Jerk System Research Articles

Initial-offset and amplitude control of multi-scroll attractor in a chaotic jerk system

Abstract With the gradual deepening of research on chaos, in order to flexibly apply chaos, offset boosting and amplitude control have become a hot topic. However, few people have studied chaotic systems with multi-scroll attractor under initial conditions.
This paper constructs and analyzes a simple four-dimensional chaotic system. Unlike other extension methods that modify the original seed system, this method does not make any changes to the original seed system, only adds a new variable in the extension dimension. It is found that the proposed system can achieve offset boosting and amplitude control by changing the initial values, and the system has characteristics such as multi-scroll attractors, infinite equilibrium points, symmetry, and so on. Finally, FPGA hardware implementation was carried out to verify the numerical results.

Interior Crisis Route to Extreme Events in a Memristor-Based 3D Jerk System

In dynamical systems, events that deviate significantly from usual or expected behavior are referred to as extreme events. This paper investigates the mechanism of extreme event generation in a 3D jerk system based on a generalized memristive device. In addition, regions of coexisting parallel bifurcation branches are explored as a way of investigating the multistability of the memristive system. The system is examined using bifurcation diagrams, Lyapunov exponents, time series, probability density functions of events, and inter-event intervals. It is found that extreme events occur via a period-doubling route and are due to an interior crisis that manifests itself as a sudden shift from low-amplitude to high-amplitude oscillations. Multistability is also identified when both control parameters and initial values are modified. Finally, an analog circuit based on the memristive jerk system is designed and experimentally realized. To our knowledge, this is the first time that extreme events have been reported in a memristive jerk system in particular and in jerk systems in general.

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.

Extreme events in a damped Korteweg–de Vries (KdV) autonomous system: A comprehensive analysis

This manuscript explores extreme events in a three-dimensional, damped Korteweg–de Vries (KdV) autonomous system that is derived from the jerk system with a modified basin. Through rigorous stability analysis, we identify the extreme events, confirmed using phase portraits, Poincaré return maps, time series, and probability distribution functions. The Dragon-king phenomenon is obtained and self-organised criticality is discussed. The regime of extreme events in a wide parameter range was obtained through two-parameter scanning. The analog circuit was constructed to mimicking dynamics of damped KdV equation and demonstrated in real-time hardware experiment as well as PSpice simulation. The experimental results show excellent agreement with the numerically obtained results. This study is the first comprehensive examination of extreme events in the damped KdV system, highlighting their nature and potential applications in chaos-based dynamical systems.

Bifurcation analysis with chaotic attractor for a special case of jerk system

This article focuses on investigating local bifurcations in a special type of chaotic jerk system. It examines the occurrence and non-occurrence of saddle-node, transcritical, zero-Hopf, Hopf, and pitchfork bifurcations at the origin. The parameters that result in a zero-Hopf equilibrium point at the origin are characterized for the proposed system. Additionally, a demonstration is provided to show that the utilization of the first-order averaging theory leads to the emergence of a single periodic solution branching out from the zero-Hopf equilibrium located at the origin. Furthermore, the focus quantities method is applied to explore the periodicity of the cubic part of the system. This method helps determine the number of periodic solutions that can emerge from the Hopf point. Due to the computational load for computing singular quantities, only three singular quantities are found. Under specific conditions, it is shown that three periodic solutions can bifurcate from the origin of the system. Finally, the study also examines the chaotic attractors of the system.

A novel resistorless memristor emulator circuit and its implementation of chaotic Jerk system

This work presents a novel resistorless memristor emulator circuit designed in both grounded and floating configurations. The grounded memristor consists of a Current-Controlled Current Conveyor Transconductance Amplifier (CC-CCTA) and a capacitor. On the other hand, a floating memristor includes a CC-CCTA and a capacitor, along with an additional Operational Transconductance Amplifier (OTA). These proposed emulators are electronically controllable owing to the internal structures of active elements. Theoretical analysis and simulation results have been obtained for the proposed designs. Simulation results have been conducted utilizing the LTspice program and TSMC 180 nm technology has been used for internal structures. In these results, the effects of the changing capacitor values, different frequencies, electronic tunability, reactions to temperature change, and pulse response have been investigated. Subsequently, a chaotic Jerk circuit has been implemented incorporating the proposed memristor emulator. The chaotic behavior of the floating memristor has been investigated using the LTspice simulation software.

Hopf Bifurcation of Three-Dimensional Quadratic Jerk System

هذا البحث مخصص لبحث تشعب هوبف لنظام الرعشة التربيعي ثلاثي الأبعاد. تم دراسة الاستقرار للنقاط المنفردة وظهور تشعب هوبف والدورات الحدية للنظام. بالإضافة إلى ذلك، يتم استخدام تقنية كميات ليابانوف لدراسة دورية النظام ومعرفة عدد دورات الحد التي يمكن تشعبها من نقاط هوبف. نظرًا للحمل الحسابي المطلوب لحساب كميات ليابانوف، تم تثبيت بعض المعلمات. حاليًا، يُظهر التحليل أنه يمكن تشعب ثلاث دورات حدية من نقاط هوبف. تم استخدام برمجة Maple للتحقق من جميع النتائج المعروضة في هذه الدراسة.

Jerk Chaotic System: Analysis, Circuit Simulation, Control and Synchronization with Application to Secure Communication

Dynamical behaviors of 3D Jerk system were examined in terms of equilibrium, stability, dissipative, and phase space attractor in this study. The system’s practical applications were demonstrated through circuit realization and synchronization scheme via active backstepping control with its effectiveness demonstrated in secure communication. The viability of the theoretical model of 3D Jerk system was confirmed using electronic circuit workbench designed in MultiSIM enviroment. A nonlinear feedback controller was designed using the recursive backstepping technique to control and track a desire function. For secure communication application, active backstepping method was adopted to synchronize two identical chaotic systems evolving from different initial conditions. It was demonstrated that when the controller was activated, the systems synchronize successfully. The results of the active backstepping designed controllers were numerically applied in the area of secure communication, with the variable of the drive being encrypted information transmitted through a coupling channel. Using an additive encryption masking scheme, the encrypted signal was a superposition of sinusoidal information specified by period function and chaotic carrier generated from a variable of the Jerk system. The transmitted information signal was successfully retrieved from the chaotic response signal using an inverse function decryption algorithm, thereby confirming the effectiveness and robustness of the designed controller.

Bifurcation analysis with self-excited and hidden attractors for a chaotic jerk system

Bifurcation analysis with self-excited and hidden attractors for a chaotic jerk system

Constructing chaotic oscillators with memory components

Memory components, including memristors, meminductors and memcapacitors, have special nonlinear dynamical properties and thus applicable for constructing a compact chaotic circuit in parallel or in series. By selecting the restriction of a memory component, chaos control can also be achieved using a single circuit parameter. In this work, three different types of circuit structures are designed to implement memory components for chaos generation. With the help of a voltage-control or a current-control memristor, within a series or a parallel structure, a memristive chaotic oscillator with partial control is built based on the variable boostable systems. The two constructed structures are equivalent and can be mutually tuned, achieving partial amplitude control as well as offset boosting. By constructing a magnetic-control meminductor or a charge-control memcapacitor, the equivalent parallel circuit is built with a meminductor, a capacitor and a resistor, corresponding to the series circuit with a memcapacitor, an inductor and a resistor. It can be applied to realizing the hypogenetic chaotic jerk system even with conditional symmetry. All the designed circuits are physically realized based on PCB, proving the effectiveness of memory components for chaos generation.

Codimension-2 bifurcations of a generalized three-dimensional cubic jerk system

Codimension-2 bifurcations of a generalized three-dimensional cubic jerk system

Dynamical behavior of a new jerk system inspired from chaotic memory oscillators

This paper constructs a six-term new simple 3D jerk system modeled by chaotic model memory oscillators with four parameters that control the behavior. The suitable choice of one of these parameters helps the system describe behavior and attractors. This means that the choice is a parameter of the associated behavior (dissipative or conservative) and attractors (self-excited or hidden). Some features of the equilibrium are observed that are related to the dependence on these parameters, such as saddle-foci, non-hyperbolic, and node-foci. This system is rich in dynamic features including chaotic, quasi-periodic (2-torus), and periodic via the utilization of bifurcation diagrams and Lyapunov spectrum. Finally, a new image encryption algorithm is introduced that utilizes the jerk system. The algorithm is assessed through statistical performance analysis, according to the results of the experiments and security tests, it has been verified that the suggested image encryption algorithm is highly secure and could be a viable option for real-world applications.

A new chaotic jerk system with a sinusoidal nonlinearity, its bifurcation analysis, multistability, circuit design and complete synchronization design via backstepping control

A new chaotic jerk system with a sinusoidal nonlinearity, its bifurcation analysis, multistability, circuit design and complete synchronization design via backstepping control

A New Chaotic Jerk System with Cubic and Hyperbolic Sine Nonlinearities and Its Application to Random Number Generation and Biomedical Image Encryption

A New Chaotic Jerk System with Cubic and Hyperbolic Sine Nonlinearities and Its Application to Random Number Generation and Biomedical Image Encryption

A new modified WINDMI jerk system with exponential and sinusoidal nonlinearities, its bifurcation analysis, multistability, circuit simulation and synchronization design

A new modified WINDMI jerk system with exponential and sinusoidal nonlinearities, its bifurcation analysis, multistability, circuit simulation and synchronization design

Modelling, bifurcation analysis, circuit design and FPGA-based implementation of a new chaotic jerk system exhibiting Hopf bifurcations

Modelling, bifurcation analysis, circuit design and FPGA-based implementation of a new chaotic jerk system exhibiting Hopf bifurcations

Energy function and complex dynamics from a jerk system

Jerk, as a mathematical concept, is used in mechanics to describe the rate of change of acceleration and plays a crucial role in the design of control systems for machines and vehicles. Therefore, it is important to master the various states and the energy released during the change of acceleration. This is why a new simple jerk function introduced afterward, energy released, is derived from a Hamilton function using the Helmholtz theorem. The condition of having a stable or unstable rate of change of acceleration is established using Hopf bifurcation theory. Some two-parameter stability charts are then computed for a suitable selection region of the study. Using some nonlinear analysis metrics, in the unstable region of the study, the occurrence of phenomena is found, such as reverse period doubling bifurcation, antimonotonicity, and hysteresis involving the coexistence of the states in the considered jerk system. An electronic circuit is built and used to implement the mathematical expression of the jerk equation and validate the result of the theoretical investigation.

Parametric fractional-order analysis of Arneodo chaotic system and microcontroller-based secure communication implementation

Chaotic systems are often used in cryptology applications thanks to their complex dynamic structures. By increasing the dynamic diversity of chaotic systems through fractional-order analysis, more complex systems for data security can be obtained. In this study, fractional calculus analyses of the Arneodo chaotic jerk system with cubic nonlinearity and its microcontroller-based chaotic masking application are presented. Numerical analyses such as bifurcation diagrams, spectral entropy complexity, and Lyapunov spectra are performed to examine the dynamic characteristics of the fractional-order system. The parametric analysis indicates the presence of chaotic states in the system for nearly all fractional-order values ranging from 0.55 to 1. In this manner, the study focuses on investigating the minimum applicable fractional-order value for use in secure communication applications, constituting the main contribution of this research. Consequently, the dynamic diversity and unpredictability of the cubic system are greatly increased. Moreover, two fractional-order Arneodo chaotic (FOAC) systems with distinct initial conditions are synchronized using a single-state fractional-order sliding mode controller (FOSMC). Finally, an innovative secure communication system based on a microcontroller is designed and implemented by employing the chaotic masking method. As anticipated, the voltage outputs of the microcontroller-based secure communication application demonstrated good agreement with the numerical simulations.

Analysis of the Dynamics of a $\phi^{6}$ Duffing Type Jerk System

A theoretically and numerically analysis on Duffing Jerk systems with a sixth-order type potential and a sixth-order potential smoothed by a gaussian function are carried out in this work. The Jerk is transformed into a dynamical system of dimension three. The dynamics and stability of the resulting system are analyzed, through phase space, bifurcation diagrams and Lyapunov exponents by varying the relevant parameters, finding the existence of a strange attractor. The dynamics of system with potential smoothed was studied by varying the smoothing parameter $\alpha$, finding that this parameter can be used to controlling chaos, since the exponential factor keeps the same fixed points and it regulates smoothly the amplitude of the potential.

Active control and electronic simulation of a novel fractional order chaotic jerk system

The chaotic jerk system represents a class of nonlinear systems with unique, varied, and interesting characteristics and behavior. In this study, we proposed a modified fractional-order chaotic Jerk system by the introduction of two novel nonlinear terms and two constant terms. The nonlinear terms introduced are the exponential and tangential functions while the two parameters are a and b. Stability analysis of the fractional-order Jerk system was used to establish the parameter ranges for chaotic behavior in the system. Chaos in the system was confirmed via the Lyapunov exponents. The bifurcation diagram with respect to the parameters, a and b, as well as the fractional order, α were examined. The synchronization of the system was implemented using the active control method. Finally, the studied novel circuit was realized using MultiSIM electronic software for experimental validation.