Oscillations and chaos in simple quadratic systems

International Journal of Bifurcation and Chaos

Volumn 18, Issue 7, 2008, Pages 1917-1937

  Innocenti, G ^a   Genesio, R ^a   Ghilardi, C ^a  

^a UNIVERSITY OF FLORENCE   (Italy)

Author keywords

Chaos; Harmonic balance; Hopf bifurcation; Jerk equations

Indexed keywords

BIFURCATION (MATHEMATICS); CHAOS THEORY;

AUTONOMOUS SYSTEMS; EQUATION MODELS; HARMONIC BALANCE; HARMONIC BALANCE TECHNIQUES; JERK EQUATION; PERIODIC OSCILLATION; QUADRATIC SYSTEMS; SYSTEM REDUCTIONS;

HOPF BIFURCATION;

EID: 52149122084     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127408021452     Document Type: Article

References (30)

1. Allwright, D. J. [1977] "Harmonic balance and the Hopf bifurcation theorem," Math. Proc. Cambridge Philos. Soc. 82, 453-467.
2. Basso, M., Genesio, R. & Tesi, A. [1997] "A frequency method for predicting limit cycle bifurcations," Non-lin. Dyn. 13, 339-360.
3. Ĉelikoský, S. & iquest; Chen, G. [2002] "On a generalized Lorenz canonical form of chaotic systems," Int. J. Bifurcation and Chaos 12, 1789-1812.
4. Chen, G. & Yang, L. [2003] "Chaotifying a continuous-time system near a stable limit cycle," Chaos Solit. Fract. 15, 245-253.
5. Eichhorn, R., Linz, S. J. & Hanggi, P. [1998] "Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flows," Phys. Rev. E 58, 7151-7164.
6. Eichhorn, R., Linz, S. J. & Hanggi, P. [2002] "Simple polynomial classes of chaotic jerky dynamics," Chaos Solit. Fract. 13, 1-15.
7. Fu, Z. & Heidel, J. [1997] "Non-chaotic behavior in three-dimensional quadratic systems," Nonlinearity ([1999] Erratum, Nonlinearity 12, 739) 10, 1290-1303.
8. Genesio, R. & Tesi, A. [1992] "Harmonic balance approach for the analysis of chaotic dynamics in nonlinear systems," Automatica 28, 531-548.
9. Genesio, R. & Ghilardi, C. [2005] "On the onset of quasi-periodic solutions in third-order dynamical systems," Int. J. Bifurcation and Chaos 15, 3165-3180.
10. Hindmarsh, J. L. & Rose, R. M. [1984] "A model of neuronal bursting using three coupled first order differential equations," Proc. Royal Soc. London B 221, 87-102.
11. Isidori, A. [1995] Nonlinear Control Systems, 3rd edition (Springer-Verlag, NY).
12. Khalil, H. K. [2002] Nonlinear Systems, 3rd edition (Prentice-Hall, Englewood Cliffs).
13. Kuznetzov, Y. A. [1995] Elements of Applied Bifurcation Theory (Springer-Verlag, NY).
14. Linz, S. J. [2000] "No-chaos criteria for certain jerky dynamics," Phys. Lett. A 275, 204-210.
15. Liu, W. & Chen, G. [2003] "A new chaotic system and its generation," Int. J. Bifurcation and Chaos 13, 261-267.
16. Lu, J., Chen, G. & Cheng, D. [2004] "A new chaotic system and beyond: The generalized Lorenz-like systems," Int. J. Bifurcation and Chaos 14, 1507-1539.
17. Madan, R. A. [1993] Chua's Circuit: A Paradigm for Chaos (World Scientific, Singapore).
18. Mees, A. I. [1981] Dynamics of Feedback Systems (Wiley, NY).
19. Nijmeijer, H. & Mareels, I. M. Y. [1997] "An observer look at synchronization," IEEE Trans. Circuits Syst. Part 144, 882-890.
20. Sprott, J. C. [1994] "Some simple chaotic flows," Phys. Rev. E 50, R647-R650.
21. Sprott, J. C. [1997a] "Simplest dissipative chaotic flow," Phys. Lett. A 228, 271-274.
22. Sprott, J. C. [1997b] "Some simple chaotic jerk functions," Am. J. Phys. 65, 537-543.
23. Sprott, J. C. & Linz, S. J. [2000] "Algebraically simple chaotic flows," Int. J. Chaos Th. Appl. 5, 3-22.
24. Sprott, J. C. [2003] Chaos and Time-Series Analysis (Oxford University Press, Oxford).
25. Vidyasagar, M. [1993] "Nonlinear Systems Analysis, 2nd edition (Prentice-Hall, Englewood Cliffs).
26. Wang, X. F., Chen, G. & Yu, X. [2000] "Anticontrol of chaos in continuous-time systems via time-delay feedback," Chaos 10, 771-779.
27. Yang, X.-S. [2000] "A technique for determining autonomous 3-D ODEs being non-chaotic," Chaos Solit. Fract. 11, 2313-2318.
28. Zhou, T. S. & Chen, G. [2004a] "A simple smooth chaotic system with a 3-layer attractor," Int. J. Bifurcation and Chaos 14, 1795-1799.
29. Zhou, T. S. & Chen, G. [2004b] "Chen attractor exists," Int. J. Bifurcation and Chaos 14, 3167-3177.
30. Zhou, T. S. & Chen, G. [2006] "Classification of chaos in 3-D continuous quadratic systems," Int. J. Bifurcation and Chaos 16, 2459-2479.