Article

Family of Colpitts-like chaotic oscillators

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Abstract

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]. ...
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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. ...
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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. ...
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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 ...
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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]. ...
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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. ...
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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. ...
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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. ...
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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]. ...
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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. ...
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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. ...
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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.
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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.