Exponent Plots 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 Difference Equation of Banking Loan with Nonlinear Deposit Interest Rate

This paper considers a banking loan model using a difference equation with a nonlinear deposit interest rate. The construction of the model is based on a simple bank balance sheet composition and a gradient adjustment process. The model produces two unstable loan equilibriums and one stable equilibrium when the parameter corresponding to the deposit interest rate is situated between its transcritical and flip bifurcations. Some numerical simulations are presented to align with the analytical findings, such as the bifurcation diagram, Lyapunov exponent, cobweb diagram, and contour plot sensitivity. The significance of our result is that the banking regulator may consider the lower and upper bounds for setting the nonlinear interest rate regulation and provide a control regulation for other banking factors to maintain loan stability.

Novel competitive fractional chaotic circuits interaction networks with hierarchical structure and encryption application

In the realm of complex networks, the challenge of ensuring secure communication amidst the vulnerabilities of conventional encryption methods has become increasingly critical. This study delves into the complex realm of synchronized behaviors in networks, employing fractional-order chaotic circuits within hierarchically structured competitive interaction networks to enhance encryption security, particularly for medical image transmission. We propose a novel paradigm that transcends traditional synchronization methods used across various disciplines, from engineering to social sciences, by unveiling the intricate dynamics of how units within networks share interactions. Our approach leverages the unique properties of fractional chaos and network hierarchy, demonstrating that the proposed model, characterized by multi-directed links and competitive strategies, significantly improves synchronization. Through detailed analysis, including bifurcation diagrams and Lyapunov exponent plots, we uncover the optimal configurations of coupling strength and fractional order that lead to enhanced network synchronization. This synchronization is pivotal for our encryption application, showcasing a high level of security and privacy in the transmission of medical images. The encryption technique benefits from the network’s complex and synchronized dynamics, rendering it a formidable challenge for potential attackers to decipher the encrypted data. While our findings offer a promising mechanism for creating robust communication networks capable of securing sensitive medical data, the implications of our work extend beyond this application. The successful application of fractional-order chaotic circuits sets a groundwork for securing diverse types of data transmissions against the evolving landscape of cyber threats. This research not only marks a significant advancement in network security but also opens new avenues for applying these principles across a spectrum of fields where data security and privacy are paramount.

Unsteady double-diffusive Brinkman–Bénard convection in cylindrical enclosure saturated with hybrid bi-viscous Bingham nanoliquid

This paper aims to present an analytical as well as a comparative study to investigate the effect of the Brinkman porous medium on the onset of regular and chaotic motion in cylindrical enclosures of different heights. The hybrid bi-viscous Bingham nanoliquid is considered as the working fluid. Modified Brinkman–Buongiorno and bi-viscous Bingham fluid models are incorporated to obtain the flow governing dynamics. The thermophysical properties of the hybrid nanoliquid are calculated using phenomenological laws and the mixture theory. The study is carried out for the axisymmetric mode, and the Bessel functions are taken as the eigenfunctions of the problem. Double Fourier–Bessel series expansions are used for weakly non-linear stability analysis. The limiting cases of the study are obtained, and the results on the onset of convection, heat, and mass transport are discussed graphically. The behavior of the dynamical system is analyzed using the maximum Lyapunov exponent plot, the bifurcation diagram, and phase plots. Outcomes suggest that convection sets in earlier in the water-based hybrid nanoliquid than in the bi-viscous Bingham hybrid nanoliquid. The use of Single-walled carbon nanotubes enhances the heat transfer rate by approximately 17%. Further, it is concluded that a tall cylindrical enclosure is the most favorable geometry for achieving a higher heat transfer rate among the others.

Theoretical study and circuit implementation of three chain-coupled self-driven Duffing oscillators.

In this paper, we describe the scenario from the birth of oscillations to multi-spiral chaos in a novel system composed of three chain-coupled self-driven Duffing oscillators. Eight of the equilibrium points develop (multiple) Hopf bifurcation when varying a parameter (e.g., coupling coefficient). Considering the computer integration of the state equations, the combined exploitation of Lyapunov exponent plots, bifurcation diagrams, basins of attraction, and phase portraits, unusual and attractive features were highlighted including the coexistence of eight bifurcation branches, Hopf bifurcations, a multitude of coexisting types of oscillations and a six-spiral chaotic attractor, just to cite a few. Using basic electronic components, the electronic circuit of the three chain-coupled Duffing oscillator system is performed. Orcad-PSpice simulated dynamics of the proposed chain-coupled analog circuit confirm the theoretically disclosed features. Moreover, the practical feasibility of the coupled system is demonstrated by considering microcontroller-based hardware realization.

Coexistence behavior of asymmetric attractors in hyperbolic-type memristive Hopfield neural network and its application in image encryption

The neuron model has been widely employed in neural-morphic computing systems and chaotic circuits. This study aims to develop a novel circuit simulation of a three-neuron Hopfield neural network (HNN) with coupled hyperbolic memristors through the modification of a single coupling connection weight. The bistable mode of the hyperbolic memristive HNN (mHNN), characterized by the coexistence of asymmetric chaos and periodic attractors, is effectively demonstrated through the utilization of conventional nonlinear analysis techniques. These techniques include bifurcation diagrams, two-parameter maximum Lyapunov exponent plots, local attractor basins, and phase trajectory diagrams. Moreover, an encryption technique for color images is devised by leveraging the mHNN model and asymmetric structural attractors. This method demonstrates significant benefits in correlation, information entropy, and resistance to differential attacks, providing strong evidence for its effectiveness in encryption. Additionally, an improved modular circuit design method is employed to create the analog equivalent circuit of the memristive HNN. The correctness of the circuit design is confirmed through Multisim simulations, which align with numerical simulations conducted in Matlab.

From signals to music: a bottom-up approach to the structure of neuronal activity.

The search for the "neural code" has been a fundamental quest in neuroscience, concerned with the way neurons and neuronal systems process and transmit information. However, the term "code" has been mostly used as a metaphor, seldom acknowledging the formal definitions introduced by information theory, and the contributions of linguistics and semiotics not at all. The heuristic potential of the latter was suggested by structuralism, which turned the methods and findings of linguistics to other fields of knowledge. For the study of complex communication systems, such as human language and music, the necessity of an approach that considers multilayered, nested, structured organization of symbols becomes evident. We work under the hypothesis that the neural code might be as complex as these human-made codes. To test this, we propose a bottom-up approach, constructing a symbolic logic in order to translate neuronal signals into music scores. We recorded single cells' activity from the rat's globus pallidus pars interna under conditions of full alertness, blindfoldedness and environmental silence. We analyzed the signals with statistical, spectral, and complex methods, including Fast Fourier Transform, Hurst exponent and recurrence plot analysis. The results indicated complex behavior and recurrence graphs consistent with fractality, and a Hurst exponent >0.5, evidencing temporal persistence. On the whole, these features point toward a complex behavior of the time series analyzed, also present in classical music, which upholds the hypothesis of structural similarities between music and neuronal activity. Furthermore, through our experiment we performed a comparison between music and raw neuronal activity. Our results point to the same conclusion, showing the structures of music and neuronal activity to be homologous. The scores were not only spontaneously tonal, but they exhibited structure and features normally present in human-made musical creations. The hypothesis of a structural homology between the neural code and the code of music holds, suggesting that some of the insights introduced by linguistic and semiotic theory might be a useful methodological resource to go beyond the limits set by metaphoric notions of "code."

Design of a New Chaotic System with Sine Function: Dynamic Analysis and Offset Boosting Control

A new chaotic system is presented in this research work.The proposed system has three nonlinear terms and one sine term which improves the complexity of the system. The basic properties of new system such as Lyapunov exponent, equilibrium point and stability are analyzed in detail. The dynamic analysis is conducted using classic tools such as bifurcation diagram and Lyapunov exponent plot to verify the chaotic nature in the proposed system. The changes in the states of the system is verified using bifurcation diagram and Lyapunov exponent plot. The proposed system presents some special features such as two wing attractors, forward and reverse periodic doubling bifurcation, and dc offset boosting control. The dc offset boosting behavior can be used to diagnosis the multistability behaviour in the dynamical system and to reduce the number of components in the communication system. This special feature converts the bipolar signal in to unipolar signal which can be used in many engineering applications. The theoretical study and the simulation results show that the proposed system has wealthy chaotic behaviour itself. Furthermore, the adaptive synchronization of identical new system is achieved for the application of secure communication system.

Biomedical Image Encryption with a Novel Memristive Chua Oscillator Embedded in a Microcontroller

In this paper, a chaos-based encryption/decryption scheme using a novel memristive Chua oscillator to protect medical images is presented. The novel Chua oscillator is constructed by using the active voltage memristor in the nonlinear branch of the Chua oscillator. The chaotic dynamics behaviors are investigated using 1-D, 2-D bifurcation diagrams, time traces, basin of attractions, and largest Lyapunov exponent plot. The study reveals that the novel memristive Chua oscillator exhibits versatile transitions to chaos with interesting dynamics like multistability, spiking, and bursting oscillations just to name a few. These remarkable features are experimentally confirmed by a laboratory microcontroller-based setup. Thereafter, a chaos-based cryptography algorithm designed for biomedical images is built using pseudorandom number generated from the oscillator. The robustness and security tests undergone by the algorithm yielded high sensitivity on the encryption keys and resisted noise contamination as well as data loss. These results are encouraging and prove that the chaos-based cryptosystem built with the memristive Chua circuit–generated pseudorandom number is suitable for securing images in a healthcare system.

The Deterministic Nature of Sensor-Based Information for Condition Monitoring of the Cutting Process

Condition monitoring of the cutting process is a core function of autonomous machining and its success strongly relies on sensed data. Despite the enormous amount of research conducted so far into condition monitoring of the cutting process, there are still limitations given the complexity underlining tool wear; hence, a clearer understanding of sensed data and its dynamical behavior is fundamental to sustain the development of more robust condition monitoring systems. The dependence of these systems on acquired data is critical and determines the success of such systems. In this study, data is acquired from an experimental setup using some of the commonly used sensors for condition monitoring, reproducing realistic cutting operations, and then analyzed upon their deterministic nature using different techniques, such as the Lyapunov exponent, mutual information, attractor dimension, and recurrence plots. The overall results demonstrate the existence of low dimensional chaos in both new and worn tools, defining a deterministic nature of cutting dynamics and, hence, broadening the available approaches to tool wear monitoring based on the theory of chaos. In addition, recurrence plots depict a clear relationship to tool condition and may be quantified considering a two-dimensional structural measure, such as the semivariance. This exploratory study unveils the potential of non-linear dynamics indicators in validating information strength potentiating other uses and applications.

FRACTIONAL QUANTUM LOGISTIC MAP AND ITS APPLICATION IN IMAGE ENCRYPTION TECHNOLOGY

In this paper, a novel fractional quantum logistic map (FQLM) based on fractional difference is proposed and image encryption algorithm based on it are exploited. First, the FQLM is proposed and its dynamical behaviors are observed with bifurcation diagram, phase portraits and largest Lyapunov exponent plots. After that, the ECC is utilized to cover up the encryption keys and the FQLM is used for the permutation and diffusion process of image encryption. Finally, the cryptosystem is used in image encryption and the algorithm is analyzed in four respects, showing a distinct advantage of the proposed cryptosystem over some of existing ones.

Four-Scroll Hyperchaotic Attractor in a Five-Dimensional Memristive Wien Bridge Oscillator: Analysis and Digital Electronic Implementation

An electronic implementation of a novel Wien bridge oscillation with antiparallel diodes is proposed in this paper. As a result, we show by using classical nonlinear dynamic tools like bifurcation diagrams, Lyapunov exponent plots, phase portraits, power density spectra graphs, time series, and basin of attraction that the oscillator transition to chaos is operated by intermittency and interior crisis. Some interesting behaviors are found, namely, multistability, hyperchaos, transient chaos, and bursting oscillations. In comparison with some memristor-based oscillators, the plethora of dynamics found in this circuit with current-voltage (i–v) characteristic of diodes mounted in the antiparallel direction represents a major advance in the knowledge of the behavior of this circuit. A suitable microcontroller based design is built to support the numerical findings as these experimental results are in good agreement.

Morphology identification of gas hydrate from pointwise Lipschitz regularity for P- and S-wave velocity

The accurate morphology identification of gas hydrate-bearing sediments (GHBS) has great significance in practical exploitation and subsequent resource evaluation. Previous studies have disclosed two main morphologies for gas hydrate in sediments: pore- and fracture-filling. However, the existing identification methods of gas hydrate’s morphology rarely consider their intrinsic differences in distribution characteristics. In this paper, a new method is proposed to identify the morphology of hydrate according to the scattered distribution of fracture dips for fracture-filling GHBS. Firstly, numerical simulations are performed to study the relationships between the morphology of hydrate and the sonic velocities. Considering the dip variation is within a certain range for fracture-filling hydrate, the theoretical curves show that the resulting mutation degrees between P- and S-wave velocities are inconsistent in fracture-filling GHBS, which is different from pore-filling GHBS. Then the modified estimation method for pointwise Lipschitz exponent α is introduced to capture their differences. The cross plots of Lipschitz exponent for P-wave velocity, α(Vp), and Lipschitz exponent for S-wave velocity, α(Vs), indicate that most of the dots representing pore-filling GHBS are evenly distributed near the line α(Vp)=α(Vs), while the dots representing fracture-filling GHBS are scattered outside the line α(Vp)=α(Vs). Based on these characteristics, a ratio method is put forward to differentiate the two types of hydrate. These hypotheses and methods are verified using the measured P- and S-wave velocities log data at different sites in Leg 204, Ocean Drilling Program (ODP), in the United States. Finally, the results of this new method agree closely with core data.

IMAGE ENCRYPTION BASED ON TWO-DIMENSIONAL FRACTIONAL QUADRIC POLYNOMIAL MAP

A new fractional two-dimensional quadric polynomial discrete chaotic map (2D-QPDM) with the discrete fractional difference is proposed. Afterwards, the new dynamical behaviors are observed, so that the bifurcation diagrams, the largest Lyapunov exponent plot and the phase portraits of the proposed map are given, respectively. The new discrete fractional map is exploited into color image encryption algorithm and it is illustrated with several examples. The proposed image encryption algorithm is analyzed in six aspects which indicates that the proposed algorithm is superior to other known algorithms as a conclusion.

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.

IMAGE ENCRYPTION TECHNOLOGY BASED ON FRACTIONAL TWO-DIMENSIONAL DISCRETE CHAOTIC MAP ACCOMPANIED WITH MENEZES-VANSTONE ELLIPTIC CURVE CRYPTOSYSTEM

A new fractional two-dimensional triangle function combination discrete chaotic map (2DTFCDM) with the discrete fractional difference is proposed in this paper. The chaos behaviors are observed through the bifurcation diagrams, the largest Lyapunov exponent plot and the phase portraits. The proposed map is applied in color image encryption with the secret keys generated by Menezes–Vanstone Elliptic Curve Cryptosystem. The image encryption system is analyzed using six aspects indicating the superiority of the proposed algorithm compared to the other algorithms.

The Effects of a Constant Excitation Force on the Dynamics of an Infinite-Equilibrium Chaotic System Without Linear Terms: Analysis, Control and Circuit Simulation

In this paper, the effects of a bias term modeling a constant excitation force on the dynamics of an infinite-equilibrium chaotic system without linear terms are investigated. As a result, it is found that the bias term reduces the number of equilibrium points (transition from infinite-equilibria to only two equilibria) and breaks the symmetry of the model. The nonlinear behavior of the system is highlighted in terms of bifurcation diagrams, maximal Lyapunov exponent plots, phase portraits, and basins of attraction. Some interesting phenomena are found including, for instance, hysteretic dynamics, multistability, and coexisting bifurcation branches when monitoring the system parameters and the bias term. Also, we demonstrate that it is possible to control the offset and amplitude of the chaotic signals generated. Compared to some few cases previously reported on systems without linear terms, the plethora of behaviors found in this work represents a unique contribution in comparison with such type of systems. A suitable analog circuit is designed and used to support the theoretical analysis via a series of Pspice simulations.

Hopf Bifurcation Analysis and Existence of Heteroclinic Orbit and Homoclinic Orbit in an Extended Lorenz System

In this paper, we have considered a Lorenz-like model with slight changes in the nonlinear terms. Here we have studied the system dynamics for different range of values of parameters $$\sigma , r$$ . The Hopf bifurcation analysis of the system has been done using center manifold theorem for $$\sigma = -1, r > 1$$ . Phase portraits of solutions of the system are plotted for various system parameters to substantiate the change in dynamics. The bifurcation diagram and the Lyapunov exponent evaluation plots also help to explain the behaviour of the system. Using Fishing principle, we have shown the existence of homoclinic orbit and consequently, observed the existence of homoclinic as well as heteroclinic orbits in the numerical simulation for $$\sigma> 0, r > 1$$ .

Complex bifurcation analysis and synchronization optimal control for Hindmarsh-Rose neuron model under magnetic flow effect.

In this contribution, the complex behaviour of the Hindmarsh-Rose neuron model under magnetic flow effect (mHR) is investigated in terms of bifurcation diagrams, Lyapunov exponent plots and time series when varying only the electromagnetic induction strength. Some exciting phenomena are found including, for instance, various firings patterns by applying appropriate magnetic strength and Hopf-fold bursting through fast-slow bifurcation. In addition to this, the interesting phenomenon of Hopf bifurcation is examined in the model. Thus, we prove that Hopf bifurcation occurs in this memristor-based HR neuron model when an appropriately chosen magnetic flux varies and reaches its critical value. Furthermore, one of the main results of this work was the optimal control approach to realize the synchronization of two mHR. The main advantage of the proposed optimal master-slave synchronization from a control point of view is that, in the practical application, the electrical activities (quiescent, bursting, spiking, period and chaos states) of a neuron can be regulated by a pacemaker (master) associated with biological neuron (slave) to treat some diseases such as epilepsy. A suitable electronic circuit is designed and used for the investigations. PSpice based simulation results confirm that the electrical activities and synchronization between coupled neurons can be modulated by electromagnetic flux.

Complex dynamics of a novel 3D autonomous system without linear terms having line of equilibria: coexisting bifurcations and circuit design

In recent years, an increasing interest has been devoted to the discovery of new chaotic systems with special properties. In this paper, a novel and singular 3D autonomous system without linear terms is introduced. The singularity of the model is that it is dissipative, possesses rotation symmetry and line of equilibria thus displays complex dynamics. The nonlinear behaviour of the introduced model is studied in terms of bifurcation diagrams, Lyapunov exponent plots, time series, frequency spectra two-parameter diagrams, Lyapunov stability diagrams as well as basins of attraction. Some interesting phenomena are found including, for instance, periodic oscillations, chaotic oscillations, periodic windows, symmetry restoring crises scenario, the coexistence of multiple bifurcations and offset-boosting property while monitoring the system parameters. Coexistence of attractors discovered in this work includes up to six competing attractors with one point attractor. Compared to some few cases previously reported (system without linear terms) (Xu and Wang in Opt Int J Light Electron Opt 125:2526–2530, 2014. https://doi.org/10.1007/s11071-016-3170-x; Kengne in Commun Nonlinear Sci Numer Simul 52:62–76, 2017. https://doi.org/10.1016/j.cnsns.2017.04.017; Mobayen et al. in Int J Syst Sci 49:1–15, 2018. https://doi.org/10.1080/00207721.2017.1410251; Zhang et al. in Int J Non-linear Mech 106:1–12, 2018. https://doi.org/10.1016/j.ijnonlinmec.2018.08.012; Pham et al. in Chaos Solitons Fractals 120:213–221, 2019. https://doi.org/10.1016/j.chaos.2019.02.003), the model considered in this work is the only one in which such type of complex dynamics has already been found. A suitable analog simulator (electronics circuit) is designed and used to support the theoretical analysis.