Hyperjerk System Research Articles

A New Hyperjerk System With a Half Line Equilibrium: Multistability, Period Doubling Reversals, Antimonotonocity, Electronic Circuit, FPGA Design, and an Application to Image Encryption

A New Hyperjerk System With a Half Line Equilibrium: Multistability, Period Doubling Reversals, Antimonotonocity, Electronic Circuit, FPGA Design, and an Application to Image Encryption

Initial-condition effects on ReLU-type hyper-jerk system and its application in image encryption

The memristive hyper-jerk system described in this study has a non-smooth Rectified Linear Unit (ReLU) memductance function. A non-smooth memductance function of this kind might cause the system to have three line equilibria within it, bring about the arrival of extreme multistability with infinitely many coexisting attractors. Analyzing the three line equilibria’s stabilities and researching the intricate dynamical effects triggered by memristor and non-memristor initial conditions are related topic. Phase portraits, bifurcation diagrams, Lyapunov exponent spectra, and the basins of attraction all show the dynamical tendencies. According to the results, the novel system’s dynamical behaviors depend on both the memristor initial condition and the non-memristor initial conditions. Moreover, PSIM circuit simulations and MCU-based hardware measurements are run to confirm the coexisting infinitely many attractors. The ReLU-type hyper-jerk system is then utilized to encrypt images. Experimental findings support its exceptional robustness against a range of potential attacks.

A new multi-stable chaotic hyperjerk system, its special features, circuit realization, control and synchronization

A new multi-stable chaotic hyperjerk system, its special features, circuit realization, control and synchronization

A 4-D chaotic hyperjerk system with a hidden attractor, adaptive backstepping control and circuit design

A 4-D chaotic hyperjerk system with a hidden attractor, adaptive backstepping control and circuit design

Dynamics and synchronization of a novel 4D-hyperjerk autonomous chaotic system with a Van der Pol nonlinearity

AbstractIn the literature, hyperjerk systems raised up meaningful interest due to their simple and elegant structure as well as their complex dynamical features. In this work, we propose a novel 4D autonomous hyperjerk system which the particularity resides on the type of its nonlinearity namely the Van der Pol nonlinearity. The dynamics of this hyperjerk system is assessed thanks to the well-known nonlinear dynamic tools such as time series, bifurcation diagrams, Lyapunov exponent spectrum, two-parameter phase diagram, and phase portraits. As important result, it is established that the system presents a particular phenomenon of hysteretic dynamics that leads to the coexistence of attractors. Next, through the calculation of the Hamiltonian energy, we show that this latter depends on all the variables of the novel hyperjerk system. Furthermore, basing on an adaptive backstepping method whose target is a function of the states of the error system, a new controller is designed to carry out fromt= 30, complete chaotic synchronization of the identical novel hyperjerk chaotic systems. Likewise, PSpice (9.2 full version) based simulations are presented in detail to confirm the feasibility of the theoretical model. One of the key points of this work is the designing in PSpice environment of this new adaptive backstepping controller to validate both theoretical and numerical synchronization results. Finally, our experimental measurements in the laboratory are in good agreement with the numerical and analog results.

Hidden and Self-Excited Collective Dynamics of a New Multistable Hyper-Jerk System with Unique Equilibrium

Nonlinear dynamical systems with hidden attractors belong to a recent and hot area of research. Such systems can exist in different forms, such as without equilibrium or with a stable equilibrium point. This paper focuses on the dynamics of a new 4D chaotic hyper-jerk system with a unique equilibrium point. It is shown that the new hyper-jerk system effectively exhibits different hidden behaviors, which are hidden point attractor, hidden periodic attractor, and hidden chaotic state. Collective behaviors of the system are studied in terms of the equilibrium point, bifurcation diagrams, phase portraits, frequency spectra, and two-parameter Lyapunov exponents. Some remarkable and exciting properties are found in the new snap system, such as period-doubling transition, asymmetric bubbles, and coexisting bifurcations. Also, we demonstrate that it is possible to generate different varieties of two, three, four, or five coexisting hidden and self-excited attractors in the introduced model. In addition, the amplitude and offset of the hidden chaotic attractors are perfectly controlled for possible application in engineering. Furthermore, a circuit design has been implemented to support the physical feasibility of the proposed model.

A Damping-Tunable Snap System: From Dissipative Hyperchaos to Conservative Chaos.

A hyperjerk system described by a single fourth-order ordinary differential equation of the form has been referred to as a snap system. A damping-tunable snap system, capable of an adjustable attractor dimension () ranging from dissipative hyperchaos () to conservative chaos (), is presented for the first time, in particular not only in a snap system, but also in a four-dimensional (4D) system. Such an attractor dimension is adjustable by nonlinear damping of a relatively simple quadratic function of the form , easily tunable by a single parameter A. The proposed snap system is practically implemented and verified by the reconfigurable circuits of field programmable analog arrays (FPAAs).

A Chaotic Quadratic Bistable Hyperjerk System with Hidden Attractors and a Wide Range of Sample Entropy: Impulsive Stabilization

Hidden attractors generated by the interactions of dynamical variables may have no equilibrium point in their basin of attraction. They have grabbed the attention of mathematicians who investigate strange attractors. Besides, quadratic hyperjerk systems are under the magnifying glass of these mathematicians because of their elegant structures. In this paper, a quadratic hyperjerk system is introduced that can generate chaotic attractors. The dynamical behaviors of the oscillator are investigated by plotting their Lyapunov exponents and bifurcation diagrams. The multistability of the hyperjerk system is investigated using the basin of attraction. It is revealed that the system is bistable when one of its attractors is hidden. Besides, the complexity of the systems’ attractors is investigated using sample entropy as the complexity feature. It is revealed how changing the parameters can affect the complexity of the systems’ time series. In addition, one of the hyperjerk system equilibrium points is stabilized using impulsive control. All real initial conditions become the equilibrium points of the basin of attraction using the stabilizing method.

Variable universe adaptive fuzzy sliding mode projective synchronization of hyperjerk system based on disturbance observer

In this paper, a sliding mode projective synchronization strategy based on disturbance observer and fuzzy system is presented to implement projective synchronization of hyperjerk system with low time-varying disturbance and white noise. Theoretical analysis and numerical calculation show that the disturbance observer can approach the low time-varying disturbance very well. The application of disturbance observer reduces the chattering of the controller. Variable universe adaptive fuzzy control (VUAFC) method is utilized to further reduce the chattering phenomenon. The simulation results demonstrate the effectiveness of the proposed controller.

Connecting Curves as a Tool to Localize Hidden Attractors in a New Chaotic Hyperjerk System with No Equilibria

Localizing hidden attractors of chaotic systems is practically and theoretically important. Differing from self-excited attractors, hidden ones do not have any equilibria on the boundaries of their basin of attraction. This characteristic makes hidden attractors hard to localize. Some theoretical and numerical methods have been developed to recognize these attractors, yet the problem remains highly uncertain. For this purpose, the theory of connecting curves is utilized in this work. These curves are one-dimensional set-points that describe the structure of chaotic attractors even in the absence of zero-dimensional fixed-points. In this study, a new four-dimensional chaotic system with hidden attractors is presented. Despite the controversial idea of connecting curves that pass through fixed-points, the connecting curves of a system with no equilibria are considered. This analysis confirms that connecting curves provide more critical information about attractors even if they are hidden.

N-th-Order Simple Hyperjerk System with Unstable Equilibrium and Its Application as RPG

This paper presents a simple mathematical model with an electronic circuit for n-th-order hyperjerk system. For demonstration, the proposed iterative methodology comes with a fourth-order parametric control hyperjerk system that relies on the integrator, a summer amplifier, and an adjustable hyperbolic sine nonlinearity function. The proposed hyperjerk system exhibits exceptional unstable behavior at equilibrium. However, the corresponding hidden attractor near to equilibrium point has a high degree of disorder and randomness. The proposed n-th-order chaotic system has some important features such as simple model without multiplication term, simple circuitry, use of n number of capacitors, 2 $$(n-1)$$ number of resistors, and sensitivity with passive components. Moreover, various dynamic behavior and typical time series chaotic responses of the proposed design are validated by incorporating bifurcation sequence, numerical simulation, and practical implementation by using Op-Amp. Also, an application perspective of the proposed hyperjerk system is extended to a random pulse generator (RPG). This concept reflects an idea to obtain the RPG by interfacing with the chaotic signals.

A novel antimonotic hyperjerk system: Analysis, synchronization and circuit design

We propose a novel chaotic oscillator derived from the generic four-dimensional autonomous jerk systems. By using the Routh–Hurwitz criterion, we analyze the equilibrium point stability. Furthermore, we use the bifurcation diagram and Lyapunov exponents to explore the chaotic regions of the oscillator. One of the novel hyperjerk systems’ main features is the presence of antimonotonicity in a region of the parameter’s space. This phenomenon occurs when periodic orbits are generated and then annihilated via reverse period-doubling bifurcation scenarios as a control parameter is slowly scanned. Next, we propose an electronic design of the hyperjerk system which is compared with the numerical simulations, from which we observe a qualitative agreement among them. Latter, we analyze the synchronization behavior of the chaotic oscillator via feedback control. By using the Lyapunov function methodology, we establish the conditions to achieve synchronization based on the positivity of a matrix written in terms of the system parameters. The numerical simulations support our theoretical results.

A new chaotic multi-stable hyperjerk system with various types of attractors

In this paper, a multi-stable chaotic hyperjerk system with both self-excited and hidden attractors is proposed. Such a system is infrequent between dynamical systems. State-space, bifurcation, and Lyapunov exponent plots are presented to show the existence of chaotic dynamics. The fractional-order model of the system and its dynamical properties are studied in this paper.

Stumped nature hyperjerk system with fractional order and exponential nonlinearity: Analog simulation, bifurcation analysis and cryptographic applications

This article presents the construction and consumption of hyperjerk system having chaotic nature with the commensurate ordered fractional derivative on rapidly digitalized emerging technologies. The system analyzed analytically using Lipschitz condition and numerically with the help of predictor corrector approaches. Originality of constructed system is confirmed using log error plots which gives satisfactory small values on finite time scale. The dynamical performances of the complex system are termed in multiple phase planes by the mean of their significant nature. Stability and bifurcation analysis around the fractional derivative is also studies for the various parameter of system to check the visibility of chaotic solution. The system is also designed in analog circuit simulator with the aid of operational amplifier, anti-parallel semiconductor diodes for validation of the system. With the help of random number generators (RNGs), binary array generated from the hyperjerk system and plugged in to the NIST 800–22 test suite to measure the randomness. The array having high randomness is used as strong cypher key for the cryptographic execution straightforwardly in both direction encryption and decryption.

A Memristive Hyperjerk Chaotic System: Amplitude Control, FPGA Design, and Prediction with Artificial Neural Network

An amplitude controllable hyperjerk system is constructed for chaos producing by introducing a nonlinear factor of memristor. In this case, the amplitude control is realized from a single coefficient in the memristor. The hyperjerk system has a line of equilibria and also shows extreme multistability indicated by the initial value-associated bifurcation diagram. FPGA-based circuit realization is also given for physical verification. Finally, the proposed memristive hyperjerk system is successfully predicted with artificial neural networks for AI based engineering applications.

No-argument memristive hyper-jerk system and its coexisting chaotic bubbles boosted by initial conditions

Abstract Antimonotonicity is used to characterize the specific dynamics for the generation of periodic orbits cascaded by their destruction, which is generally involved with the change of a bifurcation parameter. However, this phenomenon of antimonotonicity is rarely induced by the initial conditions of chaotic systems. To this end, this paper presents a no-argument memristive hyper-jerk system. When taking the initial condition of the memristor inner state (called as “memristor initial condition” for short) as a bifurcation parameter, we disclose the chaotic bubbles located in the primary interval and thereby display the coexisting infinitely many attractors with extreme multi-stability. Afterwards, when switching the memristor initial condition, we also uncover the initial condition-boosted coexisting chaotic bubbles theoretically and numerically. Therefore, the complex phenomenon of the extreme multi-stability with the initial condition-boosted coexisting chaotic bubbles is well revealed. Furthermore, a reconstituted model with the initial conditions in an explicit form is established in integral domain and the extreme multi-stability can be effectively interpreted through the stability analysis of the determined equilibrium points of the reconstituted model. Finally, a digital hardware platform is implemented to verify the memristor initial condition-dependent chaotic bubbles.

Bifurcations and chaos dynamics of a hyperjerk system with antimonotonicity

In this paper, the bifurcations and chaos dynamics of a hyperjerk system with antimonotonicity are investigated via the analytical methods and numerical calculations. We discuss the local stabilities of equilibrium points and its bifurcations depending on parameters. The predicted bifurcations of periodic orbits including flip bifurcation, fold bifurcation and symmetry-breaking bifurcation are investigated. The accurate relations between parameters are established to identify the type of bifurcation appearing in system efficiently. Such simple system has very abundant dynamical behaviors, including period-doubling cascades route to chaos, intermittency regime, antimonotonicity and boundary crisis. The coexisting dynamics is confirmed effectively by using basins of attraction, bifurcation diagram, Lyapunov exponent spectrum and phase portraits. The research reveals the richness and complexity of potential stable states to which this chaotic hyperjerk system can evolve and offers flexibility in realization of various applications including secure communications.

Coexistence of Multiple Points, Limit Cycles, and Strange Attractors in a Simple Autonomous Hyperjerk Circuit with Hyperbolic Sine Function

In this contribution, a new elegant hyperjerk system with three equilibria and hyperbolic sine nonlinearity is investigated. In contrast to other models of hyperjerk systems where either hidden or self-excited attractors are obtained, the case reported in this work represents a unique one which displays the coexistence of self-excited chaotic attractors and stable fixed points. The dynamic properties of the new system are explored in terms of equilibrium point analyses, symmetry and dissipation, and existence of attractors as well. Common analysis tools (i.e., bifurcation diagram, Lyapunov exponents, and phase portraits) are used to highlight some important phenomena such as period-doubling bifurcation, chaos, periodic windows, and symmetric restoring crises. More interestingly, the system under consideration shows the coexistence of several types of stable states, including the coexistence of two, three, four, six, eight, and ten coexisting attractors. In addition, the system is shown to display antimonotonicity and offset boosting. Laboratory experimental measurements show a very good coherence with the theoretical predictions.

A modified simple chaotic hyperjerk circuit: coexisting bubbles of bifurcation and mixed-mode bursting oscillations

Abstract A relatively simple chaotic hyperjerk circuit, which is the modified chaotic hyperjerk system [Dalkiran and Sprott, IJBC 2016] is proposed and investigated in this paper. Only one semiconductor diode modelled the nonlinear function capable of rich and complex dynamical behaviours of the system. We investigate a new kind of behaviours name “bubbles of bifurcation’’ (referred as BsB hereafter) observed here for the first time in the hyperjerk system. An interesting phenomenon of mixed-mode bursting oscillations (MMBOs) is also investigated. The complex dynamics of the novel oscillator (such as MMBOs, BsB, offset boosting and multistability) with respect to its parameters and initial conditions are uncovered using bifurcation diagrams, Lyapunov exponents (LE) and phase portraits. As another interesting property of this circuit, some parameter regions are determined for the existence of coexisting BsB and the coexistence of asymmetric mixed-mode bursting oscillations. Let us emphasized that the complex phenomena observed in this work is very rare in the literature and henceforth merit dissemination. Finally, a physical circuit is constructed to demonstrate some experimental observation of MMBOs.

Analysis, Synchronization, and Robotic Application of a Modified Hyperjerk Chaotic System

In this work, a novel hyperjerk system, with hyperbolic sine function as the only nonlinear term, is proposed, as a modification of a hyperjerk system proposed by Leutcho et al. First, a dynamical analysis on the system is performed and interesting phenomena concerning chaos theory, such as route to chaos, antimonotonicity, crisis, and coexisting attractors, are studied. For this reason, the system’s bifurcation diagrams with respect to different parameter values are plotted and its Lyapunov exponents are computed. Afterwards, the synchronization of the system is considered, using active control. The proposed system is then applied, as a chaotic generator, to the problem of chaotic path planning, using a combination of sampling and a modulo tactic technique.