Chua's Circuit Research Articles

Reliable Finite‐Time Control for Nonlinear Chaotic Semi‐Markov Jump Systems With Incomplete Transition Rates

ABSTRACTThis article studies the finite‐time control problem for a class of nonlinear chaotic semi‐Markov jump systems with incomplete transition rates described by the T‐S fuzzy model approach. As a means to depict the dynamical properties pertaining to the examined system, parametric uncertainties, faults, external disturbances and input saturation are taken into consideration. The foremost objective of this study is to come up with a composite control mechanism that effectively rejects and attenuates the repercussions of faults and disturbances. In particular, we primarily built a disturbance observer in order to obtain a precise estimation of the external disturbances. After which, the fault diagnosis observer is designated to effectively estimate the faults. Specifically, a composite reliable control mechanism is developed by fusing the output of the constructed observers with the mode‐dependent fuzzy‐rule based state feedback controller. By employing suitable Lyapunov functions in conjunction with the linear matrix inequality technique, a set of mode‐dependent conditions is derived to ensure the finite‐time stochastic boundedness of the underlying closed‐loop and estimation error systems. Following this, the anticipated controller and observer gain matrices are elicited by resolving the established linear matrix inequalities. Thereafter, intending to ascertain the efficacy and usefulness of accrued theoretical findings, simulation results performed on Chua's circuit system is endowed.

A Switching Memory-Based Event-Trigger Scheme for Synchronization of Lur'e Systems With Actuator Saturation: A Hybrid Lyapunov Method.

This article is concerned with the event-triggered synchronization of Lur'e systems subject to actuator saturation. Aiming at reducing control costs, a switching-memory-based event-trigger (SMBET) scheme, which allows a switching between the sleeping interval and the memory-based event-trigger (MBET) interval, is first presented. In consideration of the characteristics of SMBET, a piecewise-defined but continuous looped-functional is newly constructed, under which the requirement of positive definiteness and symmetry on some Lyapunov matrices is dropped within the sleeping interval. Then, a hybrid Lyapunov method (HLM), which bridges the gap between the continuous-time Lyapunov theory (CTLT) and the discrete-time Lyapunov theory (DTLT), is used to make the local stability analysis of the closed-loop system. Meanwhile, using a combination of inequality estimation techniques and the generalized sector condition, two sufficient local synchronization criteria and a codesign algorithm for the controller gain and triggering matrix are developed. Furthermore, two optimization strategies are, respectively, put forward to enlarge the estimated domain of attraction (DoA) and the allowable upper bound of sleeping intervals on the premise of ensuring local synchronization. Finally, a three-neuron neural network and the classical Chua's circuit are used to carry out some comparison analyses and to display the advantages of the designed SMBET strategy and the constructed HLM, respectively. Also, an application to image encryption is provided to substantiate the feasibility of the obtained local synchronization results.

Prescribed‐time fault‐tolerant synchronization control of complex dynamical networks using time‐varying protocols

AbstractThis research delves into the intricacies of the prescribed‐time synchronization control problem of a class of complex dynamical systems, all while factoring in the presence of time‐varying actuator faults. An innovative time‐varying protocol is initially established for the case of additive actuator faults, hinging on the solution to a particular set of parametric Lyapunov equations. Through the astute exploitation of the scalarization properties intrinsic to the solution to the parametric Lyapunov equation, the time‐varying parameter within the protocol is effectively ascertained. Furthermore, to mitigate the impact stemming from multiplicative actuator fault(s), an alternative protocol is also introduced. Through rigorous analysis rooted in the principles of Lyapunov, it is demonstrated that the synchronization errors in both scenarios remain bounded, ultimately converging to zero within a predetermined time frame. The developed protocols are employed in coupled Chua's circuit control systems with faulty actuators, and simulation results validate the efficacy of the proposed approaches.

Input–Output Finite-Time Bipartite Synchronization for Multiweighted Complex Dynamical Networks Under Dynamic Hybrid Triggering Mechanism

Abstract The problem of input–output finite-time (IO-FT) bipartite synchronization for a class of nonlinear multiweighted complex dynamical networks (CDNs) in the presence of multiple coupling delays, external disturbances, and deception attacks is explored in this study. To be precise, the limited communication resources have been mitigated with the aid of undertaken hybrid triggered strategy, which reduces the unwanted network transmission and simultaneously improves the system's performance. Specifically, in the hybrid-trigger scheme, a Bernoulli distributed random variable has been employed to switch between the two communication channels. Moreover, the event-triggered scheme involving the dynamic trigger conditions is incorporated in the sensor-to-controller, which reduces the number of triggers compared to static event-triggered strategy. Further, the adequate conditions are derived in terms of linear matrix inequalities by constructing a Lyapunov–Krasovskii functional candidate. In light of this, the required parameters involved in triggering and the gain matrix are acquired by solving the developed linear matrix inequalities. Eventually, the reliability of the developed approach is verified via the illustration of two numerical examples, including the Chua's circuit with simulation verifications.

FPGA-Based Numerical Simulation of the Chaotic Synchronization of Chua Circuits

The objective of this work was to design and implement a system based on reconfigurable hardware as a study tool for the synchronization of chaotic circuits. Mathematical models were established for one circuit, two synchronized, and multiple synchronized Chua circuits. An ordinary differential equation solver was developed applying Euler’s method using the Verilog hardware description language and synthesized on a Spartan 3E FPGA (Field-Programmable Gate Array) equipped with a 32-bit RISC processor, 64 MB of DDR SDRAM, and 4 Mb of PROM. With a step size of 0.005 and a total of 10,000 iterations, the state equations for one and three Chua circuits were solved at a time of 0.2 ms and a frequency of 50 Mhz. The logical resources used by the system did not exceed 4%. To verify the operation, a numerical simulation was carried out using the Octave V9.1.0 calculation software on an Intel(R) Core i7-9750H CPU 2.59 GHz computer, obtaining the same results but in a time of 493 ms and 3.177 s for one and three circuits, respectively.

Snapback Repellers, Computational Chaos and Extreme Multistability in Discrete-Time Memristor Murali–Lakshmanan–Chua Circuit

Discretization schemes such as Euler method and Runge–Kutta techniques are extensively used to find approximate solutions of Continuous-Time (CT) dynamical system. While the approximation is good for small discretization step sizes, as pointed out by Lorenz, when the step size increases, computational chaos and computational instability are frequently observed, the former phenomenon being a precursor to the latter. By computational instability, it is meant that there is a blow up of trajectories for the Discrete-Time (DT) system. Computational chaos instead means that for certain step sizes, the DT system displays chaos while the CT counterpart is not chaotic. This paper studies the dynamics of a class of second-order maps obtained via the discretization of a Memristor Murali–Lakshmanan–Chua Circuit (MMLCC). The discretization, which is based on the DT Flux–Charge Analysis Method (FCAM), guarantees that the first integrals of a CT-MMLCC are preserved exactly for the DT system. Hence the dynamics of DT-MMLCC evolves on invariant manifolds and it is characterized by the coexistence of infinitely many different attractors (extreme multistability). The paper uses analytic techniques introduced by Marotto, based on the concept of snapback repellers and transverse homoclinic orbits, to study the chaotic behaviors of the maps. Regions of the parameter space are singled out where there exist snapback repellers for DT-MMLCC, thus implying that the maps display chaos in the Marotto and in the Li–Yorke sense. Since the corresponding CT-MMLCC does not display chaos, the observed chaos of DT-MMLCC is genuinely a consequence of the discretization scheme used in the paper, i.e. it can be actually considered as computational chaos. It is also verified that computational chaos is a precursor to computational instability of the DT-MMLCC maps. Finally, the paper analyzes the effect on chaos obtained by changing the invariant manifold where the dynamics evolves.

Jump resonance in the driven Chua's circuit to design frequency selective devices

Jump resonance in the driven Chua's circuit to design frequency selective devices

Involution symmetry quantification using recurrences.

Symmetries are ubiquitous in science, aiding theoretical comprehension by discerning patterns in mathematical models and natural phenomena. This work introduces a method for assessing the extent of symmetry within a time series. We explore both microscopic and macroscopic features extracted from a recurrence plot. By analyzing the statistics of small recurrence matrices, our approach delves into microscale dynamics, facilitating the identification of symmetric time series segments through diagonal macroscale structures on a recurrence plot. We validate our approach by successfully quantifying involution symmetries for three-dimensional dynamical models, specifically, order-2 rotational symmetry in the Lorenz '63 model, and inversion symmetry in the Chua circuit. Our quantifier also detects symmetry breaking in the modified Lorenz model for El Niño phenomenon. The method can be applied in a versatile manner, not only to three-dimensional trajectories but also to univariate time series. Symmetry quantification in time series is promising for enhancing dynamical system modeling and profiling.

Modeling and dynamics analysis of a novel fractional-order meminductive multi-stable chaotic circuit and its FPGA implementation

Integrating a meminductor into a Chua's circuit and extending it to fractional order can produce chaotic circuits with complex dynamical behaviour. First, a fractional-order meminductor model is developed and it's meminductive characteristic curve is investigated. Subsequently, the dynamical behaviour of the fractional-order meminductive chaotic circuit is analysed by means of phase diagrams, bifurcation diagrams, and Lyapunov exponential diagrams. The results show that the circuit generates a large number of coexisting attractors, has multistability, state transition phenomenon, achieves amplitude control and offset boosting control. Finally, circuit simulations were performed and FPGA experiments were conducted using the NI-Labview platform. The experimental results are in good agreement with the numerical simulation results, confirming the accuracy and feasibility of the circuit.

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.

Practical Fixed-Time Bipartite Synchronization of Uncertain Coupled Neural Networks Subject to Deception Attacks via Dual-Channel Event-Triggered Control.

This article investigates the practical fixed-time synchronization of uncertain coupled neural networks via dual-channel event-triggered control. Contrary to some previous studies, the bipartite synchronization of signed graphs representing cooperative and antagonistic interactions is studied. The communication channel is introduced into deception attacks, which are described by Bernoulli's stochastic variables. Based on the concept of two channels, event-triggered mechanisms are designed for sensor-to-controller and controller-to-actuator channels to reduce communication consumption and controller update consumption as much as possible. Lyapunov and comparison theories are used to derive synchronization criteria and explicit expression of settling time. An example of Chua's circuit system is presented to demonstrate the feasibility of the obtained theoretical results.

Physical implementation of cobalt ferrite memristor in Chua's circuit for chaotic encryption

Memory resistor, or memristor, has been realized as a discrete electronic device and has a perspective application in the field of cryptography. The physical implementation of the memristor in chaotic circuits has been scarcely explored. In this paper, a memristor is fabricated by spin-coating a cobalt ferrite precursor on a processed silicon and is then electro-sputtered with silver to act as the anode with the base silicon as the cathode. This fabrication process has a scalability potential in conjunction with integrated circuit fabrication techniques and complementary metal oxide semiconductor (CMOS) technologies. The fabricated cobalt ferrite memristor has shown a ratio between the on and off resistance of >1000 and has been implemented in a chaotic Chua's circuit, making it one of few physical implementations of a physical memristor in a physical circuit. The analysis and characterization of this circuit using bifurcation diagrams and Lyapunov exponent prove the chaotic behavior of a real Chua's circuit. This chaotic behavior can be useful in chaotic cryptography as nonperiodic oscillations can be leveraged to make sensitive information more difficult to interpret by bad actors.

Finite‐time stabilization control of nonlinear systems with full‐state constraints

AbstractThis paper studies the finite time stabilization control (FTSC) problem of nonlinear systems with unknown functions and full‐state constraints (FSCs). A lemma dealing with unknown functions is put forward so that the assumptions and the “explosion of terms” (EOT) of backstepping are avoided. Compared with approximation algorithms, this algorithm can make the system states converge to the origin in a finite time. On the other hand, a new logarithmic constraint function is utilized to solve the drawback of barrier Lyapunov functions (BLFs). Then, combined with finite time control (FTC), an FTSC algorithm with FSCs is proposed. Finally, simulations of Chua's circuit system are given to verify the effectiveness and superiority of this algorithm. Compared with the approximation algorithm, this algorithm can theoretically make the states of Chua's circuit system to the origin in a finite time.

To the Derivation of a Mathematical Model of a Chaotic System Using the Example of a Chua Circuit of Fractional Order

To the Derivation of a Mathematical Model of a Chaotic System Using the Example of a Chua Circuit of Fractional Order

Observer‐based hybrid triggered control design for cluster synchronization of nonlinear complex dynamical networks

SummaryThe issues of cluster synchronization and state estimation for a class of nonlinear complex dynamical networks are concurrently focused in this study. In precise, led by the design of complex dynamical networks, a proportional integral‐based observer is implemented for achieving robustness, by the virtue of which the states of examined networks are estimated. Thereupon, a proportional integral observer‐based hybrid triggered control scheme is proposed, wherein, the switch‐over between time‐ and event‐ triggered scheme is governed through a Bernoulli distributed random variable. Further, with the application of mixed and passivity performance technique, the external disturbances for the addressed complex dynamical networks are reduced. Based on the Lyapunov stability theory, the requisite sufficient conditions in respect of linear matrix inequalities are provided which ensures the discussed complex dynamical network's synchronization phenomenon. In a further, the presented criteria are exploited for the attainment of required controller, proportional and integral gain matrices. In conclusion, the applicability of proposed results and efficacy of the undertaken strategy are evidently proven with the illustration of two numerical examples with Chua's circuit and Goodwin oscillator model.

Reconstructing bifurcation diagrams of chaotic circuits with reservoir computing.

Model-free reconstruction of bifurcation diagrams of Chua's circuits using the technique of parameter-aware reservoir computing is investigated. We demonstrate that (1) reservoir computer can be utilized as a noise filter to restore the system dynamics from noisy signals; (2) for a single Chua circuit, a machine trained by the noisy time series measured at several sampling states is capable of reconstructing the whole bifurcation diagram of the circuit with a high precision; and (3) for two coupled chaotic Chua circuits with mismatched parameters, the machine trained by the noisy time series measured at several coupling strengths is able to anticipate the variation of the synchronization degree of the coupled circuits with respect to the coupling strength over a wide range. Our studies verify the capability of the technique of parameter-aware reservoir computing in learning the dynamics of chaotic circuits from noisy signals, signifying the potential application of this technique in reconstructing the bifurcation diagram of real-world chaotic systems.

The Dynamics and Theoretical Analysis Underlying Periodic Bursting in the Nonsmooth Murali–Lakshmanan–Chua Circuit

Motivated by the study of multiple time scale system dynamics, we analyze the Murali–Lakshmanan–Chua (MLC) dissipative circuit exhibiting periodic bursting with different types of oscillations in this paper. Such patterns can be analyzed by treating the slow variable as a parameter of the nonsmooth fast subsystem. The discontinuous bifurcation structure of the subsystem can be described by the generalized differential of Clarke and the Jacobian matrix of the system. Bursting patterns as well as the oscillation mechanisms can be clarified by the discontinuous bifurcations, time series and phase portraits. For the chosen circuit parameter values, this designed circuit admits both MLC type bursting attractors and Duffing–van der Pol circuit type bursting attractors. It is found that not only the features combined with the circuit parameters of the system, but also the two switching boundaries may have an important impact on the origin of the bursting behaviors. In particular, the explicit analytic solutions of the proposed circuit are investigated in detail. We also show that the bursting can arise from either discontinuous saddle-node bifurcation or slow passage through a discontinuous Hopf bifurcation. Finally, the validity of the theoretical analysis has been well verified by the numerical simulations.

Data-driven learning of chaotic dynamical systems using Discrete-Temporal Sobolev Networks

We introduce the Discrete-Temporal Sobolev Network (DTSN), a neural network loss function that assists dynamical system forecasting by minimizing variational differences between the network output and the training data via a temporal Sobolev norm. This approach is entirely data-driven, architecture agnostic, and does not require derivative information from the estimated system. The DTSN is particularly well suited to chaotic dynamical systems as it minimizes noise in the network output which is crucial for such sensitive systems. For our test cases we consider discrete approximations of the Lorenz-63 system and the Chua circuit. For the network architectures we use the Long Short-Term Memory (LSTM) and the Transformer. The performance of the DTSN is compared with the standard MSE loss for both architectures, as well as with the Physics Informed Neural Network (PINN) loss for the LSTM. The DTSN loss is shown to substantially improve accuracy for both architectures, while requiring less information than the PINN and without noticeably increasing computational time, thereby demonstrating its potential to improve neural network forecasting of dynamical systems.

Slow Invariant Manifolds of Memristor-Based Chaotic Circuits

This work presents an efficient approach for computing the slow invariant manifold of the fourth-order canonical memristor-based Chua circuits using the flow curvature method. First, the magnetic-flux and charge characteristic curve is generated from the classical circuit with a piecewise-linear function. Then, the characteristic curve is generated from the circuit with the piecewise-linear function replaced by a cubic function. Further, the duality principle is applied to studying such memristor-based circuits in the three-dimensional flux-linkage and charge phase space and then in the four-dimensional current–voltage phase space. It is demonstrated that the slow invariant manifolds of these fourth-order memristor-based chaotic circuits can be more directly computed for the first case than the second.

Adaptive fuzzy finite-time PID backstepping control for chaotic systems with full states constraints and unmodeled dynamics

To solve the problem that the controlled object and system model cannot be fully mastered or the system parameters cannot be well obtained through conventional measurement methods, in this paper, a novel adaptive fuzzy finite-time PID backstepping control method is developed for a class of nonlinear chaotic systems with full state constraints and unmodeled dynamics. The unmodeled dynamics are handled by employing a dynamic signal. To avoid the violations of full state constraints (FSCs), a barrier Lyapunov function is used to implement a PID backstepping framework, whose merit consists in that three control gain constants KP, KI and KD with corresponding proportional relationships are designed to further improve control precision. Besides, the PID control method also embodies three dynamic gain components ΔKP, ΔKI and ΔKD, all of which are free functions that can be self-tuning online, and have a significant link with the adaptive law and modify accordingly. Finally, theoretical analysis confirms that all signals are ultimately uniformly bounded, the error signal converges to a nearby area near the origin in a finite-time, and the FSCs are not smashed. The practicality of the proposed method is confirmed by two simulation studies using the second-order Duffing chaotic system and third-order Chua's circuit.