Lorenz equation and Chua's equation

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 6, Issue 12 A/B, 1996, Pages 2443-2489

  Pivka, Ladislav ^a   Wu, Chai Wah ^a   Huang, Anshan ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0345847877     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/s0218127496001594     Document Type: Article

References (53)

1. Afraimovich, V. S., Bykov, V. V. & Shilnikov, L. P. [1983] "On structurally unstable attracting limit sets of Lorenz attractor type," Trans. Moscow Math. Soc. 44(2), 153-216.
2. Anishchenko, V. S., Neiman, A. B. & Chua, L. O. [1994] "Chaos-chaos intermittency and 1/f noise in Chua's circuit," Int. J. Bifurcation and Chaos 4(1), 99-107.
3. Anishchenko, V. S., Safonova, M. A. & Chua, L. O. [1993] "Stochastic resonance in the nonautonomous Chua's circuit," J. Circ. Syst. Comp. 3(2), 553-578.
4. Belykh, V. N. & Chua, L. O. [1992] "New type of strange attractor from a geometrical model of Chua's circuit," Int. J. Bifurcation and Chaos 2(3), 697-704.
5. Birman, J. S. & Williams, R. F. [1983] "Knotted periodic orbits in dynamical systems - I: Lorenz equations," Topology 22(1), 47-82.
6. Boldrighini, C. & Franceschini, V. [1979] "A five-dimensional truncation of the plane incompressible Navier-Stokes equations," Commun. Math. Phys. 64, 159-170.
7. Chua, L. O. [1992] "The genesis of Chua's circuit," Archiv für Elektronik und Übertragungstechnik 46(4), 250-257.
8. Chua, L. O. [1993] "Global unfolding of Chua's circuit," IEICE Trans. Fund. Electron., Comm. Comput. Sci. E76-A(5), 704-734.
9. Chua, L. O. (Guest Editor) [1995] Special Issue on Non-linear Waves, Patterns and Spatio-Temporal Chaos in Dynamic Arrays, IEEE Trans. Circuits Syst. 42(10).
10. Chua, L. O., Komuro, M. & Matsumoto, T. [1986a] "The double scroll family," Part I: Rigorous Proof of Chaos. IEEE Trans. Circuits Syst. 33(11), 1072-1097.
11. Chua, L. O., Komuro, M. & Matsumoto, T. [1986b] "The double scroll family," Part II: Rigorous Analysis of Bifurcation Phenomena IEEE Trans. Circuits Syst. 33(11), 1097-1118.
12. Chua, L. O. & Huynh, L. T. [1992] "Bifurcation analysis of Chua's circuit," Memorandum No. UCB/ERL M92/96, University of California, Berkeley, USA.
13. Chua, L. O., Wu, C. W., Huang, A. S. & Zhong, G-Q. [1993a] "A universal circuit for studying and generating chaos - Part I: Routes to chaos," IEEE Trans. Circ. Syst. I: Fundamental Theory and Applications 40(10), 732-744.
14. Chua, L. O., Wu, C. W., Huang, A. S. & Zhong, G-Q. [1993b] "A universal circuit for studying and generating chaos - Part II: Strange attractors," IEEE Trans. Circ. Syst. I: Fundamental Theory and Applications 40(10), 745-761.
15. Cruz, J. M. & Chua, L. O. [1993] "An IC chip for Chua's circuit," IEEE Trans. Circ. Syst., 40(10), 614-625.
16. Curry, J. H. [1978] "A generalized Lorenz system," Commun. Math. Phys. 60, 193-204.
17. Feldmann, U., Hasler, M. & Schwarz, W. [1995] "On the design of a synchronizing inverse of a chaotic system," Proc. of ECCTD'95 European Conference on Circuit Theory & Design, 479-482.
18. Franceschini, V. & Tebaldi, C. [1979] "Sequences of infinite bifurcations and turbulence in a five-mode truncation of the Navier-Stokes equations," J. Stat. Phys. 21(6), 707-726.
19. Franceschini, V. & Tebaldi, C. [1980] "A seven-mode truncation of the Navier-Stokes equations," preprint, University di Modena, Italy.
20. Franz, M., Feldmann, U., Schwarz, W. & Wu, C. W. [1995] "Bifurcation phenomena in Chua's circuit with a cubic nonlinearity," preprint, University of Dresden.
21. Galias, Z. "Positive topological entropy of Chua's circuit: A computer assisted proof," Int. J. Bifurcation and Chaos to appear in 1997.
22. Genot, M. [1993] "Applications of 1-D map from Chua's circuit," J. Circ. Syst. Comput. 3, 375-409.
23. Ghrist, R. W. & Holmes, P. J. [1996] "An ODE whose solutions contain all knots and links," Int. J. Bifurcation and Chaos 6(5), 779-800.
24. Gibbon, J. D. & McGuinness, M. J. [1982] "The real and complex Lorenz equations in rotating fluids and lasers," Physica 5D, 108-122.
25. Guckenheimer, J. & Williams, R. F. [1979] "Structural stability of Lorenz attractors," Inst. Hautes Etudes Sci. Publ. Math. 50, 59-72.
26. Haken, H. [1975] "Analogy between higher instabilities in fluids and lasers," Phys. Lett. 53A(1), 77-78.
27. Hassard, B., Zhang, J., Hastings, S. P. & Troy, W. C. [1994] "A computer proof that the Lorenz equations have "chaotic" solutions," Appl. Math. Lett. 7(1), 79-83.
28. Hastings, S. P. [1994] "A "shooting" approach to chaos," Int. J. Bifurcation and Chaos 4(1), 17-32.
29. Huang, A. S., Pivka, L., Wu, C. W. & Franz, M. [1996] "Chua's circuit with cubic nonlinearity," Int. J. Bifurcation and Chaos 6(12A), 2175-2222.
30. Kennedy, M. P. [1992] "Robust OP amp realization of Chua's circuit," Frequenz 6(3-4), 66-80.
31. Knobloch, E. [1981] "Chaos in the segmented disc dynamo," Phys. Lett. 82A(9), 439-440.
32. Kocarev, Lj., Tasev, Z., Dimovski, D. & Chua, L. O. [1994] "Knotted periodic orbit in Chua's circuit," Int. J. Bifurcation and Chaos 4(3), 609-621.
33. Kuznetsov, A. P., Kuznetsov, S. P., Sataev, I. R. & Chua, L. O. [1993] "Self-similarity and universality in Chua's circuit via the approximate Chua's 1-D map," Int. J. Bifurcation and Chaos 3, 943-962.
34. Kuznetsov, A. P., Kuznetsov, S. P., Sataev, I. R. & Chua, L. O. [1996] "Multi-parameter criticality in Chua's circuit at period doubling transition to chaos," Int. J. Bifurcation and Chaos 6, 119-148.
35. Lorenz, E. N. [1963] "Deterministic non-periodic flows," J. Atmos. Sc. 20, 130-141.
36. Lukin, K. A. [1993] "High frequency oscillations from Chua's circuit," J. Circ. Syst. Comp. 3(2), 627-643.
37. Madan, R. [1993] Chua's Circuit: A Paradigm for Chaos (World Scientific, Singapore).
38. Malkus, W. V. R. [1972] "Non-periodic convection at high and low Prandtl number," Mempoirs Societe Royale des Sciences de Lige, Series 6, VI, 125-128.
39. Mischaikow, K. & Mrozek, M. [1995] "Chaos in the Lorenz equations: A computer-assisted proof," Bull. (New Series) Am. Math. Soc. 32(1), 66-72.
40. Misiurewicz, M. [1993] "Unimodal interval maps obtained from the modified Chua's equation," Int. J. Bifurcation and Chaos 3, 323-332.
41. Pivka, L. & Spany, V. [1993] "Boundary surfaces and basin bifurcations in Chua's circuit," J. Circ. Syst. Comp. 3(2), 441-470.
42. Sharkovsky, A. N., Maistrenko, Yu., Deregel, P. & Chua, L. O. [1993] "Dry turbulence from a time-delayed Chua's circuit," J. Circ. Syst. Comp. 3(2), 645-668.
43. Shil'nikov, L. P. [1994] "Chua's circuit: Rigorous results and future problems," Int. J. Bifurcation and Chaos 4(3), 489-519.
44. Shimizu, T. & Morioka, N. [1978] "Chaos and limit cycles in the Lorenz model," Phys. Lett. 66A, 182-184.
45. Sparrow, C. [1982] The Lorenz Equation: Bifurcations, Chaos, and Strange Attractors (Springer-Verlag, New York).
46. Tesi, A., De Angeli, A. & Genesio, R. [1994] "On system decomposition for synchronizing chaos," Int. J. Bifurcation and Chaos 4(6), 1675-1685.
47. Wu, C. W. & Chua, L. O. [1994] "A unified framework for synchronization and control of dynamical systems," Int. J. Bifurcation and Chaos, 4(4), 979-998.
48. Wu, C. W. & Chua, L. O. [1996a] "On adaptive synchronization and control of nonlinear dynamical systems," Int. J. Bifurcation and Chaos 6(3), 455-471.
49. Wu, C. W. & Chua, L. O. [1996b] "On the generality of the unfolded Chua's circuit," Int. J. Bifurcation and Chaos 6(6), 801-832.
50. Wu, C. W. & Rul'kov, N. F. [1993] "Studying chaos via 1-D maps - A tutorial," IEEE Trans. Circuits and Systems, 40, 707-721.
51. Yorke, E. D. & Yorke, J. A. [1979] "Metastable chaos: Transition to sustained chaotic behavior in the Lorenz model," J. Stat. Phys. 21(3), 263-277.
52. Zhong, G.-Q. & Ayrom, F. [1985] "Experimental confirmation of chaos from Chua's circuit," Int. J. Cir. Theor. Appl. 13(1), 93-98.
53. Zhong, G.-Q. [1994] "Implementation of Chua's circuit with a cubic nonlinearity," IEEE Trans. Circ. Syst. I: Fundamental Theory and Applications 41(12), 934-941.