Simple Nonlinear Circuit Research Articles

Dynamics of memristive circuit driven by Josephson junction

The nonlinear circuit with charge-controlled memristor (CCM) can capture the external electric field effect. The nonlinear circuit with Josephson junction (JJ) can estimate the external magnetic field effect. Therefore, we propose an enhanced functional circuit by connecting a CCM and a JJ into a simple RLC nonlinear circuit. This enchanced circuit can estimate the external electromagnetic fields concurrently. The dynamical equations of the new memristive circuit and its Hamilton energy function are obtained by using the Kirchhoff’s law and the Helmholtz’s theorem. Furthermore, the complex dynamics of memristive circuit are investigated by applying bifurcation diagrams, Lyapunov exponents and time sampled series. The simulation experiment results indicate that the electromagnetic field has a great influence on complex dynamics of memristive circuit. In fact, this new nonlinear circuit is also a functional neural circuit, and it can be used to study the collective dynamic of functional neural network under the condition of an external electromagnetic fields.

A memristive map neuron under noisy electric field

From a neural circuit to a biophysical neuron model, standard scale transformation requires the combination of physical variables including capacitance and resistance associated with a capacitor C and a resistor R, respectively. The voltage across the capacitor is used to mimic the membrane potential for neurons. Therefore, most of the neuron models contain one or two capacitive variables for the membrane potential. In fact, the reference time scale (RC) can be replaced by combinating parameters for inductor and resistor during approach of scale transformation for physical variables in the neural circuit. In this study, a simple nonlinear circuit is built by applying two inductors, a charge-controlled memristor (CCM), a nonlinear resistor and a resistor. The circuit equations are derived to develop a memristive oscillator with exact description of energy function. Furthermore, an equiavlent memristive map is obtained by applying linear transformation on the sampled variables from the memristive oscillator. The Hamilton energy function for memristive oscillator is calculated by using the Helmholtz's theorem, and then the memristive map is given in energy function in discrete form. The memristive map has rich dynamic characteristics and can identify coherence resonance under applying noisy electric field. This scheme will provide a theoretical guidance for building nonlinear circuits and maps without using capacitors.

NASA DART mission: A preliminary mathematical dynamical model and its nonlinear circuit emulation

Abstract On September 22, 2022, a spacecraft, designed by the Double Asteroid Redirection Test (DART) team, successfully attempted to deflect the orbit of the asteroid Dimorphos, which together with Didymos, constitutes a binary system of near-Earth asteroids orbiting around the Sun. The effect of the impact of the spacecraft was to shorten the orbit of Dimorphos of about 33 min with respect to the original one. In this communication, a simple nonlinear circuit emulator based on a mathematical model allowing the emulation of the DART mission behavior is presented. The modeling is approached referring to the Kepler problem that leads to a highly nonlinear dynamical model. The problem is approached numerically, by using appropriate integration algorithms for both the two-body and three-body formulations of the problem, and experimentally, by means of an analog/digital electronic circuit emulator of the system that allows us to realize faster and qualitative more efficient experiments.

Realization of all logic gates and memory latch in the SC-CNN cell of the simple nonlinear MLC circuit

We investigate the State-Controlled Cellular Neural Network framework of Murali–Lakshmanan–Chua circuit system subjected to two logical signals. By exploiting the attractors generated by this circuit in different regions of phase space, we show that the nonlinear circuit is capable of producing all the logic gates, namely, or, and, nor, nand, Ex-or, and Ex-nor gates, available in digital systems. Further, the circuit system emulates three-input gates and Set–Reset flip-flop logic as well. Moreover, all these logical elements and flip-flop are found to be tolerant to noise. These phenomena are also experimentally demonstrated. Thus, our investigation to realize all logic gates and memory latch in a nonlinear circuit system paves the way to replace or complement the existing technology with a limited number of hardware.

Realisation of parallel logic elements and memory latch in a quasiperiodically-driven simple nonlinear circuit

We investigate the effect of two aperiodic square waves in a quasiperiodically-driven Murali–Lakshmanan–Chua circuit. It is found that the response of the circuit produces logical output in both strange nonchaotic and chaotic regions. Changing the biasing of the circuit changes the response of the circuit into another kind of logic operation and SR flip flop. Further, we show how this circuit produces two logical elements as its outputs which are complementary to each other. It is also shown that the logical nature of the circuit persists even when experimental noise is present. Thus, we confirm that both the dynamical behaviours, namely strange nonchaos and chaos, can be efficient tools to construct computer architecture.

Different transitions of bursting and mixed-mode oscillations in Liénard system

In this paper, we present the emergence of bursting oscillations, mixed-mode oscillations and transient chaos in the forced Liénard system. The system exhibits intriguing dynamical transitions such as from chaotic bursting oscillations to spiking trains via periodic bursting oscillations, period doubling mixed-mode oscillations route to chaos, and intermittent mixed-mode oscillations on variation of the nonlinear damping coefficient as the control parameter. In addition, the system transits from spiking trains to chaotic bursting oscillations through successive period adding sequences of bursting oscillations upon variation of the strength of nonlinearity. Besides the system exhibits transient chaos also. We believe that it is for the first time that these multiplicity of dynamical behaviors and transitions are reported in a single model. We validate our numerical investigations with the help of experimental observations using a simple nonlinear electronic circuit implementation of the model.

Nonlinear effect of forced harmonic oscillator subject to sliding friction and simulation by a simple nonlinear circuit

We study the nonlinear behaviors of mass-spring systems damped by dry friction using simulation by a nonlinear LC circuit damped by anti-parallel diodes. We show that the differential equation for the electric oscillator is equivalent to that of the mechanical system when a piecewise linear model is used to simplify the diodes' I–V curve. We derive series solutions to the differential equation under weak nonlinear approximation which can describe the resonant response as well as amplitudes of superharmonic components. The experimental results are consistent with the series solutions. We also present the phenomenon of hysteresis. A theoretical analysis along with numerical simulations is conducted to explore the stick-slip boundary. The correspondence between the mechanical and electric oscillators makes it easy to demonstrate the behaviors of this nonlinear oscillator on a digital oscilloscope. It can be used to extend the linear RLC experiment at the undergraduate level.

Wide-range precision RF peak detectors

A significant increase in the accuracy of a conventional bipolar monolithic radio-frequency bipolar peak detector is robustly achieved over a wide amplitude range using a simple nonlinear loading circuit. The loading circuit provides two voltage-controlled auxiliary currents which cancel the errors by altering the original DC currents flowing through the core transistors, according to the level of the detected voltage. Expressions are derived for the auxiliary currents for perfect error cancellation; a good approximation is provided by a simple cross-coupled transistor pair, but more complex circuit structures can be employed for enhanced range and/or robustness. Because the loading circuit operates essentially at DC, the high-frequency nature of the original peak detector is preserved. The presented concept and its various circuit implementations are discussed and validated in simulation, and are expected to work in a large variety of modern bipolar or BiCMOS integrated-circuit technologies, where parasitics-free, simple exponential-law models accurately represent the operation of the transistors up to very high frequencies.

Generalized analytical solutions and experimental confirmation of complete synchronization in a class of mutually coupled simple nonlinear electronic circuits

A novel generalized analytical solution for the normalized state equations of a class of mutually coupled chaotic systems is presented. To the best of our knowledge, for the first time synchronization dynamics of mutually coupled chaotic systems is studied analytically. The coupled dynamics obtained through the analytical solutions has been validated by numerical simulation results. Furthermore, we provide a suitable condition for the occurrence of synchronization in mutually coupled, second-order, non-autonomous chaotic systems through the analysis of the difference system on the stability of fixed points. The bifurcation of the eigenvalues of the difference system as a function of the coupling parameter in each of the piecewise-linear regions, revealing the existence of stable synchronized states, is presented. The stability of synchronized states in each coupled system discussed in this article is studied using the Master Stability Function. Finally, the electronic circuit experimental results confirming the analytical and numerical results are presented.

A Locally Active Memristor and Its Application in a Chaotic Circuit

In this brief, we propose a novel locally active memristor based on a voltage-controlled generic memristor model and use the analysis methods of standard nonlinear theory to analyze its characteristics and illustrate the concept of local activity via the dc v-i loci of memristor and non-volatile memory via the power-off plot of memristor. A chaotic attractor is observed with a simple nonlinear circuit that only includes three circuit elements in parallel: 1) a nonlinear locally active memristor; 2) a linear passive inductor; and 3) a linear passive capacitor. Then, we analyze the dynamical characteristics of the above circuit and show complex bifurcation behaviors.

A Simple Nonlinear Circuit Contains an Infinite Number of Functions

The complex dynamics of a simple nonlinear circuit contains an infinite number of functions. Specifically, this brief shows that the number of different functions that a nonlinear or chaotic circuit can implement exponentially increases as the circuit evolves in time, and this exponential increase is quantified with an exponent that is named the computing exponent. This brief argues that a simple nonlinear circuit that illustrates rich complex dynamics can embody infinitely many different functions, each of which can be dynamically selected. In practice, not all of these functions may be accessible due to factors such as noise or instability of the functions. However, these infinitely many functions do exist within the dynamics of the nonlinear circuit regardless of accessibility or inaccessibility of the functions in practice. This nonlinear-dynamics-based approach to computation opens the door for implementing extremely slim low-power circuits that are capable of performing many different types of functions.

Synchronization of strange non-chaotic attractors via unidirectional coupling of quasiperiodically-forced systems

In this paper, we present a numerical investigation on the robust synchronization phenomenon observed in a unidirectionally-coupled quasiperiodically-forced simple nonlinear electronic circuit system exhibiting strange non-chaotic attractors (SNAs) in its dynamics. The SNA obtained in the simple quasiperiodic system is characterized for its SNA behavior. Then, we studied the nature of the synchronized state in unidirectionally coupled SNAs by using the Master-Slave approach. The stability of the synchronized state is studied through the master stability functions (MSF) obtained for coupling different state variables of the drive and response system. The property of robust synchronization is analyzed for one type of coupling of the state variables through phase portraits, conditional lyapunov exponents and the Kaplan-Yorke dimension. The phenomenon of complete synchronization of SNAs via a unidirectional coupling scheme is reported for the first time.

Master stability functions for a class of coupled simple nonlinear electronic circuits

The stability of the synchronized state of coupled, identical, simple, nonlinear electronic circuits is analyzed using the Master Stabilty Functions of the variational equations. The Master Stability Functions for different types of coupling schemes of the circuit variables are arrived at, and the stability of the synchronized state is observed using the simulation results.

Implementation of fractional slot concentrated winding technique to large salient-pole synchronous generators & development with permanent magnets

This paper presents an implementation of the fractional slot concentrated winding (FSCW) technique into large salient-pole synchronous generators (SPSGs) and the development of the generators by using additional permanent magnets (PMs). Compared to conventional synchronous generators, the FSCW technique makes it possible to achieve lighter, cheaper and higher-efficiency generators with a simpler structure. To overcome saturation of the rotor pole-body problem well known in the SPSGs, the designed FSCW SPSG has been developed by inserting PMs between the rotor-pole shoes. The insertion of PMs between rotor poles increases the generator power output while reducing magnetic saturation in the rotor pole body. Using these features of the PMs, more compact generators can be designed while keeping the same output parameters. In order to achieve these objectives, the FSCW and PM-FSCW SPSGs are designed by using finite element simulation software (MAXWELL®). Furthermore, designed machines are compared with conventional SPSGs in order to realize the degree of the improvement. The machines are modeled not only by using MAXWELL® but also by using a simple nonlinear magnetic circuit to describe the effects of PMs. Simple theoretical expressions and detailed comparisons of performance characteristics of the designed machines are presented.

IMPLEMENTATION OF A CHUA CIRCUIT TO DEMONSTRATE BIFURCATIONS AND STRANGE ATTRACTORS IN A CLASS

This paper describes the design and implementation of a Chua double-scroll circuit to demonstrate chaos in dynamical systems to students in a graduate course in order to enhance their visualization and understanding of strange attractor and Feigenbaum bifurcation trees.Teaching dynamical systems (i.e., nonlinear systems that can exhibit chaos) is often considered difficult because of the mathematical modeling involved and the inclusion of the fourth strange-attractor state, in addition to the traditional point stability, cyclic stability, and toroidal stability, as found in dynamic systems. A graduate course has been offered at the University of Manitoba for many years to provide both (i) a unified theory of fractal dimensions, together with many practical implementations of algorithms to compute the fractal dimensions, including the Rényi dimension spectrum that is required for characterization of the strange attractors using multifractal analysis.Leon Chua developed a simple nonlinear circuit capable of producing a rich collection of dynamic phenomena, ranging from fixed points to cycle points, standard bifurcations (period doubling), other standard routes to chaos, and chaos itself. The reason for selecting this specific circuit as a class demonstration tool is threefold: (i) the circuit has an analytical model and can be simulated, (ii) the circuit is implementable using available commercial off-the-shelf components, and (iii) the signals in the circuit can be acquired without affecting and altering its operation significantly.This paper describes the architecture, implementation, verification, and testing of the Chua system, as well as an analysis of the data obtained during the current phase of the development. Although there are many possible implementations of Chua’s circuit, our implementation has several innovative design features to make it more applicable to enhance students’ learning in the classroom.

Modeling of Salient-Pole Synchronous Machines Assisted by Permanent Magnets

In this paper, we investigate the mechanism of output increase by inserting additional permanent magnets in salient-pole synchronous machines. The machines are modeled not only by finite element methods but also by a simple nonlinear magnetic circuit in order to express the assist effects of permanent magnets by simple theoretical expressions. The no-load saturation curves obtained by the finite element method and the magnetic circuit are compared with those of measurements in order to confirm the validities. It is clarified that the accuracy of the proposed magnetic circuit is nearly identical to that of the finite element method. Furthermore, the mechanism of the output increase by the permanent magnets is revealed by the theoretical expressions derived from the magnetic circuit equations.

High‐order sensitivity invariants

AbstractThe aim of the research reported in this paper is to extend the notion of invariant sensitivity sum, widely known for electrical networks, from first‐order sensitivities to high‐order sensitivities. The results are high‐order invariant sums of sensitivities of the first and the second kind, formulated for nonlinear lumped circuit, which consists of one‐port and two‐ports only. One‐ports are generalized resistances, capacitances, inductances, voltages, and current sources, whereas two‐ports are nonlinear generalizations of four types of controlled sources. It is observed that the invariant sums actually found for nonlinear lumped networks are generalizations of sums given earlier by authors for linear lumped networks. The article is illustrated by numerical sensitivity analysis of simple linear and nonlinear circuits. Copyright © 2010 John Wiley & Sons, Ltd.

Stability Analysis of Nonlinear Circuits Driven With Modulated Signals

Existing methods for the large-signal stability analysis of microwave circuits assume a periodic excitation, usually consisting of a single tone. However, practical circuits such as power amplifiers will generally be driven by modulated signals or multiple input tones. This paper presents a procedure for the stability analysis of linear time-varying systems, applicable to circuits excited with nonperiodic input signals. The procedure is based on the determination of the time-varying poles associated to an input-output representation of the system. Under modulated signals, this representation is obtained by linearizing the envelope-transient system that rules the circuit behavior. The formulation and methodology have been applied to a simple nonlinear circuit with arbitrary excitation and a practical power-combined amplifier at 3 GHz with very good agreement in comparison with independent simulations and experimental results.

ENRICHED DYNAMICS OF A SIMPLE NONLINEAR NONAUTONOMOUS PARALLEL LCR CIRCUIT

We present the results of a very simple nonlinear nonautonomous parallel LCR circuit which exhibits a rich variety of bifurcations and chaos. This simple experimental chaotic oscillator is made up of a few passive components and a uniquely designed nonlinear element comprising of a junction diode and Op-amp based negative conductor. The system exhibits a rich dynamics such as chaos via torus breakdown, period-doubling, period three-doubling, period-adding, Farey sequences, etc., and the results of the investigation are vividly presented along with experimental, numerical and analytical evidences. The laboratory experimental outcomes are in agreement with the numerical and analytical results.

A Local Nonlinear Model for the Approximation and Identification of a Class of Systems

The special form of the Laplace-domain Volterra kernels for linear-analytic systems is exploited to obtain an approximate model that obeys an appealingly simple feedforward block structure. It comprises a composition of the linearization and the multivariate nonlinear function of the original system. The model does not involve a truncation in the power series expansion nor in the memory depths and offers an economic parameterization. It is shown to be linearly identifiable in one step if <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">priori</i> information about the linearized dynamics is provided. We present simulation results for a simple nonlinear circuit showing the validity of the model.