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A family of chaotic oscillators with qualitative dynamics similar to the chaotic Colpitts oscillator is introduced. The oscillators use a single current feedback op amp, configured as a noninverting voltage-controlled voltage source, as the active building block, and a nonlinear element with an antisymmetrical current—voltage characteristic. A procedure for obtaining the chaotic oscillators by modifying simple harmonic oscillators is demonstrated. These chaos generators are suitable for high-frequency operation; device parasitics have negligible effect. Experimental results, PSpice circuit simulations and numerical simulations of the derived mathematical models are included.

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... Even for those simple piecewise-linear circuits, the knowledge of the essential structures required to get rich nonlinear behavior is not completely well understood. Recently, several works have appeared about decomposition of circuits in functional blocks ( [13], [14]), looking for a systematic procedure suitable for designing oscillators with prescribed properties. Our approach has some points in common with the quoted works, but we are rather interested in the analysis of the dynamics of the systems and so we search for simpler equations, i.e., canonical forms, as a starting point to describe the corresponding dynamics and its eventual bifurcations. ...

... A key observation is that systems (4) for are particular instances of the control systems (13) where is the control signal and is the output. Then, some concepts of classical linear time invariant control systems are useful in obtaining reduced canonical forms for systems with parallel boundaries. ...

... As is well known, the rank of controllability and observability matrices are invariant under linear changes of variables. Now, we will give a canonical form for observable systems (13), which is slightly different from that of [16] but totally equivalent. ...

A basic methodology to understand the dynamical behavior of a
system relies on its decomposition into simple enough functional blocks.
In this work, following that idea, we consider a family of
piecewise-linear systems that can be written as a feedback structure. By
using some results related to control systems theory, a simplifying
procedure is given. In particular, we pay attention to obtain equivalent
state equations containing both a minimum number of nonzero coefficients
and a minimum number of nonlinear dynamical equations (canonical forms).
Two new canonical forms are obtained, allowing to classify the members
of the family in different classes. Some consequences derived from the
above simplified equations are given. The state equations of different
electronic oscillators with two or three state variables and two or
three linear regions are studied, illustrating the proposed methodology

... amplifiers, and thus do not satisfy the second criteria of simplicity presented above. LC based oscillators, among which the celebrated Chua's oscillator, can be suitably designed to satisfy the second criteria of simplicity [1][2][3][4][19][20][21][22][23][24][25][26]. At this point, we categorize autonomous circuits based on symmetry considerations. ...

... However, Chua's diode presents a relatively complex structure that justifies the need for other LC oscillator structures with simpler nonlinear component. Unfortunately, the literature devoted to LC oscillators is dominated with models without symmetry [3,4,9,19,[22][23][24][25][26], and only few examples of those with symmetry [1,2,20,21,24] are reported. Motivated by the above results, this work introduces a novel autonomous single amplifier-based LC oscillator obtained via replacing the single semiconductor diode of the circuit in [19] by two antiparallel semiconductor diodes. ...

... The novel oscillator uses only offthe shelf and affordable electronic components, and may be re-scaled over a wide range of frequencies by properly selecting the values of circuit components. The novel chaotic oscillator introduced in this paper uses only a single op amplifier chip without any analog multiplier; and thus represents, to the best of our knowledge, one of the simplest LC circuit reported to date, with the ability to develop such type of multistability [1][2][3][4][19][20][21][22][23][24][25][26]. A combination of features including the simplicity of the mathematical model, the simplicity of the electronic circuit, and an extremely rich dynamic behavior including antimonotonicity, chaos, crises, hysteresis, and multiple attractors are demonstrated in the proposed chaotic circuit, and deserves dissemination [45]. ...

A novel autonomous RLCC-Diodes-Opamp chaotic oscillator with a pair of antiparallel semiconductor diodes implementing hyperbolic sine nonlinearity is introduced. Basic dynamic properties of the new system are categorized numerically with respect to its parameters by exploiting standard nonlinear analysis tools such as time series, bifurcation diagrams, plots of largest Lyapunov exponent, phase portraits, Poincaré sections, and basins of attraction. Some striking phenomena are reported including antimonotonicity, period doubling, crises, chaos, hysteresis, and coexisting bifurcations. More importantly, one of the most interesting results is the finding of various regions in the parameters’ space in which the proposed oscillator develops the phenomenon of multiple attractors characterized by the coexistence of up to four disconnected periodic and chaotic attractors for the same values of parameters. Laboratory measurements are consistent with the theoretical results.

... One of the most active fields for chaos and applications remains that of electronic circuits. Apart from the groundbreaking Chua's circuit [3], many other chaotic circuits have been found, or existing oscillators used to introduce sinusoidal oscillations in curricula have been modified for chaos [16,17,18,19,20,21], so that proposing new circuits can now make sense according to Sprott [22], only if they fulfill at least one of the following requirements: ...

... In several works the expression of current in the transistor is a function of the voltage between the base and the emitter [13,16,17,18,21,31,32,36]. But this is actually a simplified form and can hide some phenomena. ...

A simple driven bipolar junction transistor (BJT) based two-component circuit is presented, to be used as didactic tool by Lecturers, seeking to introduce some elements of complex dynamics to undergraduate and graduate students, using familiar electronic components to avoid the traditional black-box consideration of active elements. Although the effect of the base-emitter (BE) junction is practically suppressed in the model, chaotic phenomena are detected in the circuit at high frequencies (HF), due to both the reactant behavior of the second component, a coil, and to the birth of parasitic capacitances as well as to the effect of the weak nonlinearity from the base-collector (BC) junction of the BJT, which is otherwise always neglected to the favor of the predominant but now suppressed base-emitter one. The behavior of the circuit is analyzed in terms of stability, phase space, time series and bifurcation diagrams, Lyapunov exponents, as well as frequency spectra and Poincar e map section. We find that a limit cycle attractor widens to chaotic attractors through the splitting and the inverse splitting of periods known as antimonotonicity. Coexisting bifurcations confirm the existence of multi-stability behaviors, marked by the simultaneous apparition of different attractors (periodic and chaotic ones) for the same values of system parameters and different initial conditions. This contribution provides an enriching complement in the dynamics of simple chaotic circuits functioning at high frequencies. Experimental lab results are completed with PSpice simulations and theoretical ones.

... However, with the aid of the composites presented in Section III and with sufficient experience, this design methodology proves to be indeed systematic. The authors have demonstrated the flexibility of this procedure by modifying families of sinusoidal oscillators for chaos [33] [35]. ...

... Starting with a second-order oscillator, it remains to add an extra capacitor or inductor. Although several chaotic oscillators have been designed using the FET-C composite after adding an inductor [33], [34], we demonstrate two configurations that require the addition of a single capacitor. The result is an inductorless chaotic oscillator which is advantageous in many respects. ...

A design procedure for producing chaos is proposed. The procedure aims to transfer design issues of analog autonomous chaotic oscillators from the nonlinear domain back to the much simpler linear domain by intentionally modifying sinusoidal oscillator circuits in a semisystematic manner. Design rules that simplify this procedure are developed and then two composite devices, namely, a diode-inductor composite and a FET-capacitor composite are suggested for carrying out the modification procedure. Applications to the classical Wien-bridge oscillator are demonstrated. Experimental results, PSpice simulations, and numerical simulations of the derived models are included Enterprise Ireland (Basic Research Programme under Grant SC/98/740) Published Version Peer reviewed

... Even though chaotic systems are extremely sensitive, the sensitivity of these systems is depend on the initial conditions. The chaotic character is one of the qualitative [7], [8] properties of a dynamical system [9], [10], [11], [12]. ...

... Find Ψ along with the trajectories associated with (8). It follows that ...

In this paper, two logarithmic non-linearities are proposed for a new four-dimensional chaotic system. The phase portrait, Lyapunov exponent, bifurcation, stability, and other dynamical features of the new chaotic system are all discussed. The multi-stability of the new chaotic system with coexisting attractors has been established. The adaptive backstepping control approach with proper Lyapunov functions is used in the control application to retrieve the unknown parameters of the system. To synchronise the states between the drive-response system, non-linear feedback control is used, as well as back-stepping control to synchronise the states on the system's error dynamics. Op-amp circuits are used to create the electronic circuit design for a new chaotic system. The system's efficiency is confirmed using MATLAB numerical simulation.

... Some of these systems have unique features from a nonlinear dynamical point of view [1] while others are more focused on simplicity and suitability for circuit implementation [2]. Some of the simplest chaotic circuits include the Wien-type oscillator of [3] and the Colpitts-based family of [4]. Generation of chaos requires the existence of at least one nonlinear function which can be asymmetric (typical of diode characteristics for example) [4], oddsymmetric [5], even-symmetric [6,7], periodic [8], containing hysteresis [9], or based on discrete maps [10]. ...

... Some of the simplest chaotic circuits include the Wien-type oscillator of [3] and the Colpitts-based family of [4]. Generation of chaos requires the existence of at least one nonlinear function which can be asymmetric (typical of diode characteristics for example) [4], oddsymmetric [5], even-symmetric [6,7], periodic [8], containing hysteresis [9], or based on discrete maps [10]. Chaotic dynamics are widely used to produce pseudo-random number generators and for secure communications and encryption applications [11,12]. ...

We propose a mathematical system capable of exhibiting chaos with a chaotic attractor which is odd symmetrical in the x − y phase plane but even symmetrical in the x − z and y − z phase planes respectively. A hardware implementation of the system is done on a digital FPGA platform for verification. The system is also attractive in the sense that (i) its dynamics are single-parameter controlled and (ii) it inherently generates two chaotic clock signals. As an application, an FPGA design methodology using this oscillator for speech encryption is demonstrated. The security of the proposed encryption scheme is evaluated and results confirm its robustness. Due to the efficient hardware resource utilization, the encrypted system delivers a throughput of 1.3Gbit/sec using a Xilinx Kintex 7.

... Many operational amplifier-based Colpitts oscillators use a built-in negative resistor to inject the non-linear signal responsible for generating oscillations in the feedback loop [31,37]. Although the use of negative resistance generally leads to impressive results [38,39], we would challenge to use only the CLC resonator and one Op-Amp to produce an autonomous Colpitts like chaotic and hyperchaotic oscillator, which is one of the aims of this paper. ...

In the framework of a project on simple circuits with unexpected high degrees of freedom, we report an autonomous microwave oscillator made of a CLC linear resonator of Colpitts type and a single general purpose operational amplifier (Op-Amp). The resonator is in a parallel coupling with the Op-Amp to build the necessary feedback loop of the oscillator. Unlike the general topology of Op-Amp-based oscillators found in the literature including almost always the presence of a negative resistance to justify the nonlinear oscillatory behavior of such circuits, our zero resistor circuit exhibits chaotic and hyperchaotic signals in GHz frequency domain, as well as many other features of complex dynamic systems, including bistability. This simplest form of Colpitts oscillator is adequate to be used as didactic model for the study of complex systems at undergraduate level. Analog and experimental results are proposed.

... Elwakil and Kennedy presented family of chaotic oscillator circuits with dynamics qualitatively similar to the Col Fig. 1. The chaotic Colpitts Oscillator Circuit [16] pitts oscillator [29]. The oscillators used a single current feedback non-inverting op-amp used as voltage controlled voltage source along with a nonlinear component with nonsymmetrical current versus voltage characteristics. ...

This paper presents a comparative study of the Colpitts oscillator circuit using circuit simulations and experimental results. Different techniques of dynamical systems theory like time series plots, phase portraits and Lyapunov exponents were employed. The time series plots and phase portraits for different state variables of Colpitts oscillator circuit, obtained from PSpice simulation and experimental implementation, were compared with each other. This compasison showed that both the results are identical. It is evident that chaotic Colpitts oscillator exhibited periodic and aperiodic behavior for different values of the circuit parameters.

... Although the Colpitts oscillator was originally designed to be an almost-sinusoidal oscillator [Sedra & Smith, 1998], it has been shown to exhibit a rich dynamical behavior at certain parameter values [Kennedy, 1994;Maggio et al., 1999;Elwakil & Kennedy, 1999]. Recently, it was demonstrated in Maggio et al., 1999;] how bifurcation theory, normal forms, and numerical continuation techniques can be usefully employed to characterize qualitatively the different dynamical behaviors exhibited by this oscillator. ...

This paper presents an experimental verification of the theoretical predictions, recently published in [Maggio et al., 1999; De Feo et al., 2000], about the bifurcation phenomena occurring in the Colpitts oscillator. Specifically, we performed an automated series of simulations based on the Spice model and, more importantly, a computer-assisted set of measurements on a concrete realization of the oscillator. It turns out that the bifurcation phenomena exhibited by the oscillator are relatively independent of the simplifying assumptions on the transistor model. Moreover, it is shown that the predicted behaviors can be reproduced experimentally, both qualitatively and quantitatively, in a robust way.

... Our design is superior in that it provides a buffered output voltage that directly represents a state variable, in addition to a current output signal. It should also be noted that other extended frequency chaotic oscillators using the CFOA have also been introduced recently [20], [21]. ...

... Various electronic circuits exhibiting chaos have been proposed and intensively studied in the last four decades [1][2][3][4][5][6][7][8][9][10]. Mostly, the dynamics of circuits used to introduce sinusoidal oscillations in basic electronics courses such as the Colpitts oscillator [6,[11][12][13], the Wien-Bridge oscillator [14][15][16][17], the Twin-T oscillator [5] etc. have been widely discussed in their nonlinear and chaotic behavior too, as indicated in the mentioned references. Surprisingly, as far as the Hartley's oscillator is concerned, which indeed also belongs to the previous category of circuits contained in some general electronics books, in the best of our knowledge, very few case studies on its ability to generate chaotic signals have been done, though the Hartley's oscillator is easy to construct, harmonic-rich and well used in telecommunication [18][19][20][21][22]. ...

This paper shows an experimental evidence of chaos in one of the simplest imaginable autonomous implicit Hartley’s oscillator made simply of a junction field effect transistor (JFET) and a tapped coil. The experimental setup is implemented. The variations of amplitude of the oscillations through the control element are obtained showing the domain of existence of chaos. Phase portraits of the PSpice simulation, of the numerical integration and of the experiment are displayed, confirming a good agreement between theory and praxis.

... Designing chaotic oscillators is a research topic that has received considerable interest during the past few years [1][2][3][4]. This has been motivated by possible commercial applications of chaotic signals, particularly in communication systems. ...

A simple relaxation oscillator is designed by directly coupling a RC timing network to a passive S-shaped current-controlled nonlinear resistor and is then modified for chaos. The resulting chaotic oscillator inherits the main features of the relaxation oscillator, which are its low-power consumption and low-voltage operation from single or dual power supplies. These features are attributed to a simple two-bipolar-transistor passive nonlinear resistor. PSpice circuit simulations, experimental results and simulations of the derived mathematical models are included.

... The occurrence of chaos in the Colpitts oscillator was firstly discovered by Kennedy in 1994 [1]. Since then it is always a hot top in the area of nonlinear circuits and systems [2][3][4][5][6][7][8]. Unlike the famous Chua's circuit [9], whose bandwidth is greatly limited by the nonlinear negative resistance commonly built with operational amplifier, the upper limit fundamental frequency of a Colpitts oscillator is generally determined by the threshold frequency of the bipolar junction transistors (BJTs) employed. ...

Randomness test was conducted on signal sampled from the output of a microwave chaotic Colpitts oscillator operating with fundamental frequency of 960 MHz. Experimental data analysis shows the signal has an irregular distribution and a low degree of autocorrelation. Binary sequences converted from the signal can pass the four but one randomness tests in FIPS standard. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 1981–1984, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22565

A collection of novel chaotic oscillators displaying behavior similar to that of the chaotic Colpitts oscillator and requiring the same number and type of energy storage elements is proposed. The oscillators use as an active element the current feedback op amp (CFOA) mostly employed as a current negative impedance converter (INIC). Nonlinearity is introduced through a two-terminal voltage-controlled nonlinear device with an antisymmetric driving-point characteristic. The chaos generators are designed based on sinusoidal oscillators that have been modified for chaos in a semi-systematic manner. By using CFOAs, several attractive features are attained, in particular suitability for high frequency operation. Systems of third- and fourth-order ordinary differential equations describing the chaotic behaviors are derived. Experimental results, PSpice circuit simulations and numerical simulations of the derived mathematical models are included.

We propose a novel autonomous system for chaos generation based on
a third-order abstract canonical mathematical model. Nonlinearity in
this system is introduced by a bipolar switching constant which reflects
the behaviour of a simple inverter circuit. Two implementations of the
system are given. The first uses commercially available components while
the second was designed on a CMOS chip. Numerical simulations and
experimental results are provided

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 Enterprise Ireland (Basic Research Program Grant SC/98/740) Published Version Peer reviewed

An improved implementation of Chua's chaotic oscillator is proposed. The new realization combines attractive features of the current feedback op amp (CFOA) operating in both voltage and current modes to construct the active three-segment voltage-controlled nonlinear resistor. Several enhancements are achieved: the component count is reduced and the chaotic spectrum is extended to higher frequencies. In addition, a buffered and isolated voltage output directly representing a state variable is made available. Based on a linearized model of Chua's circuit, the useful tuning range of the major bifurcation parameter (G) and the expected frequency of oscillation, are estimated Enterprise Ireland (Grant SC/98/740) Published Version Peer reviewed

The aim of this article is to present a new chaotic oscillator. Although several chaotic systems have been formulated, despite that a few chaotic systems exhibit chaotic behavior. A new chaotic system with chaotic attractor is introduced for the nonlinearity of triangular waves. It is interesting to note that this striking phenomenon occurs rarely compared to chaotic systems. The results from the numerical simulation indicate the feasibility of the proposed chaotic system. In addition, the chaos control, stability, diffusion and synchronization of such a system were discussed.

A chaotic oscillator configuration employing a frequency-dependent negative resistor (FDNR) as the only active element is proposed. The configuration relies on a simple two-terminal passive device; namely a general purpose signal diode, to provide the necessary non-linearity. The structure requires no floating elements and is independent of any circuit specific realization of the FDNR. Experimental results, PSpice simulations and numerical simulations of the derived mathematical models are included. Copyright © 2000 John Wiley & Sons, Ltd.

In this paper, we continue our study of rank one chaos in switch-controlled circuits. Periodically controlled switches are added to Chua's original piecewise linear circuit to generate rank one attractors in the vicinity of an asymptotically stable periodic solution that is relatively large in size. Our previous investigations relied heavily on the smooth nonlinearity of the unforced systems, and were, by large, restricted to a small neighborhood of supercritical Hopf bifurcations. Whereas the system studied in this paper is much more feasible for physical implementation, and thus the corresponding rank one chaos is much easier to detect in practice. The findings of our purely numerical experiments are further supported by the PSPICE simulations.

Using two-port network representations, we classify Colpitts oscillators into three categories based on the specific active device terminal which is grounded. General and accurate characteristic equations which are independent of any particular transistor (or active device) model are derived for all three classes. Possible two-impedance two-port network oscillators are also analysed.

A novel broadband meander-line antenna with vertical lines is presented. The proposed antenna has a broadband impedance bandwidth of ∼785 MHz, which covers DCS, PCS, UMTS, WiBro, and WLAN bands. For frequencies across the operating bands, the proposed antenna displays omni-directional radiation patterns. The proposed antenna has compact dimensions of 12 × 23.1 mm2. The antenna is flexible due to the F-PCB material used. For these reasons, a cylindrical and folded structure is presented, as transformed from planar. Experimental results using the constructed prototype are presented and discussed. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 1984–1987, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22564

The fact that Chua's circuit can be physically decomposed into a sinusoidal oscillator coupled to an active voltage-controlled nonlinear resistor is demonstrated. The sinusoidal oscillator is the beating heart of Chua's circuit and many novel implementations can be obtained by using di!erent sinusoidal oscillator engines. In particular, inductorless realizations can be derived not by replacing the passive inductor in the classical Chua's circuit con"guration with an active RC emulation, but rather by replacing the passive LC tank resonator by a sinusoidal oscillator. We provide several circuit-design examples and verify our results experimentally. Finally, we state a conjecture which we believe forms a basis for the design of autonomous analog chaotic oscillators.

In this paper, a new chaotic oscillator consists of a single op-amp, two capacitors, one resistor, one inductor, and memristive diode bridge cascaded with an inductor is proposed. The proposed chaotic oscillator has a line of equilibria. In the new oscillator circuit, negative feedback, i.e. inverting terminal of the op-amp is used, and the non-inverting terminal is grounded. The new oscillator has chaotic, periodic, quasi-periodic behaviours, as seen from the Lyapunov spectrum plots. Some more theoretical and numerical tools are used to present the dynamical behaviours of the new oscillator like bifurcation diagram, phase plot. Further, a non-singular terminal sliding mode control (N-TSMC) is designed for the suppression of the chaotic states of the new oscillator. An application of the new oscillator is shown by designing a chaos-based random number generator. Raspberry Pi 3 is used for the realisation of the random number generator.

A systematic method for realizing a class of hysteresis RC chaotic oscillators is described. The method is based on direct coupling of a general second-order sinusoidal oscillator structure to a passive non-monotone current-controlled non-linear resistor. Owing to this passive non-linearity, the power consumption, supply voltage and bandwidth limitations imposed upon the chaotic oscillator are mainly those due to the active sinusoidal oscillator alone. Tunability of the chaotic oscillator can be achieved via a single control parameter and the evolution of the two-dimensional sinusoidal oscillator dynamics into a three-dimensional state-space is clearly recognized. The flexibility of this method is demonstrated by two examples using PSpice simulations and experimental results. Numerical simulations of derived mathematical models are also shown. Copyright © 2000 John Wiley & Sons, Ltd.

This chapter deals with miscellaneous linear and nonlinear applications of CFOAs which include electronically-variable gain amplifier, cable driver, video distribution amplifier, a variety of Schmitt Triggers, nonlinear wave form generators, Precision rectifiers, Analog divider, Pseudo exponential circuits and both autonomous and non-autonomous chaotic non-linear circuits.

In this chapter, it has been demonstrated how CFOAs have been used by a number of researchers in realizing other analog circuit building blocks such as various types of current conveyors, unity gain voltage and current followers, four terminal floating nullors, Current differencing buffered amplifiers, operational trans-resistance amplifiers, Current differencing transconductance amplifiers, third generation Current conveyors (CCIII), differential voltage second generation Current Conveyors, Current follower transconductance amplifiers, current controlled current conveyor transconductance amplifier, differential-input buffered transconductance amplifier and voltage differencing differential input buffered amplifier etc. These applications further establish the flexibility and versatility of CFOAs in analog circuit design.

We propose a nonautonomous version of Wien-bridge oscillator with diode nonlinearity. It is a kind of simple circuit which exhibits chaotic behaviour. This oscillator circuit contains an operational amplifier, four resistors, two capacitors, a diode as a nonlinear element and external periodic force. This system exhibits various interesting dynamical phenomena like periodic, quasiperiodic and chaotic oscillations. The detailed analysis is carried out numerically by using two-parameter phase diagram in the forcing amplitude-frequency plane, one-parameter bifurcation diagram, Lyapunov exponents and phase portraits. Most of these numerical studies are in good agreement with observations from experiments.

This paper implements two kinds of memristor-based colpitts oscillators, namely, the circuit where the memristor is added into the feedback network of the oscillator in parallel and series, respectively. First, a MULTISIM simulation circuit for the memristive colpitts oscillator is built, where an emulator constructed by some off-the-shelf components is utilized to replace the memristor. Then the physical system is implemented in terms of the MULTISIM simulation circuit. Circuit simulation and experimental study show that this memristive colpitts oscillator can exhibit periodic, quasi-periodic, and chaotic behaviors with certain parameter’s variances. Besides, in a sense, the circuit is robust with circuit parameters and device types.

A novel three-dimensional chaotic attractor derived from Colpitts equation is proposed in this paper. When the given parameter varies in a broad range, the amplitude of the signals of the first two dimensions changes linearitly while the third one keeps its amplitude in the same range. At the same time, the Lyapunov exponent spectrum keeps invariable. This chaotic system is developed by substituting the absolute term for the exponent term in normalized Colpitts equation. Lyapunov exponent, Poincaré mapping, phase portrait and spectrum are given to verify that the attractors are chaotic. In addition, some basic dynamical characteristics of the new system are investigated briefly. Based on Lyapunov exponent spectrum analysis, it is demonstrated that the new system can go into periodic and chaotic behaviors. At last, the Jerk function of the new system is put forward and its circuit implementation is designed. The feature that the chaotic characteristic of this system has nothing to do with the given parameter while the amplitude of some state variables can be changed linearly makes it reasonable to predict that the chaotic system will have tremendous potential applications in chaotic radar, secure communications and other information processing systems.

Two complementary sinusoidal oscillator circuits based on the Twin-T network are modified for chaos using a single discrete nonlinear device with antisymmetrical current-voltage characteristics, namely a junction field effect transistor (JFET) operating in its triode region. The two oscillators' design equations are used as a starting point for chaos modification. Mathematical models that describe the observed behaviour in both circuits are derived. Experimental results, PSpice circuit simulations and numerical simulations of the mathematical models agree well and are included.

The current feedback operational amplifiers (CFOAs) are receiving increasing attention as basic building blocks in analog circuit design. This paper gives an overview of the applications of the CFOAs, in particular several new circuits employing the CFOA as the active element are given. These circuits include differential voltage amplifiers, differential integrators, nonideal and ideal inductors, frequency dependent negative resistors and filters. The advantages of using the CFOAs in realizing low sensitivity universal filters with grounded elements will be demonstrated by several new circuits suitable for VLSI implementation. PSPICE simulations using the AD844-CFOA which indicate the frequency limitations of some of the proposed circuits are included.

An RC oscillator exhibiting chaotic behaviour is described. It
contains two opamps, a Wien bridge and a diode used as a nonlinear
device. The typical waveforms, the phase portraits, the power spectra
and the correlation dimension of the attractor are presented to
illustrate the chaotic oscillations

A nonlinear Wien-bridge based circuit generating chaotic
oscillations is reported. The generator contains a single opamp and a
single nonlinear device displaying a current saturation characteristic.
The oscillator is described by a set of three ordinary differential
equations. Experimental results are included demonstrating the circuit
performance

Chua's circuit can produce a very rich variety of signals that are
both periodic and chaotic. The authors explore some classes of these
attractors with respect to their auditory display and musical
properties. They discuss the fast control of the circuit through a
specially developed computer-controlled electronic resistor and how
chaotic control methods might be applied to optimally switch between
different attractors. The Chua circuit has parameter regions where noisy
frequency and amplitude modulated sounds are generated, each of which is
related to a certain transition to chaos. The authors discovered a
period-adding sequence of bassoon-like sounds that produces interesting
almost harmonic pitch changes. Finally, they emphasize the importance of
transient dynamics especially in the context of percussion-like sounds

This work describes a new CMOS current-feedback operational
amplifier (CFOA) with an on-chip continuous-time current-mode input
offset voltage compensation circuit. The proposed compensation method is
based on a combination of two techniques: the error integration and the
current feedback. In addition, this method is irrespective of process
and temperature parameters because of its fully symmetrical
architecture. HSPICE simulations of the designed CMOS CFOA layout show
that the input offset voltage could be reduced to lass than 1 mV, and a
gain of around 112 dB and a power consumption of less than 3 mW are
achievable

In this letter, we show that the two-region third-order
piecewise-linear dynamics of the chaotic Colpitts oscillator may be
mapped to a Chua's oscillator with an asymmetric nonlinearity

In this work we derive a general non-conservative model for any RC sinusoidal oscillator independent of its particular passive topology or the employed active devices. We consider an arbitrary unknown second-order passive RC network terminated at one port by a negative resistor and proceed to impose oscillation start-up and frequency constraints on a derived state-matrix.

New sinusoidal oscillators with single element control using a current-feedback amplifier (CFA) are presented. The oscillation frequency of the proposed sinusoidal oscillators can be controlled by a resistor or capacitor. The oscillation frequency of one of the proposed sinusoidal oscillators is insensitive to the input and output voltage tracking errors of the CFA. Another of the proposed oscillators uses only grounded capacitors. Passive and active sensitivities are all low. Experimental results that confirm the theoretical analysis are obtained.

A novel CMOS realization of the second generation current
conveyor is given. A circuit which compensates the voltage offset due to
channel length modulation effect is then developed. The CCII is then used
to realize a new electronically tunable low-pass-hand-pass filter suitable
for VLSI. Simulation results taking the second-order effects into account
indicate the excellent performance of both the CC11 circuit and the filter
over a wide dynamic range.

A family consisting of four Wien-type oscillator circuits are modified for chaos by direct replacement of one of the linear resistors with an asymmetrical-type non-linearity introduced by a junction field effect transistor (JFET) operating in its triode region and the addition of a single capacitor. The internal op amp dominant pole is found to play a major role in understanding the chaotic behaviour of the proposed circuits. Mathematical models that describe the observed behaviours are derived. The well known Wien bridge oscillator design equations are shown to be useful as a starting point for chaos modification. Experimental laboratory results, PSpice simulations and numerical simulations of the mathematical models are provided for this family of autonomous RC chaos generators. © 1997 John Wiley & Sons, Ltd.

First Page of the Article

A new variable frequency sinusoidal oscillator configuration is
proposed. Only two current feedback operational amplifiers (CFOAs), two
grounded capacitors and three resistors are used. The frequency and
condition of oscillation can be independently adjusted by means of
single grounded resistors. Also, very high frequency operation is
demonstrated and an example application of an oscillator which does not
require an external capacitor presented

A new circuit, which is formed by coupling a Chua diode with a
Wien bridge oscillator in parallel, is presented. This circuit contains
only resistors, capacitors and operational amplifiers. By choosing
element values appropriately, this circuit is shown experimentally to
exhibit various forms of chaotic behaviour

New CMOS rail to rail second generation current conveyor circuits
are proposed. First a class A current conveyor circuit which operates
from a single supply of 1.5 V with a rail to rail voltage swing
capability is given. The circuit is then modified to work as a class AB
while maintaining the rail to rail swing capability. The class AB
circuit works from supply voltages down to +1.1 V with standby current
of 56 μA. These new current conveyor realizations are insensitive to
the threshold voltage variation caused by the body effect, which
minimizes the layout area and makes both circuits a valuable addition to
the analog VLSI libraries. PSpice simulation confirms the attractive
properties of the proposed circuits

In this work, we present experimental results and SPICE
simulations of chaos in a Colpitts oscillator. We show that the
nonlinear dynamics of this oscillator may be modeled by a third-order
autonomous continuous-time circuit consisting of a linear inductor, two
linear capacitors, two linear resistors, two independent voltage
sources, a linear current-controlled current source, and a single
voltage-controlled nonlinear resistor. The nonlinear resistor has a
two-segment piecewise-linear DP characteristic. With the appropriate
choice of parameters, the piecewise-linear circuit model has a positive
Lyapunov exponent

A multiloop active filter circuit of Kerwin is shown to yield a VCO whose oscillation frequency is conveniently controlled with one variable resistor. Using an FET to obtain the voltage variable resistor, experimental results are shown to agree with the theory developed.

The results of a comprehensive investigation into the
characteristics and optimization of inductors fabricated with the
top-level metal of a submicron silicon VLSI process are presented. A
computer program which extracts a physics-based model of microstrip
components that is suitable for circuit (SPICE) simulation has been used
to evaluate the effect of variations in metallization, layout geometry,
and substrate parameters upon monolithic inductor performance.
Three-dimensional (3-D) numerical simulations and experimental
measurements of inductors were also used to benchmark the model
accuracy. It is shown in this work that low inductor Q is primarily due
to the restrictions imposed by the thin interconnect metallization
available in most very large scale integration (VLSI) technologies, and
that computer optimization of the inductor layout can be used to achieve
a 50% improvement in component Q-factor over unoptimized designs

Chaos in the Colpitts oscillator, 1EEE Trans. Circuits and Systems -I 41

- M P Kennedy

M. P. Kennedy, Chaos in the Colpitts oscillator, 1EEE Trans. Circuits and Systems -I 41 (1994) 77l 774.

Emerging Techniques for High Frequency BJT Amplifier Design: A Current Mode Perspective

- C Toumazou
- J Lidgey
- A Payne

C. Toumazou, J. Lidgey, A. Payne, Emerging Techniques for High Frequency BJT Amplifier Design: A Current Mode Perspective, Parchment Press, Oxford, 1994.