Dynamical analysis of the generalized Sprott C system with only two stable equilibria

Nonlinear Dynamics

Volumn 68, Issue 4, 2012, Pages 543-554

  Wei, Zhouchao ^a   Yang, Qigui ^b  

^a SOUTH CENTRAL UNIVERSITY FOR NATIONALITIES   (China)

^b SOUTH CHINA UNIVERSITY OF TECHNOLOGY   (China)

Author keywords

Chaotic attractors; Coexisting attractors; Degenerate heteroclinic cycles; Lyapunov exponent; Sil'nikov theorem

Indexed keywords

CHAOTIC ATTRACTORS; CO-EXISTING ATTRACTORS; HETEROCLINIC CYCLES; LYAPUNOV EXPONENT; SIL'NIKOV THEOREM;

COMPUTER SIMULATION; DIFFERENTIAL EQUATIONS; DYNAMICAL SYSTEMS; FRACTAL DIMENSION; LYAPUNOV FUNCTIONS;

LYAPUNOV METHODS;

EID: 84862837546     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-011-0235-8     Document Type: Article

References (31)

1. E.N. Lorenz 1963 Deterministic non-periodic flow J. Atmos. Sci. 20 130 141 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
2. O.E. Rössler 1976 An equation for continuous chaos Phys. Lett. A 57 397 398 10.1016/0375-9601(76)90101-8
3. J.C. Sprott 1994 Some simple chaotic flows Phys. Rev. E 50 647 650 1381868 10.1103/PhysRevE.50.R647
4. J.C. Sprott 2000 A new class of chaotic circuit Phys. Lett. A 266 19 23 10.1016/S0375-9601(00)00026-8
5. J.C. Sprott 1997 Simplest dissipative chaotic flow Phys. Lett. A 228 271 274 1442639 1043.37504 10.1016/S0375-9601(97)00088-1 (Pubitemid 127428843)
6. G.R. Chen T. Ueta 1999 Yet another chaotic attractor Int. J. Bifurc. Chaos Appl. Sci. Eng. 9 1465 1466 1729683 0962.37013 10.1142/S0218127499001024 (Pubitemid 129292251)
7. J.H. Lü G.R. Chen 2002 A new chaotic attractor coined Int. J. Bifurc. Chaos Appl. Sci. Eng. 12 659 661 1063.34510 10.1142/S0218127402004620 (Pubitemid 34862041)
8. Q.G. Yang G.R. Chen K.F. Huang 2007 Chaotic attractors of the conjugate Lorenz-type system Int. J. Bifurc. Chaos Appl. Sci. Eng. 17 3929 3949 2384392 1149.37308 10.1142/S0218127407019792 (Pubitemid 351269037)
9. G. van der Schrier L.R.M. Maas 2000 The diffusionless Lorenz equations: Sil'nikov bifurcations and reduction to an explicit map Physica D 141 19 36 1764166 0956.37038 10.1016/S0167-2789(00)00033-6
10. R. Shaw 1981 Strange attractor, chaotic behavior and information flow Z. Naturforsch. 36A 80 112
11. Q.G. Yang G.R. Chen 2008 A chaotic system with one saddle and two stable node-foci Int. J. Bifurc. Chaos Appl. Sci. Eng. 18 1393 1414 1147.34306 10.1142/S0218127408021063 (Pubitemid 352025678)
12. F.S. Dias L.F. Mello J.G. Zhang 2010 Nonlinear analysis in a Lorenz-like system Nonlinear Anal., Real World Appl. 11 3491 3500 2683807 1208.34066 10.1016/j.nonrwa.2009.12.010
13. Sparrow, C.: The Lorenz Equations: Bifurcation, Chaos, and Strange Attractor. Springer, New York (1982)
14. T.S. Zhou G.R. Chen Y. Tang 2003 Complex dynamical behaviors of the chaotic Chen's system Int. J. Bifurc. Chaos Appl. Sci. Eng. 13 2561 2574 2013113 1046.37018 10.1142/S0218127403008089 (Pubitemid 37396706)
15. Q.G. Yang G.R. Chen T.S. Zhou 2006 A unified Lorenz-type system and its canonical form Int. J. Bifurc. Chaos Appl. Sci. Eng. 16 2855 2871 2282907 1185.37088 10.1142/S0218127406016501 (Pubitemid 44924965)
16. H. Kokubu R. Roussarie 2004 Existence of a singularly degenerate heteroclinic cycle in the Lorenz system and its dynamical consequences: Part 1J. Dyn. Differ. Equ. 16 513 557 2105787 1061.34036 10.1007/s10884-004-4290-4 (Pubitemid 39519642)
17. M. Messias 2009 Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system J. Phys. A, Math. Theor. 42 115101 2485832 10.1088/1751-8113/42/11/115101
18. L.F. Mello S.F. Coelho 2009 Degenerate Hopf bifurcations in the Lü system Phys. Lett. A 373 1116 1120 2489562 1228.70014 10.1016/j.physleta.2009. 01.049
19. L.F. Mello M. Messias D.C. Braga 2008 Bifurcation analysis of a new Lorenz-like chaotic system Chaos Solitons Fractals 37 1244 1255 2411546 1153.37356 10.1016/j.chaos.2007.11.008
20. J. Li J. Zhang 1993 New treatment on bifurcation of periodic solutions and homoclinic orbits at high r in the Lorenz equations SIAM J. Appl. Math. 53 1059 1071 1232167 0781.34031 10.1137/0153053
21. Z. Wang 2009 Existence of attractor and control of a 3D differential system Nonlinear Dyn. 60 369 373 10.1007/s11071-009-9601-1
22. S.Q. Pang Y.J. Liu 2011 A new hyperchaotic system from the Lü system and its control J. Comput. Appl. Math. 235 2775 2789 2763185 1217.37031 10.1016/j.cam.2010.11.029
23. Z.C. Wei Q.G. Yang 2009 Controlling the diffusionless Lorenz equations with periodic parametric perturbation Comput. Math. Appl. 58 1979 1987 2557520 1189.34118 10.1016/j.camwa.2009.07.058
24. J.H. Bao Q.G. Yang 2010 Complex dynamics in the stretch-twist-fold flow Nonlinear Dyn. 61 773 781 2718359 1204.37083 10.1007/s11071-010-9686-6
25. D.Y. Chen C. Wu C.F. Liu X.Y. Ma Y.J. You R.F. Zhang 2011 Synchronization and circuit simulation of a new double-wing chaos Nonlinear Dyn. 10 1 24
26. L.P. Sil'nikov 1965 A case of the existence of a countable number of periodic motions Sov. Math., Dokl. 6 163 166 0136.08202
27. L.P. Sil'nikov 1970 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 10.1070/SM1970v010n01ABEH001588
28. C.P. Silva 1993 Sil'nikov theorem-a tutorial IEEE Trans. Circuits Syst. I 40 657 682 10.1109/81.246142
29. Y. Hardy W.H. Steeb 1999 The Rikitake two-disk dynamo system and domains with periodic orbits Int. J. Theor. Phys. 38 2413 2417 1722023 0980.86005 10.1023/A:1026640221874
30. J. Llibre M. Messias 2009 Global dynamics of Rikitake system Physica D 238 241 252 2590447 1162.37017 10.1016/j.physd.2008.10.011
31. J. Llibre X. Zhang 2000 Invariant algebraic surfaces of the Rikitake system J. Phys. A, Math. Gen. 33 7613 7635 1802113 0967.34002 10.1088/0305-4470/33/42/310