:10.1103/PhysRevB.50.647
[4] Sprott, J.C.: A new class of chaotic circuit. Phys. Lett. A 266, 19–23 (2000) · doi:10.1016/S0375-9601(00)00026-8
[5] Sprott, J.C.: Simplest dissipative chaotic flow. Phys. Lett. A 228, 271–274 (1997) · Zbl 1043.37504 · doi:10.1016/S0375-9601(97)00088-1
[6] Chen, G.R., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos Appl. Sci. Eng. 9, 1465–1466 (1999) · Zbl 0962.37013 · doi:10.1142/S0218127499001024
[7] Lü, J.H., Chen, G.R.: A new chaotic attractor coined. Int. J. Bifurc. Chaos Appl. Sci. Eng. 12, 659–661 (2002) · Zbl 1063.34510 · doi:10.1142/S0218127402004620
[8] Yang, Q.G., Chen, G.R., Huang, K.F.: Chaotic attractors of the conjugate Lorenz-type system. Int. J. Bifurc. Chaos Appl. Sci. Eng. 17, 3929–3949 (2007) · Zbl 1149.37308 · doi:10.1142/S0218127407019792
[9] van der Schrier, G., Maas, L.R.M.: The diffusionless Lorenz equations: Sil’nikov bifurcations and reduction to an explicit map. Physica D 141, 19–36 (2000) · Zbl 0956.37038 · doi:10.1016/S0167-2789(00)00033-6
[10] Shaw, R.: Strange attractor, chaotic behavior and information flow. Z. Naturforsch. 36A, 80–112 (1981) · Zbl 0599.58033
[11] Yang, Q.G., Chen, G.R.: A chaotic system with one saddle and two stable node-foci. Int. J. Bifurc. Chaos Appl. Sci. Eng. 18, 1393–1414 (2008) · Zbl 1147.34306 · doi:10.1142/S0218127408021063
[12] Dias, F.S., Mello, L.F., Zhang, J.G.: Nonlinear analysis in a Lorenz-like system. Nonlinear Anal., Real World Appl. 11, 3491–3500 (2010) · Zbl 1208.34066 · doi:10.1016/j.nonrwa.2009.12.010
[13] Sparrow, C.: The Lorenz Equations: Bifurcation, Chaos, and Strange Attractor. Springer, New York (1982) · Zbl 0504.58001
[14] Zhou, T.S., Chen, G.R., Tang, Y.: Complex dynamical behaviors of the chaotic Chen’s system. Int. J. Bifurc. Chaos Appl. Sci. Eng. 13, 2561–2574 (2003) · Zbl 1046.37018 · doi:10.1142/S0218127403008089
[15] Yang, Q.G., Chen, G.R., Zhou, T.S.: A unified Lorenz-type system and its canonical form. Int. J. Bifurc. Chaos Appl. Sci. Eng. 16, 2855–2871 (2006) · Zbl 1185.37088 · doi:10.1142/S0218127406016501
[16] Kokubu, H., Roussarie, R.: Existence of a singularly degenerate heteroclinic cycle in the Lorenz system and its dynamical consequences: Part 1*. J. Dyn. Differ. Equ. 16, 513–557 (2004) · Zbl 1061.34036 · doi:10.1007/s10884-004-4290-4
[17] Messias, M.: Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system. J. Phys. A, Math. Theor. 42, 115101 (2009) · Zbl 1181.37019
[18] Mello, L.F., Coelho, S.F.: Degenerate Hopf bifurcations in the Lü system. Phys. Lett. A 373, 1116–1120 (2009) · Zbl 1228.70014 · doi:10.1016/j.physleta.2009.01.049
[19] Mello, L.F., Messias, M., Braga, D.C.: Bifurcation analysis of a new Lorenz-like chaotic system. Chaos Solitons Fractals 37, 1244–1255 (2008) · Zbl 1153.37356 · doi:10.1016/j.chaos.2007.11.008
[20] Li, J., Zhang, J.: New treatment on bifurcation of periodic solutions and homoclinic orbits at high r in the Lorenz equations. SIAM J. Appl. Math. 53, 1059–1071 (1993) · Zbl 0781.34031 · doi:10.1137/0153053
[21] Wang, Z.: Existence of attractor and control of a 3D differential system. Nonlinear Dyn. 60, 369–373 (2009) · Zbl 1189.70103 · doi:10.1007/s11071-009-9601-1
[22] Pang, S.Q., Liu, Y.J.: A new hyperchaotic system from the Lü system and its control. J. Comput. Appl. Math. 235, 2775–2789 (2011) · Zbl 1217.37031 · doi:10.1016/j.cam.2010.11.029
[23] Wei, Z.C., Yang, Q.G.: Controlling the diffusionless Lorenz equations with periodic parametric perturbation. Comput. Math. Appl. 58, 1979–1987 (2009) · Zbl 1189.34118 · doi:10.1016/j.camwa.2009.07.058
[24] Bao, J.H., Yang, Q.G.: Complex dynamics in the stretch-twist-fold flow. Nonlinear Dyn. 61, 773–781 (2010) · Zbl 1204.37083 · doi:10.1007/s11071-010-9686-6
[25] Chen, D.Y., Wu, C., Liu, C.F., Ma, X.Y., You, Y.J., Zhang, R.F.: Synchronization and circuit simulation of a new double-wing chaos. Nonlinear Dyn. 10, 1–24 (2011) · Zbl 1256.94082
[26] Sil’nikov, L.P.: A case of the existence of a countable number of periodic motions. Sov. Math., Dokl. 6, 163–166 (1965)
[27] Sil’nikov, L.P.: A contribution of the problem of the structure of an extended neighborhood of rough equilibrium state of saddle-focus type. Math. USSR. Sbornik 10, 91–102 (1970) · Zbl 0216.11201 · doi:10.1070/SM1970v010n01ABEH001588
[28] Silva, C.P.: Sil’nikov theorem-a tutorial. IEEE Trans. Circuits Syst. I 40, 657–682 (1993) · Zbl 0844.58059 · doi:10.1109/81.246141
[29] Hardy, Y., Steeb, W.H.: The Rikitake two-disk dynamo system and domains with periodic orbits. Int. J. Theor. Phys. 38, 2413–2417 (1999) · Zbl 0980.86005 · doi:10.1023/A:1026640221874
[30] Llibre, J., Messias, M.: Global dynamics of Rikitake system. Physica D 238, 241–252 (2009) · Zbl 1162.37017 · doi:10.1016/j.physd.2008.10.011
[31] Llibre, J., Zhang, X.: Invariant algebraic surfaces of the Rikitake system. J. Phys. A, Math. Gen. 33, 7613–7635 (2000) · Zbl 0967.34002 · doi:10.1088/0305-4470/33/42/310
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