Chua's Diode Research Articles

A method to design a fast chaotic oscillator using CCTA

This article introduces a novel method for designing a fast chaotic oscillator using a CCTA (Current Conveyor Transconductance Amplifier) based on Chua's circuit. The proposed method uses innovative configurations and advanced simulation techniques to overcome challenges in high-speed operation, nonlinear dynamics, and Analog Building Block (ABB) selection. The design begins with nonlinear negative resistance, essential for Chua's diode characteristics, including two negative resistances, NR1 and NR2. The circuit integrates one CCTA block, two grounded capacitors, two fixed resistors, one inductor, and one potentiometer. It is simulated using PSPICE with IC (Integrated Circuit) macro-models and 180nm CMOS (Complementary Metal Oxide Semiconductor) technology. Various chaotic waveforms and attractors are produced, validating the theoretical and mathematical predictions. By varying the resistance values (1450Ω, 1650Ω, 1800Ω, 1950Ω), the circuit exhibits different chaotic behaviors, such as large limit cycles, double-scroll attractors, Rossler-type attractors, and I-periodic attractors. FFT (Fast Fourier Transform) analysis confirms the highest dominant operating frequency of 37.5MHz. A Monte Carlo simulation with 100 runs shows maximum voltage variations in the chaotic waveforms of 5.21 % and 4.61 % across the capacitors, demonstrating robustness and reliability. This design offers significant advancements in implementing high-frequency chaotic oscillators, with potential applications in various fields requiring chaotic signal generation.•A novel design of Chua's diode and Chua's chaotic oscillator using only one CCTA block is presented in this paper.•The proposed chaotic oscillator achieves the highest operating frequency of 37.5MHz.•The proposed circuit is simulated using commercially available ICs (MAX435 and AD844) and CMOS 180nm technology in PSPICE to confirm its workability.

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

After the discovery in early 1960s by E. Lorenz and Y. Ueda of the first example of a chaotic attractor in numerical simulation of a real physical process, a new scientific direction of analysis of chaotic behavior in dynamical systems arose. Despite the key role of this first discovery, later on a number of works have appeared supposing that chaotic attractors of the considered dynamical models are rather artificial, computer-induced objects, i.e., they are generated not due to the physical nature of the process, but only by errors arising from the application of approximate numerical methods and finite-precision computations. Further justification for the possibility of a real existence of chaos in the study of a physical system developed in two directions. Within the first direction, effective analytic-numerical methods were invented providing the so-called computer-assisted proof of the existence of a chaotic attractor. In the framework of the second direction, attempts were made to detect chaotic behavior directly in a physical experiment, by designing a proper experimental setup. The first remarkable result in this direction is the experiment of L. Chua, in which he designed a simple RLC circuit (Chua circuit) containing a nonlinear element (Chua diode), and managed to demonstrate the real evidence of chaotic behavior in this circuit on the screen of oscilloscope. The mathematical model of the Chua circuit (further, Chua system) is also known to be the first example of a system in which the existence of a chaotic hidden attractor was discovered and the bifurcation scenario of its birth was described. Despite the nontriviality of this discovery and cogency of the procedure for hidden attractor localization, the question of detecting this type of attractor in a physical experiment remained open. This article aims to give an exhaustive answer to this question, demonstrating both a detailed formulation of a radiophysical experiment on the localization of a hidden attractor in the Chua circuit, as well as a thorough description of the relationship between a physical experiment, mathematical modeling, and computer simulation.

CHUA’S CIRCUITS DYNAMIC WITH NON-IDEAL COILS AND HYSTERESIS

A complete dynamic study (theoretical, experimental and numerical) of a chaotic Chua-type circuit was carried out using a non-ideal coil with non-negligible values of its internal resistance. This set up means that the circuit does not show chaotic behavior. In order to observe chaoticity, a modification that introduces hysteresis to the Chua diode is proposed. Using spaces of up to three control parameters it is shown how this modification adapts to a wide range of non-ideal inductors presenting more extensive chaotic regions than the classical Chua circuit. These simulations are tested with a physical circuit and demonstrate how the modifications result in a simpler experimental design of the inductor. By simplifying the experimental design the difficulties caused by the electronic component is eliminated and results in the absence of Rössler type attractors and the asymmetry of the double-scroll attractors.

Resistorless Chua's diode implementation for chaotic oscillation employing single EXCCCII

A passive resistor-free Chua’s Diode with smooth nonlinearity is introduced using a single Extra X Current Controlled Current Conveyors (EX-CCCII). The proposed Chua’s Diode can generate all fundamental Chua’s chaotic responses like Rossler type attractor, double scroll attractor, and large limit cycle. Chua’s chaotic response can be easily observed in two ways: (i) by tuning the bridging resistor between two capacitors present in a Chua's circuit and (ii) by adjusting the bias current of EX-CCCII present in the proposed Chua’s diode. The proposed Chua’s model is verified with three possible combinations: Chua’s oscillation with a passive inductor, tunable Chua’s oscillation by varying the bias current present in EX-CCCII based negative resistance, and inductor less Chua’s circuit using active inductor based on EX-CCCII. Moreover, the effect of non-idealities present in EXCCCII on negative resistance and dynamical investigation of theproposed Chua’s circuit is described for better understanding. The functionality of the proposed Chua’s circuit is well verified through PSPICE simulation using 0.25 μm TSMC CMOS technology parameters and Current Feedback operational amplifier (CFOA) as IC AD844 based EX-CCCII. In addition to computer simulation, an experimental investigation is also performed by CFOA as IC AD844 to realize EX-CCCII. Finally, a comparison between the existing Chua’s circuit and the proposed one is illustrated for a fair comparison.

DC-offset-induced hidden and asymmetric dynamics in Memristive Chua's circuit

Hidden attractors have been discovered in classical Chua's circuits with special Chua's diode nonlinearities. But designing such physical Chua's circuits is a challenge due to the distinct slopes of Chua's diodes. The analog-circuit-based measurements of hidden dynamical behaviors are also difficult since their attraction basins are relatively small and independent of any equilibrium points. In this paper, a DC-offset method is proposed for obtaining hidden dynamics from preexisting nonlinear circuits. An improved memristive Chua's circuit with hidden dynamical behaviors is constructed by inserting a DC voltage source into the inductor branch therein. The DC-offset-induced hidden and asymmetric dynamical behaviors are revealed via equilibrium point analyses, numerical simulations, and experimental measurements. The coexistence of hidden attractors is demonstrated when the modified memristive Chua's circuit possesses at least one unstable equilibrium point or only one stable equilibrium point. Furthermore, a reconstituted model is formulated using the incremental integral transformation method to facilitate the measurements of hidden and coexisting attractors having relatively small attraction basins. With this model, the self-excited and hidden properties of the coexisting attractors are maintained, while their attraction basins are transferred to the neighborhood of the easily circuit-implemented origin point. Thus, the existence of hidden and coexisting attractors is readily verified via hardware circuit measurements.

Chua’s Circuit Simulation Experiment Based on Saturation Function

This paper designs a Chua's diode circuit based on saturation function. It uses Matlab/Simulink to model Chua's circuit, and simulates and analyzes the dynamic behavior of chaos, the model simply generates efficient chaotic signals, and intuitively displays processes of the chaotic attractor, chaotic synchronization, period doubling and the road to chaos through the virtual oscilloscope, which is conducive to chaotic beginners' understanding of the basic characteristics and application of chaos. The electrical and mathematical analysis of the instrument is carried out by simulation, to explore the functions of each parameter, and helps beginners to understand its working principle more quickly and conduct experimental operations. It provides theoretical support for the improvement and optimization of Chua's circuit.

Active topolectrical circuits

The transfer of topological concepts from the quantum world to classical mechanical and electronic systems has opened fundamentally different approaches to protected information transmission and wave guidance. A particularly promising emergent technology is based on recently discovered topolectrical circuits that achieve robust electric signal transduction by mimicking edge currents in quantum Hall systems. In parallel, modern active matter research has shown how autonomous units driven by internal energy reservoirs can spontaneously self-organize into collective coherent dynamics. Here, we unify key ideas from these two previously disparate fields to develop design principles for active topolectrical circuits (ATCs) that can self-excite topologically protected global signal patterns. Realizing autonomous active units through nonlinear Chua diode circuits, we theoretically predict and experimentally confirm the emergence of self-organized protected edge oscillations in one- and two-dimensional ATCs. The close agreement between theory, simulations, and experiments implies that nonlinear ATCs provide a robust and versatile platform for developing high-dimensional autonomous electrical circuits with topologically protected functionalities.

Chua's diode and strange attractor: a three‐layer hardware–software co‐design for medical image confidentiality

Medical image security relies upon the arrival of improved techniques for enhancing the confusion as well as diffusion processes in the pixel level manipulations. Non-linear circuits constructed with hardware components can be employed for hardware–software co-design in medical image encryption. The proposed approach has adopted Chua diode circuit for the generation of 256 × 256 random synthetic image with the entropy of 7.9972. This synthetic image has been utilised to scramble the DICOM image for the second level of diffusion as the cellular automata provides the first level of scrambling and diffusion. The average correlation coefficients of encrypted pixels are 0.00204, −0.00298 and −0.00054 in three directions. The encrypted pixels pass the NIST SP 800-22 test suite and offer resistance to statistical, differential and chosen plain text attacks.

Estimating the dynamics of systems with noisy boundaries

In a smooth dynamical system the characteristics of a given reference trajectory can be determined, to lowest order, by examining the linearised system about the reference trajectory. In other words, we can approximate the deviations of trajectories after a given time, with starting points in a neighbourhood of the reference trajectory, by multiplying the initial deviations by the corresponding fundamental matrix solution.This form of analysis cannot be used directly in nonsmooth systems as the vector field is either not everywhere differentiable or the flow function is not continuous. To account for this, one can derive the zero-time discontinuity mapping (ZDM) associated with the discontinuity boundary. The Jacobian of this mapping is known as the saltation matrix and its properties can tell us how the crossing of the discontinuity boundary affects the deviations of trajectories from a reference trajectory. In particular, this matrix can be composed with the fundamental matrix solutions of the individual flows on either side of the discontinuity boundary in order to determine the overall fundamental matrix solution of a trajectory that crosses the boundary.In this paper we derive a saltation matrix for a piecewise-smooth dynamical system in which the position of the discontinuity boundary oscillates according to a mean-reverting stochastic process. The derived saltation matrix contains the entire effect of both the discontinuity and the uncertainty introduced into the system by the noisy boundary, and is composable with the deterministic fundamental matrix solutions of the individual flows to give the overall fundamental matrix solution of a crossing trajectory.We also present some simple examples of piecewise-smooth systems with stochastically varying boundaries, analysed using the derived noisy saltation matrix. In particular we focus on the analysis of a discontinuous variant of the Chua circuit. In this case we apply noise to the system’s discontinuity boundaries which are generated by the piecewise-linear nature of the voltage–current response of the Chua diode. We find that our method allows us to analyse the effects of boundary noise on periodic attractors close to bifurcation points. In particular we show that we can use the method to accurately predict the noise amplitudes required to destroy or merge periodic attractors.

Generating Multi-Scroll Chua’s Attractors via Simplified Piecewise-Linear Chua’s Diode

High implementation complexity of multi-scroll circuit is a bottleneck problem in real chaos-based communication. Especially, in multi-scroll Chua's circuit, the simplified implementation of piecewise-linear resistors with multiple segments is difficult due to their intricate irregular breakpoints and slopes. To solve the challenge, this paper presents a systematic scheme for synthesizing a Chua's diode with multi-segment piecewise-linearity, which is achieved by cascading even-numbered passive nonlinear resistors with odd-numbered ones via a negative impedance converter. The traditional voltage mode op-amps are used to implement nonlinear resistors. As no extra DC bias voltage is employed, the scheme can be implemented by much simpler circuits. The voltage-current characteristics of the obtained Chua's diode are analyzed theoretically and verified by numerical simulations. Using the Chua's diode and a second-order active Sallen-Key high-pass filter, a new inductor-free Chua's circuit is then constructed to generate multi-scroll chaotic attractors. Different number of scrolls can be generated by changing the number of passive nonlinear resistor cells or adjusting two coupling parameters. Besides, the system can be scaled by using different power supplies, satisfying the low-voltage low-power requirement of integrated circuit design. The circuit simulations and hardware experiments both confirmed the feasibility of the designed system.

A physical SBT-memristor-based Chua's circuit and its complex dynamics.

SBT memristor is a physical memristor built on nanometer film, described by a deterministic flux-controlled mathematical model. A new physical SBT-memristor-based Chua's circuit is presented in this paper, using the memristor to replace the Chua diode to connect with a negative conductance in parallel. The circuit can be well modeled by a fourth-order nonlinear system in the voltage-current domain and a third-order nonlinear system in the flux-charge domain, respectively. The dependences of the circuit stability on both the initial state value of the SBT memristor and the linear resistance in the circuit are analyzed theoretically, verified by numerical simulations, which shows that the circuit system has complex dynamics with multiple behaviors, including sink, periodic cycle, limit cycle, chaos, and some complicated transient dynamical behaviors. The new findings shed light on future research on the physical realization of the SBT-memristor-based Chua's circuit and other similar devices.

Bi-stability in a fifth-order voltage-controlled memristor-based Chua's chaotic circuit

Generally, the occurrence of multiple attractors indicates that the multi-stability existing in a nonlinear dynamical system and the long-time motion behavior are essentially different, depending on which basin of attraction the initial condition belongs to. Up to now, due to the emergence of multi-stability, some particular memristor-based nonlinear circuits whose dynamical behaviors are extremely related to memristor initial conditions or other initial conditions have attracted considerable attention. By replacing linear or nonlinear resistors with memristor emulators in some already-existing oscillating circuits or introducing memristor emulators with different nonlinearities into these oscillating circuits, various memristor-based nonlinear dynamical circuits have been constructed and broadly investigated. Motivated by these considerations, we present a novel fifth-order voltage-controlled memristor-based Chua's chaotic circuit in this paper, from which a wonderful phenomenon of bi-stability is well demonstrated by numerical simulations and PSIM circuit simulations. Note that the bi-stability is a special kind of multi-stability, which is rarely reported in the memristor-based chaotic circuits.<br/>The proposed memristor-based Chua's chaotic circuit is constructed by inserting an inductor into the coupled resistor branch in series and substituting the Chua's diode with a voltage-controlled memristor in the classical Chua's circuit. Five-dimensional system model is established, of which the equilibrium point and its stability are investigated. Theoretical derivation results indicate that the proposed circuit owns one or three equilibrium points related to the circuit parameters. Especially, unlike the newly reported memristive circuit with bi-stability, the proposed memristor-based Chua's chaotic circuit has only one zero equilibrium point under the given parameters, but it can generate coexistent chaotic and periodic behaviors, and the bi-stability occurs in such a memristive Chua's circuit. By theoretical analyses, numerical simulations and PSIM circuit simulations, the bi-stability phenomenon of coexistent chaotic attractors and periodic limit cycles with different initial conditions and their formation mechanism are revealed and expounded. Besides, with the dimensionless system equations, the corresponding initial condition-dependent dynamical behaviors are further numerically explored through bifurcation diagram, Lyapunov exponents, phased portraits and attraction basin. Numerical simulation results demonstrate that the proposed memristive Chua's system can generate bi-stability under different initial conditions. The PSIM circuit simulations and numerical simulations are consistent well with each other, which perfectly verifies the theoretical analyses.

Sallen–Key low‐pass filter‐based inductor‐free simplified Chua's circuit

This present study reports a new type of inductor-free simplified Chua's circuit only using two-op-amps-based realisation, which is implemented by cascading a Sallen–Key low-pass filter with a simplified Chua's diode. With mathematical model-based numerical simulations and hardware circuit-based experimental measurements, it is demonstrated that the reported circuit exhibits the similar dynamical behaviours of the classical Chua's circuit such as point attractor, limit cycles, period doubling route, chaotic spiral attractors, double-scroll chaotic attractor, boundary crisis, coexisting bifurcation modes, and so on. Furthermore, the proposed circuit enriches the realisation forms of Chua's circuit family.

Multistability in Chua's circuit with two stable node-foci.

Only using one-stage op-amp based negative impedance converter realization, a simplified Chua's diode with positive outer segment slope is introduced, based on which an improved Chua's circuit realization with more simpler circuit structure is designed. The improved Chua's circuit has identical mathematical model but completely different nonlinearity to the classical Chua's circuit, from which multiple attractors including coexisting point attractors, limit cycle, double-scroll chaotic attractor, or coexisting chaotic spiral attractors are numerically simulated and experimentally captured. Furthermore, with dimensionless Chua's equations, the dynamical properties of the Chua's system are studied including equilibrium and stability, phase portrait, bifurcation diagram, Lyapunov exponent spectrum, and attraction basin. The results indicate that the system has two symmetric stable nonzero node-foci in global adjusting parameter regions and exhibits the unusual and striking dynamical behavior of multiple attractors with multistability.

Hidden attractors in a practical Chua's circuit based on a modified Chua's diode

Hidden attractors were reported to find in a normalised Chua's equation. To find the hidden attractors in a practical Chua's circuit, a modified Chua's diode is presented and its equivalent realisation circuit is designed. On the basis of proposed modified Chua's diode, a new Chua's circuit is rebuilt and hidden attractors under special initial conditions are captured by numerical simulations and circuit simulations.

Hidden dynamics and multi‐stability in an improved third‐order Chua's circuit

By utilising an active diode pair to substitute the classical Chua's diode, an improved third-order Chua's circuit is proposed in this study. This circuit has an unstable zero saddle point and two stable non-zero saddle-foci in a local parameter region, from which hidden dynamics and multi-stability are discovered. The parameter region of hidden dynamics is figured out and hidden attractors are captured numerically and experimentally. Furthermore, multi-stability in the proposed circuit are also exhibited by numerical and circuit simulations.

CCII+ Based on QFGMOS for Implementing Chua s Chaotic Oscillator

A new design of a positive-type second-generation current conveyor (CCII+) based on quasi-floating gate MOSFETs (QFGMOS) is introduced. The proposed CCII+ is compared with others showing bandwidth advantages, e.g. it reaches 257MHz and 523MHz for the voltage tracking (between ports Y to X) and current tracking (between ports X to Z), respectively, while the parasitic resistance at port X is low as Rx=5.1794Ω. This new CCII+ is used to implement Chua's diode that is modeled by a piecewise-linear (PWL) function consisting of two negative slopes and two break points. That way, Chua's chaotic oscillator is designed using standard CMOS technology of 0.5μm. SPICE simulation results are provided to show the suitability of the proposed CCII+ based on QFGMOS for generating chaotic phenomena through implementing Chua's circuit.

Self-Excited and Hidden Attractors Found Simultaneously in a Modified Chua's Circuit

By replacing the Chua's diode in Chua's circuit with a first-order hybrid diode circuit, a fourth-order modified Chua's circuit is presented. The circuit has an unstable zero saddle point and two nonzero saddle-foci. By Routh–Hurwitz criterion, it is found that in a narrow parameter region, the two nonzero saddle-foci have a transition from unstable to stable saddle-foci, leading to generations of self-excited and hidden attractors in the modified Chua's circuit simultaneously, which have not been previously reported. Complex dynamical behaviors are investigated both numerically and experimentally. The results indicate that the proposed circuit exhibits complicated nonlinear phenomena including self-excited attractors, coexisting self-excited attractors, hidden attractors, and coexisting hidden attractors.

Research on a chaotic circuit based on an active TiO 2 memristor

The memristor is the fourth fundamental circuit element besides the resistor, inductor and capacitor. As a two-terminal nonlinear resistor, the memristor has a broad application prospect. In this paper, a negative memconductance expression of a flux-controlled memristor is derived from the relationship between voltage and current for the Hewlett-Packard memristor. By replacing Chua's diode with the active flux-controlled TiO 2 memristor, a chaotic circuit is obtained. By means of the conventional dynamic analysis method, dynamic behaviors of the chaotic circuit are investigated. Software simulation and theoretical analysis all indicate that this active memristor-based chaotic circuit has more complex behaviors. Furthermore, the integrated circuit experiment on the digital signal processor chip of this circuit was also realized.

NONSMOOTH BIFURCATIONS, TRANSIENT HYPERCHAOS AND HYPERCHAOTIC BEATS IN A MEMRISTIVE MURALI–LAKSHMANAN–CHUA CIRCUIT

In this paper, a memristive Murali–Lakshmanan–Chua (MLC) circuit is built by replacing the nonlinear element of an ordinary MLC circuit, namely the Chua's diode, with a three-segment piecewise-linear active flux controlled memristor. The bistability nature of the memristor introduces two discontinuity boundaries or switching manifolds in the circuit topology. As a result, the circuit becomes a piecewise-smooth system of second order. Grazing bifurcations, which are essentially a form of discontinuity-induced nonsmooth bifurcations, occur at these boundaries and govern the dynamics of the circuit. While the interaction of the memristor-aided self oscillations of the circuit and the external sinusoidal forcing result in the phenomenon of beats occurring in the circuit, grazing bifurcations endow them with chaotic and hyperchaotic nature. In addition, the circuit admits a codimension-5 bifurcation and transient hyperchaos. Grazing bifurcations as well as other behaviors have been analyzed numerically using time series plots, phase portraits, bifurcation diagram, power spectra and Lyapunov spectrum, as well as the recent 0–1 K test for chaos, obtained after constructing a proper Zero Time Discontinuity Map (ZDM) and Poincaré Discontinuity Map (PDM) analytically. Multisim simulations using a model of piecewise linear memristor have also been used to confirm some of the behaviors.