An unusual 3D autonomous quadratic chaotic system with two stable node-foci

International Journal of Bifurcation and Chaos

Volumn 20, Issue 4, 2010, Pages 1061-1083

  Yang, Qigui ^a   Wei, Zhouchao ^a   Chen, Guanrong ^b  

^a SOUTH CHINA UNIVERSITY OF TECHNOLOGY   (China)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

bifurcation; Chaotic attractor; heteroclinic orbit; homoclinic orbit; Lorenz system; singularly degenerate heteroclinic cycle

Indexed keywords

BIFURCATION (MATHEMATICS); DYNAMICAL SYSTEMS; HOPF BIFURCATION;

CHAOTIC ATTRACTORS; HETEROCLINIC ORBIT; HOMOCLINIC ORBITS; LORENZ SYSTEM; SINGULARLY DEGENERATE HETEROCLINIC CYCLES;

CHAOTIC SYSTEMS;

EID: 77954210201     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127410026320     Document Type: Article

References (31)

1. Barnett, S. [1983] Polynominals and Linear Control Systems (Marcel Dekker, NY).
2. Chen, G. R. & Ueta, T. [1999] "Yet another chaotic attractor," Int. J. Bifurcation and Chaos 9, 1465-1466.
3. Doedel, E. J., Krauskopf, B. & Osinga, H. M. [2006] "Global bifurcations of the Lorenz manifold," Nonlinearity 19, 2947-2972.
4. Guckenheimer, J. & Holms, P. [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector (Springer, NY).
5. Hassard, B., Kazarinoff, N. &Wan, Y. [1981] Theory and Application of Hopf Bifurcation (Cambridge University Press, Oxford).
6. Huang, D. B. [2003] "Periodic orbits and bomoclinic orbits of the diffusionless Lorenz equations," Phys. Lett. A 309, 248-253.
7. Kokubu, H. & Roussarie, R. [2004] "Existence of a singularly degenerate heterclinic cycle in the Lorenz system and its dynamical consequences: Part 1," J. Dyn. Diff. Eqns. 16, 513-557.
8. Krauskopf, B. & Rie, T. [2008] "A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits," Nonlinearity 21, 1655-1690.
9. Letellier, C., Dutertre, P., Reizner, J. & Gousbet, G. [1996] "Evolution of a multimodal map included by an equivariant vector field," J. Phys. A: Math. Gen. 29, 5359-5373. (Pubitemid 126689333)
10. Li, J. B. & Zhang, J. [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.
11. Lorenz, E. N. [1963] "Deterministic non-periodic flow," J. Atmos. Sci. 20, 130-141.
12. Lu, J. H. & Chen, G. R. [2002] "A new chaotic attractor coined," Int. J. Bifurcation and Chaos 12, 659-661.
13. Panchev, S., Spassova, T. & Vitanov, N. K. [2007] "Analytical and numerical investgation of Lorenz-like dynamical systems," Chaos Solit. Fract. 33, 1658-1671.
14. Rossler, O. E. [1976] "An equation for continuious chaos," Phys. Lett. A 57, 397-398.
15. Shaw, R. [1981] "Strange attractor, chaotic behaviour and information flow," ZNatur. A 36, 80-112.
16. Silva, C. P. [1993] " Silnikov theorem - A tutorial," IEEE. Trans. Circ. Syst.-I 40, 657-682.
17. Sparrow, C. [1982] The Lorenz Equations: Bifurcation, Chaos, and Strange Attractors (Springer, NY).
18. Sprott, J. C. [1994] "Some simple chaotic flows," Phys. Rev. E 50, 647-650.
19. Sprott, J. C. [2000a] "Simple chaotic systems and circuits," Amer. J. Phys. 68, 758-763.
20. Sprott, J. C. [2000b] "A new class of chaotic circuits," Phys. Lett. A 266, 19-23.
21. Sprott, J. C. [2003] Chaos and Time-Series Analysis (Oxford University Press, Oxford).
22. Ueta, T. & Chen, G. R. [2000] "Bifurcation and chaos in Chen's equation," Proc. IEEE Int. Symp. Circuits and Systems 29-31, 505-508.
23. Vanderschrier, G. & Maas, L. [2000] "The diffusionless Lorenz equations; Silnikov bifurcations and reduction to an explicit map," Physica D 141, 19-36.
24. Viswanath, D. [2003] "Symbolic dynamics and periodic orbits of the Lorenz attractor," Nonlinearity 16, 1035-1056.
25. Wiggins, S. & Holmes, P. [1987] "Homoclinic orbits in slowly varying oscillators," SIAM J. Math. Anal. 18, 612-629.
26. Yang, Q. G., Chen, G. R. & Zhou, T. S. [2006] "A unified Lorenz-type system and its canonical form," Int. J. Bifurcation and Chaos 16, 2855-2871.
27. Yang, Q. G., Chen, G. R. & Huang, K. F. [2007] "Chaotic attractors of the conjugate Lorenz-type system," Int. J. Bifurcation and Chaos 17, 3929-3949.
28. Yang, Q. G. & Chen, G. R. [2008] "A chaotic system with one saddle and two stable node-foci," Int. J. Bifurcation and Chaos 18, 1393-1414.
29. Zhou, T. S., Chen, G. R. & Tang, Y. [2003] "Complex dynamical behaviors of the chaotic Chen's system," Int. J. Bifurcation and Chaos 13, 2561-2574.
30. Zhou, T. S., Chen, G. R. & Yang, Q. G. [2004] "Constructing a new chaotic system based on the Silnikov criterion," Chaos Solit. Fract. 19, 985-993.
31. Zhou,T. S., Chen, G. R. & Celikovsky, S.[2005] " Silnikov chaos in the generalized Lorenz canonical form of dynamics systems," Nonlin. Dyn. 39, 319-334.