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Art of electronics

The field of chaotic synchronization has grown considerably since its advent in 1990. Several subdisciplines and “cottage industries” have emerged that have taken on bona fide lives of their own. Our purpose in this paper is to collect results from these various areas in a review article format with a tutorial emphasis. Fundamentals of chaotic synchronization are reviewed first with emphases on the geometry of synchronization and stability criteria. Several widely used coupling configurations are examined and, when available, experimental demonstrations of their success (generally with chaotic circuit systems) are described. Particular focus is given to the recent notion of synchronous substitution—a method to synchronize chaotic systems using a larger class of scalar chaotic coupling signals than previously thought possible. Connections between this technique and well-known control theory results are also outlined. Extensions of the technique are presented that allow so-called hyperchaotic systems (syste...

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Fundamentals of synchronization in chaotic systems, concepts, and applications.

This paper addresses the adaptive synchronization problem of the drive–driven type chaotic systems via a scalar transmitted signal. Given certain structural conditions of chaotic systems, an adaptive observer-based driven system is constructed to synchronize the drive system whose dynamics are subjected to the system’s disturbances and/or some unknown parameters. By appropriately selecting the observer gains, the synchronization and stability of the overall systems can be guaranteed by the Lyapunov approach. Two well-known chaotic systems: Rossler-like and Chua’s circuit are considered as illustrative examples to demonstrate the effectiveness of the proposed scheme. Moreover, as an application, the proposed scheme is then applied to a secure communication system whose process consists of two phases: the adaptation phase in which the chaotic transmitter’s disturbances are estimated; and the communication phase in which the information signal is transmitted and then recovered on the basis of the estimated parameters. Simulation results verify the proposed scheme’s success in the communication application.

Adaptive synchronization of chaotic systems and its application to secure communications

Many new chaotic systems with algebraically simple representations are described. These systems involve a single third-order autonomous ordinary differential equation (jerk equation) with various nonlinearities. Piecewise linear functions are emphasized to permit easy electronic implementation with diodes and operational amplifiers. Several new simple and robust chaotic electrical circuits are described and evaluated.

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Simple chaotic systems and circuits

Two generic classes of chaotic oscillators comprising four different configurations are constructed. The proposed structures are based on the simplest possible abstract models of generic second-order RC sinusoidal oscillators that satisfy the basic condition for oscillation and the frequency of oscillation formulas. By linking these sinusoidal oscillator engines to simple passive first-order or second-order nonlinear composites, chaos is generated and the evolution of the two-dimensional sinusoidal oscillator dynamics into a higher dimensional state space is clearly recognized. We further discuss three architectures into which autonomous chaotic oscillators can be decomposed. Based on one of these architectures we classify a large number of the available chaotic oscillators and propose a novel reconstruction of the classical Chua's circuit. The well-known Lorenz system of equations is also studied and a simplified model with equivalent dynamics, but containing no multipliers, is introduced.

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Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices

A fourth-order hyperchaotic oscillator is described. It contains a negative impedance converter, two gyrators, two capacitors and a diode. The dynamics of the oscillator is shown to be characterised by two positive Lyapunov exponents. The performance of the circuit is investigated by means of numerical integration of appropriate differential equations, hardware experiments and PSPICE simulations.

Hyperchaotic oscillator with gyrators

Experimental results demonstrating chaotic performance of the Colpitts oscillator in the very high frequency (30–300 MHz) and the ultra-high frequency (300–1000 MHz) ranges are reported. Period-doubling route to chaos is revealed, confirming the dynamical nature of the chaotic oscillations. Broadband continuous power spectra are presented.

Experimental demonstration of chaos from Colpitts oscillator in VHF and UHF ranges

A novel version of the chaotic Colpitts oscillator is proposed. It contains two bipolar junction transistors coupled in series. The resonance loop consists of an inductor and three capacitors. The two-stage oscillator, compared with the classical circuit, enables the fundamental frequency of chaotic oscillations to be increased by a factor of three. The PSpice simulations performed with 9 GHz threshold frequency transistors demonstrate that the highest fundamental frequencies of chaotic behaviour are 1 and 3 GHz for the classical and the two-stage Colpitts oscillator, respectively.

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Two-stage chaotic Colpitts oscillator

We describe an extremely simple second order analogue electrical circuit for simulating the two-well Duffing-Holmes mathematical oscil lator. Numerical results and analogue electrical simulations are illustrated with the s napshots of chaotic waveforms, with the phase portraits (the Lissajous figures) and with the stroboscopic maps (the Poincare sections).

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Analogue Electrical Circuit for Simulation of the Duffing-Holmes Equation

We demonstrate theoretically and experimentally that the unstable delayed feedback controller is an efficient tool for stabilizing torsion-free unstable periodic orbits in nonautonomous chaotic systems. To improve the global control performance we introduce a two-step control algorithm. The problem of a linear stability of the system under delayed feedback control is treated analytically. Theoretical results are confirmed by electronic circuit experiments for a forced double-well oscillator.

Gytis Mykolaitis

Papers

Hyperchaotic oscillator with gyrators

Experimental demonstration of chaos from Colpitts oscillator in VHF and UHF ranges

Two-stage chaotic Colpitts oscillator

Analogue Electrical Circuit for Simulation of the Duffing-Holmes Equation

Delayed feedback control of periodic orbits without torsion in nonautonomous chaotic systems: theory and experiment.