Emergence of complex dynamic behaviours in the Chua's circuit with a nonlinear inductor

In this paper, we consider an infinite-dimensional extension of Chua's circuit (Fig. 1) obtained by replacing the left portion of the circuit composed of the capacitance C2 and the inductance L by a lossless transmission line as shown in Fig. 2. As we shall see, if the remaining capacitance C1 is equal to zero, the dynamics of this so-called time-delayed Chua's circuit can be reduced to that of a scalar nonlinear difference equation. After deriving the corresponding 1-D map, it will be possible to determine without any approximation the analytical equation of the stability boundaries of cycles of every period n. Since the stability region is nonempty for each n, this proves rigorously that the time-delayed Chua's circuit exhibits the "period-adding" phenomenon where every two consecutive cycles are separated by a chaotic region.

Download Full-text

CHUA’S CIRCUIT: AN OVERVIEW TEN YEARS LATER

Journal of Circuits System and Computers ◽

10.1142/s0218126694000090 ◽

1994 ◽

Vol 04 (02) ◽

pp. 117-159 ◽

Cited By ~ 142

Author(s):

LEON O. CHUA

Keyword(s):

Secure Communication ◽

Vector Fields ◽

Piecewise Linear ◽

International Workshop ◽

Noise Spectrum ◽

Chaotic Regime ◽

Practical Significance ◽

Chua’S Circuit ◽

Chua's Circuit ◽

New Phenomena

More than 200 papers, two special issues (Journal of Circuits, Systems, and Computers, March, June, 1993, and IEEE Trans. on Circuits and Systems, vol. 40, no. 10, October, 1993), an International Workshop on Chua’s Circuit: chaotic phenomena and applica tions at NOLTA’93, and a book (edited by R.N. Madan, World Scientific, 1993) on Chua’s circuit have been published since its inception a decade ago. This review paper attempts to present an overview of these timely publications, almost all within the last six months, and to identify four milestones of this very active research area. An important milestone is the recent fabrication of a monolithic Chua’s circuit. The robustness of this IC chip demonstrates that an array of Chua’s circuits can also be fabricated into a monolithic chip, thereby opening the floodgate to many unconventional applications in information technology, synergetics, and even music. The second milestone is the recent global unfolding of Chua’s circuit, obtained by adding a linear resistor in series with the inductor to obtain a canonical Chua’s circuit— now generally referred to as Chua’s oscillator. This circuit is most significant because it is structurally the simplest (it contains only 6 circuit elements) but dynamically the most complex among all nonlinear circuits and systems described by a 21-parameter family of continuous odd-symmetric piecewise-linear vector fields. The third milestone is the recent discovery of several important new phenomena in Chua’s circuits, e.g., stochastic resonance, chaos-chaos type intermittency, 1/f noise spectrum, etc. These new phenomena could have far-reaching theoretical and practical significance. The fourth milestone is the theoretical and experimental demonstration that Chua’s circuit can be easily controlled from a chaotic regime to a prescribed periodic or constant orbit, or it can be synchronized with 2 or more identical Chua’s circuits, operating in an oscillatory, or a chaotic regime. These recent breakthroughs have ushered in a new era where chaos is deliberately created and exploited for unconventional applications, e.g. secure communication.

Download Full-text