Abstract
We investigate a scenario for the creation of irregular chaotic attractors in Chua’s system. We show that irregular attractors in Chua’s system are created by those and only those mechanisms that characterize Lorenz, Rössler, and other dissipative nonlinear systems described by ordinary differential equations. These mechanisms include cascades of Feigenbaum period doubling bifurcations, subharmonic cascades of cycle bifurcations in Sharkovskii’s order, and homoclinic cascades of bifurcations.
Similar content being viewed by others
References
L. O. Chua, “Nonlinear circuits,” IEEE Trans. Circuits Syst. (Centennial special issue), CAS-31, 69–87 (Jan. 1984).
G.-Q. Zhong and F. Ayrom, “Experimental confirmation of chaos from Chua’s circuit,” Int. J. Circuit Theory and Appl., 13, 93–98 (1985).
L. O. Chua, M. Komuro, and T. Matsumoto, “The double scroll family,” IEEE Trans. Circuits Syst., CAS-33, 1072–1118 (Nov. 1986).
A. L. Mahla and A. G. Badan-Palhares, “Chua’s circuit with a discontinuous nonlinearity,” J. Circuits, Systems and Computers, 3, 231–237 (1993).
L. P. Shilnikov, “Chua’s circuit: rigorous results and future problems,” Int. J. Bifurcation Chaos, 4, No. 3, 489–519 (1994).
A. F. Gribov and A. P. Krishchenko, “Analytical existence conditions for a homoclinic loop in Chua’s circuits,” Nonlinear Dynamics and Control, No. 1, 263–268, Fizmatlit, Moscow (2001).
M. J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations,” J. Stat. Phys., 19, 25–52 (1978).
A. Yu. Loskutov and A. S. Mikhailov, An Introduction to Synergetics [in Russian], Nauka, Moscow (1990).
P. Berge, Y. Pomeau, C. Vidal, Order Within Chaos [Russian translation], Merkurii Press, Moscow (2000).
H. G. Schuster, Deterministic Chaos: An Introduction [Russian translation], Mir, Moscow (1988).
T. S. Akhromeeva, S. P. Kurdyumov, G. G. Malinetskii, and A. A. Samarskii, Nonstationary Structures and Diffusion Chaos [in Russian], Nauka, Moscow (1992).
A. N. Sharkovskii, “Coexistence of cycles of a continuous transformation of the straight line into itself,” Ukr. Mat. Zh., 26, No. 1, 61–71 (1964).
Ya. G. Sinai, Modern Topics in Ergodic Theory [in Russian], Fizmatlit, Moscow (1995).
N. A. Magnitskii and S. V. Sidorov, “A new view of the Lorentz attractor,” Different. Uravnen., 37, No. 11, 1494–1506 (2001).
N. A. Magnitskii and S. V. Sidorov, “Onset of chaos in a Lorenz system through a complete double homoclinic bifurcation cascade,” Nonlinear Dynamics and Control, No. 2, 179–194, Fizmatlit, Moscow (2002).
N. A. Magnitskii and S. V. Sidorov, “Actual problems of chaotic dynamics in dissipative systems of nonlinear ordinary differential equations,” Dynamics of Non-Homogeneous Systems, No. 7, 5–45 (2004).
N. A. Magnitskii and S. V. Sidorov, “On scenarios of the onset of chaos in nonlinear dynamical systems described by ordinary differential equations,” Nonlinear Dynamics and Control, No. 3, 73–98, Fizmatlit, Moscow (2003).
D. A. Kaloshin, N. A. Magnitskii, and S. V. Sidorov, “On some features of the onset of chaos in the system of Lorenz equations,” Nonlinear Dynamics and Control, No. 3, 99–106, Fizmatlit, Moscow (2003).
Additional information
__________
Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 135–140, 2004.
Rights and permissions
About this article
Cite this article
Magnitskii, N.A., Sidorov, S.V. A scenario for the creation of chaotic attractors in Chua’s system. Comput Math Model 19, 39–44 (2008). https://doi.org/10.1007/s10598-008-0004-9
Issue Date:
DOI: https://doi.org/10.1007/s10598-008-0004-9