Jerk Oscillator Research Articles

Extreme multi-stability and circuit implementation for a two-ReLU-memristor-based jerk oscillator

Memristor-based oscillation circuits are prone to produce coexisting infinite attractors depending on the initial conditions of memristors, leading to the appearance of extreme multi-stability. In this paper, we propose a novel memristive jerk oscillator by bringing two ReLU-type memristors in a simple jerk oscillator and investigate its dynamical behaviors associated with the coupling parameters using bifurcation plots and Lyapunov exponent plots. Further, we discuss the planar equilibrium state and its stability, and then numerically explore the coexisting infinite attractors driven by the initial conditions of two ReLU-type memristors. Because of the intervention of the two ReLU-type memristors, the memristive jerk oscillator has a planar equilibrium state whose stability closely relies on the initial conditions of two ReLU-type memristors, and different initial conditions cause different attractors to coexist, resulting in bidirectional extreme multi-stability. Finally, the memristive jerk oscillator is implemented by analog circuit and digital hardware platform, and the numerical results are confirmed by circuit simulations and hardware experiments.

A novel memristive chaotic jerk circuit and its microcontroller-based sliding mode control

Due to the experimental realization of memristor circuit elements, research on memristors and memristor-based circuits has surged. Because of their nonvolatile and nonlinear behavior, memristors can be easily applied to chaotic circuits. This study introduces a novel memristive 3D chaotic jerk system, comprising only seven terms, along with its electronic model and microcontroller-based control. The flux-controlled memristor-based jerk system exhibits complex dynamics, which were analyzed through various properties such as phase portraits, the Jacobian matrix, equilibria, eigenvalues, Lyapunov spectra, bifurcation diagrams, and transient chaos behavior. Three controllers, namely, nonlinear feedback, classical sliding mode, and integral sliding mode were designed to control the chaotic jerk oscillator. Lyapunov functions were used to synthesize the nonlinear feedback controller and ensure system stability with the sliding mode technique. Numerical tests under various performance criteria and disturbance conditions showed that the sliding mode controller outperforms the nonlinear feedback controller due to its single-state control structure. The chaotic jerk oscillator hardware circuit was designed and implemented, operating easily with initial conditions set to zero and low DC supply voltages, with all output voltages within ±6V. Both theoretical and simulation results demonstrate the system’s complexity and applicability, with experimental results aligning well with simulations. Consequently, effective microcontroller-based control was achieved using a single-state controller.

Haque’s approach with mickens’ iteration method to find a modified analytical solution of nonlinear jerk oscillator containing displacement time velocity and time acceleration

AbstractHaque’s approach with Mickens’ iteration method has been used to obtain the modified analytical solutions of the nonlinear jerk oscillator, including displacement time velocity and acceleration. The jerk oscillator represents the features of chaotic behavior in numerous nonlinear phenomena, cosmological analysis, kinematical physics, pendulum analysis, etc., such as electrical circuits, laser physics, mechanical oscillators, damped harmonic oscillators, and biological systems. In this paper, we have used different harmonic terms for different iterative stages using the truncated Fourier series. A comparison is made between the iteration method, the improved harmonic balance method, and the homotopy perturbation method. After comparison, the suggested approach has been shown to be more precise, efficient, simple, and easy to use. Furthermore, there was remarkable accuracy in the comparison between the numerical results and the generated analytical solutions. For the third approximate period, the maximum percentage error is 0.014.

A chaotic jerk oscillator with interlocked offset boosting

A chaotic jerk oscillator with interlocked offset boosting

Autonomous piecewise damping Josephson junction jerk oscillator: microcontroller implementation, controls, and combination synchronization

This paper evaluates the microcontroller implementation, offset boosting control, suppression of chaos, and combination of three types of synchronization in the autonomous piecewise damping Josephson junction (JJ) jerk oscillator (APDJJJO). The APDJJJO exhibits vast shapes of chaotic behaviors, bistable limit circles, bistable period-2-oscillation, and the coexistence of regular and chaotic behaviors exposed by numerical simulations. The microcontroller realization scheme of APDJJJO validates simulated dynamics. Proceeding, two constants are outlined in the rate equations of APDJJJO to achieve the linear offset boosting of constants based on the second and third state variables, respectively. The polarity of the chaotic signal of the second or third state variable can be flexibly altered by changing any of the two introduced constants while the other constant is kept at zero. When the two constants are equal, the second and third state variables can swap between bipolar and unipolar signals flexibly by altering the unique constant parameter. Moreover, theoretical probing is performed to validate the efficacy of the configured single controller engrossed in subduing chaos in APDJJJO. Lastly, the combination of three types of synchronization between two chaotic APDJJJO are analytically and numerically investigated.

Hyperchaotic multiscroll dynamics, complex behavior in a simple homogeneous dynamical network of jerk oscillators: bidirectional coupling scheme method, dynamical study, analog circuit and microcontroller-based implementation

Hyperchaotic multiscroll dynamics, complex behavior in a simple homogeneous dynamical network of jerk oscillators: bidirectional coupling scheme method, dynamical study, analog circuit and microcontroller-based implementation

A new chaotic jerk system with hidden heart-shaped attractor: dynamical analysis, multistability, connecting curves and its application in image encryption

Application of chaos in modeling natural phenomena and encryption encourages researchers to design new chaotic systems with exciting features. Here a chaotic jerk oscillator with different properties is proposed. Previous studies mainly used non-polynomial and piecewise linear terms to design the attractors’ shape. In the paper, the heart-shaped attractor is designed using just polynomial terms. This system is studied by considering its bifurcation diagram, Lyapunov exponent, and basin of attraction. These tools show that the proposed system has chaotic and periodic attractors that coexist in some parameter intervals. The oscillator does not have an equilibrium and has a heart-shaped attractor. Moreover, the connecting curves of the oscillator are considered to explore other structural properties. Numerical results confirm the analytical solutions for the system’s connecting curves. The interesting dynamics of the oscillator are used in an encryption and compression application.

Quasi-periodic structure in chaotic bursting attractor for a controlled jerk oscillator

The main purpose of the paper is to show that parts of the trajectory for chaotic bursting oscillations may oscillate according to different limit cycles, which implies orderly movement exists in chaos. For a typical jerk oscillator with a cubic nonlinearity, when a slow-varying controlling term is introduced, bursting oscillations may occur, appearing in the combination of large-amplitude and small-amplitude oscillations. Two coexisting stable equilibrium points may evolve to chaos via different cascades of period-doubling bifurcation, respectively, which means different types of stable attractors may coexist with the variation of slow-varying parameter, while several periodic windows exist in the chaotic region. It is found that the spiking oscillations of the full system may change according to different cycles not only for 2-D torus bursting oscillations but also for the chaotic bursting attractor. Furthermore, whether the influence of the short periodic window of the fast subsystem on the bursting oscillations appears or not depends on the size of the slow scale, since the inertia of the movement may cause the delay effect of the bifurcations at both two edges of the window. The findings suggest that coupling of multiple scales may be one of key factors for the diversity of real world.

Nonlinear dynamics, coexistence of attractors and microcontroller implementation of a modified Helmholtz Jerk oscillator

In this work, we converted a two-dimensional modified Helmholtz oscillator into a three-dimensional modified Helmholtz jerk oscillator. The study of the stability of the fixed points is made and by using the theorem of Hopf, the condition of existence of the bifurcation of Hopf is sought. By numerical simulations relating to the diagrams of the basin of parameters, attraction, bifurcation, the Lyapunov exponents and the phase portrait, the global dynamics as well as the coexistence of the attractors of the system are analyzed. This study revealed that the considered modified Jerk Helmholtz oscillator can generate Hopf bifurcation, bistable limit cycles, coexistence of chaotic and periodic attractors for appropriate choices of system parameter values. The microcontroller based implementation of the modified Jerk Helmholtz oscillator is proposed to experimentally verify the obtained analytical and numerical results. Finally, to control the amplitude of the Lyapunov attractor and exponent, we added two new parameters in the modified Helmholtz jerk oscillator.

Galerkin’s method to solve a fractional time-delayed jerk oscillator

Galerkin’s method to solve a fractional time-delayed jerk oscillator

Medical image encryption based on RNG with an autonomous piecewise damping Josephson junction jerk oscillator embedded in FPGA

This paper explore the dynamics, Field Programmable Gate Array (FPGA) validation of an autonomous piecewise damping Josephson junction jerk oscillator (APDJJJO) and uses it to protect medical images based on random number generator (RNG). APDJJJO is derived from a piecewise damping Josephson junction (JJ) model and has either no equilibrium points or two unstable equilibrium points. One-scroll chaotic hidden attractor, one-scroll sovereign complex attractor, periodic and one-scroll complex self-driven attractors coexisting, and bistable limit cycles are found in APDJJJO during the investigation based on numerical simulations. Additionally, the FPGA validation of the APDJJJO shows similar complex characteristics to those obtained during the investigation via numerical simulations. Lastly, the chaotic characteristics depicted by the APDJJJO are used to design a RNG for the encryption of medical images. The generated random bits are validated successfully by standard statistical tool set by the National Institute of Standards and Technology (NIST-800-22). Encryption algorithm is developed to secure a medical image by exploring the unpredicted bits generated. The safety and performances analysis are done to prove the robustness and efficiency of the image encryption algorithm.

FPGA implementation, controls and synchronization of autonomous Josephson junction jerk oscillator

This paper reports the field programmable gate array (FPGA) implementation of an autonomous Josephson junction jerk oscillator (AJJJO) and the investigated coexisting and chaotic attractors besteered, with different synchronization achieved. AJJJO displays bistability, megastability, and coexistence between limit cycle and self-excited chaotic attractors, which is validated by the FPGA implementation. The elimination of chaotic dynamics exhibited by AJJJO was done by using a double single controller via numerical simulations. The coexisting attractors between limit cycle and self-excited chaotic attractors are destroyed and besteered to the wanted path via the linear augmentation control method. Moreover, controllers are employed to achieve the difference synchronization of three identical coupled chaotic AJJJOs resulting from various starting conditions. Finally, numerical simulations of three identical coupled chaotic AJJJOs indicate the effectiveness of the difference synchronization.

A novel image encryption scheme based on compressive sensing, elliptic curves and a new jerk oscillator with multistability

Compression and encryption are the two main steps generally used to ensure fast and secure transmission of digital images over public communication channels. Among the existing crypto compression approaches, those based on compressive sensing and chaotic systems possess good potential. Unfortunately, the present literature presents some limits including speed issue, security issue and reconstruction problem. These problems are mainly due to the poor performances of the measurement matrix exploited. Note that in some cases these measurement matrices are obtained from chaotic sequences with very poor dynamics. To solve this problem, we proposed in this paper, a novel image encryption scheme based on 2D compressive sensing, elliptic curves and a new septic jerk oscillator with multistability. The proposed scheme adopts the compression before encryption topology to reduce the processing time and is divided into two main parts. The first part concerns image compression using compressive sensing. The input image data is divided into 4 blocks. Then, each block is compressed using 2D compressing sensing based on the measurements matrices generated using chaotic sequences coming from the novel septic jerk system. Next, the compressed blocks are combined to form the compressed image. In the second part, to strengthen the security, elliptic curve theory is used to construct a simple and secure S-box for digital image substitution. This S-box is combined with the septic chaotic solutions to change the pixel’s positions and values of the compressed image. The analysis of simulation test indicates that the proposed cipher has good compression effect, good reconstruction capability even for low compression ratio, is fast and highly secure compared to some recent schemes. As such the proposed work can be exploited to compress and secure data in Wireless Body Area Network (WBAN) applications.

Multiscroll in Bidirectionally Coupled Jerk Oscillators: Theoretical Analysis and PSpice Verification

It is known that the unidirectional coupling between a double-scroll system (drive) in the oscillation regime with another system (response) in the steady-state, gives rise to more complex dynamics such as multiscroll attractors [Dana et al., 2008]. This paper shows the way to move from n-scroll to 2n-scroll in a bidirectional coupling scheme between two jerk oscillators. The coupling considered can be interpreted as a perturbation to each oscillator’s amplitude through a signal proportional to the amplitude of the other. Indeed, starting from a situation of two oscillators which can generate double-scroll dynamics individually, we perform a bidirectional coupling of both oscillators which results in more complex dynamics such as multistability and 4-scroll chaotic attractor in the coupled system (not observed in an isolated oscillator). The underlined coupling scheme provides an efficient method for generating higher-order multiscroll attractors. The route to the 4-scroll chaotic attractor is explained with the help of bifurcation diagrams, plots of several phase portraits, and basins of attractions. Indirect validation of the theoretical analysis is carried out by using the PSpice simulation software.

An efficient approach to solving fractional Van der Pol–Duffing jerk oscillator

The motive behind the current work is to perform the solution of the Van der Pol–Duffing jerk oscillator, involving fractional-order by the simplest method. An effective procedure has been introduced for executing the fractional-order by utilizing a new method without the perturbative approach. The approach depends on converting the fractional nonlinear oscillator to a linear oscillator with an integer order. A detailed solving process is given for the obtained oscillator with the traditional system.

Dynamical analysis of autonomous Josephson junction jerk oscillator with cosine interference term embedded in FPGA and investigation of its collective behavior in a network

Dynamical analysis of autonomous Josephson junction jerk oscillator with cosine interference term embedded in FPGA and investigation of its collective behavior in a network

Autonomous jerk oscillator with sine nonlinearity and logistic map for sEMG encryption

This research discusses the dynamical behaviors of the autonomous jerk oscillator with sine nonlinearity (AJOSN) and its application to secure s-EMG (Surface ElectroMyoGraphic) data. The AJOSN has two or no-equilibrium points, and the stability of the two equilibrium points indicates that one is stable and the other is unstable. The AJOSN reveals fast-spiking and periodic bursting, relaxing and periodic oscillations, hidden chaotic attractors and coexisting attractors. The numerical analysis results are validated by the Field Programmable Gate Array (FPGA) implementation. Finally, the AJOSN’s chaotic behavior coupled with the logistic map is exploited to encrypt the s-EMG signals. The proposed encryption and decryption method consists of simultaneously encrypting the sEMG signal. The sEMG signal, once transformed into 2D as a grayscale image, undergoes first encryption based on the chaotic signal generated by the AJOSN (Autonomous jerk oscillator with sine nonlinearity) model. The encrypted signal obtained from this first part is called ‘Jerk encrypted signal.’ The second part is to encrypt the ‘Jerk Encrypted Signal’ by the chaotic logistics map. The encryption and decryption results obtained are quite encouraging and offer a great prospect.

PIC micro-controller based synchronization of two fractional order jerk systems

The paper studies a 3D Chaotic Jerk oscillator with fractional derivatives. An approach is proposed to implement it on a PIC16F877A microcontroller in order to reduce the requirements for multiple analogue electronic components such as resistors, capacitors, coils, multipliers, operational amplifiers, which are very bulky and consume a lot of power. The behaviours of the underlying system are analysed analytically, numerically and experimentally. It comes from this analysis that the fractional model exhibits chaotic dynamics when for parameters for which the equivalent integer derivative system exhibits limit-cycles. The synchronization under two closed initial conditions is also studied, highlighting one of the most common applications of the chaos concept.

Dynamical behaviors of a chaotic jerk circuit based on a novel memristive diode emulator with a smooth symmetry control

This paper presents a new chaotic memristive circuit, which is derived from the autonomous jerk circuit by replacing the first-order generalized memristive diode bridge with a simple second-order memristive emulator, which consists of six elementary electronic elements. This second-order memristive emulator model addresses the problem of changing the configuration of the diode-bridge memristor model when used for symmetry control of chaotic systems. It also allows to reduce the gap between the experimental and the numerical simulations in that during the experimental, the symmetry breaking is done by a discrete modification of the number of diodes on the branches of the diode bridge, whereas during the numerical simulations, the variation of the symmetry control parameter corresponding to the number of diodes is continuous. Thus, a new memristor emulator consisting of two diodes, two resistors and two capacitors is developed and proposes a method of control by continuous variations of a potentiometer to perform symmetry breaking. Using tools for characterizing dynamic systems such as bifurcation diagrams, Lyapunov spectrum and phase portraits, to name just a few, it is shown how the new memristive emulator model can be used to check symmetry. The first major contribution of this work is the control of the symmetry of a jerk oscillator when the resistors of the memristive emulator take different values without modifying its structure. The second major contribution of this work is the ability to change the nature of an attractor as a function of its position in phase space without changing the nature of its associated symmetric attractor, which is demonstrated using phase portraits and initial condition diagrams. An electronic circuit is presented showing how symmetry control is achieved by varying a potentiometer. And an experimental study is performed to validate the results.

A unity‐gain approach to a simple FDNR‐based chaotic jerk oscillator

A unity-gain approach is proposed for a simple chaotic jerk oscillator based on frequency-dependent negative resistance. Both unity-gain and frequency-dependent negative resistance techniques are demonstrated together, for the first time, for a simple chaotic jerk circuit. The oscillator is attractive due to the use of a grounded diode and circuit simplicity of only seven electronic components, only one of which is an active device. The simplicity results in fewer components of both active and passive devices compared to existing frequency-dependent negative resistance-based chaotic circuits. The maximised value of the largest Lyapunov exponent is relatively high at 14.62.