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We report the appearance of supertransient chaos in a single and two-coupled Liénard system with the influence of external periodic force. The existence of transient dynamics in a model is significantly long before it settles into the asymptotic steady state of periodic dynamics understood as supertransient chaos. The two diffusively coupled forced Liénard systems exhibit extremely long transient dynamics when their frequencies of the external forcing are slightly mismatched. Additionally, the coupled system signifies supertransient hyperchaotic dynamics for a specific set of system parameters. This study involves different numerical characterizations, statistical analysis, and hardware implementation using an analog electronic circuit. [ABSTRACT FROM AUTHOR]

We present a hyperchaotic oscillator with two linear terms and seven nonlinear terms that displays special algebraic properties. Notably, the introduced oscillator features distinct equilibrium types: single-point, line, and spherical equilibria. The introduced oscillator exhibits attractive dynamics like hyperchaos with two wing attractors. To gain a better understanding, we provide the bifurcation and Lyapunov exponents. The Kolmogorov–Sinai entropy is applied to show the complexity of the oscillator. A microcontroller realization confirms the reliability of the oscillator. The proposed oscillator is successfully applied for RNG. [ABSTRACT FROM AUTHOR]

In this paper, a new 2D discrete chaotic system is proposed, which is called two-dimensional closed-loop modulation map with infinite collapse (2D-CLMIC). The dynamic analyses show that the 2D-CLMIC system has a large parameter space for hyper chaos, when the parameter θ = 0.01 , the trajectory can occupy all phase plane which is beneficial for application in cryptography. Based on the 2D-CLMIC system, a novel encryption scheme is proposed. In this scheme, we first generate the initial values of 2D-CLMIC by global plaintext related method to ensure the image cryptosystem has enough plaintext sensitivity. Then, a pixel level permutation is done by a control matrix which is generated by the iteration sequence of 2D-CLMIC. After that, random templates which are used for overlapping diffusion are generated by the iteration sequence of 2D-CLMIC. In addition, a templates select control sequences, x-coordinate and y-coordinate select control sequences are generated by two other 2D-CLMIC systems. Finally, do the random overlapping diffusion and output the cipher image. Furthermore, simulation and analysis are given to verify the proposed image encryption system has an acceptable security performance. [ABSTRACT FROM AUTHOR]

Abstract This paper describes a novel 4-D hyperchaotic system with a high level of complexity. It can produce chaotic, hyperchaotic, periodic, and quasi-periodic behaviors by adjusting its parameters. The study showed that the new system experienced the famous dynamical property of multistability. It can exhibit different coexisting attractors for the same parameter values. Furthermore, by using Lyapunov exponents, bifurcation diagram, equilibrium points’ stability, dissipativity, and phase plots, the study was able to investigate the dynamical features of the proposed system. The mathematical model’s feasibility is proved by applying the corresponding electronic circuit using Multisim software. The study also reveals an interesting and special feature of the system’s offset boosting control. Therefore, the new 4D system is very desirable to use in Chaos-based applications due to its hyperchaotic behavior, multistability, offset boosting property, and easily implementable electronic circuit. Then, the study presents a voice encryption scheme that employs the characteristics of the proposed hyperchaotic system to encrypt a voice signal. The new encryption system is implemented on MATLAB (R2023) to simulate the research findings. Numerous tests are used to measure the efficiency of the developed encryption system against attacks, such as histogram analysis, percent residual deviation (PRD), signal-to-noise ratio (SNR), correlation coefficient (cc), key sensitivity, and NIST randomness test. The simulation findings show how effective our proposed encryption system is and how resilient it is to different cryptographic assaults.

The design of chaotic systems with complex dynamic behaviors has always been a key aspect of chaos theory in engineering applications. This study introduces a novel fractional-order system characterized by hidden dynamics, hyperchaotic behavior, and multi-scroll attractors. By employing fractional calculus, the system's order is extended beyond integer values, providing a richer dynamic behavior. The system's hidden dynamics are revealed through detailed numerical simulations and theoretical analysis, demonstrating complex attractors and bifurcations. The hyperchaotic nature of the system is verified through Lyapunov exponents and phase portraits, showing multiple positive exponents that indicate a higher degree of unpredictability and complexity. Additionally, the system's multi-scroll attractors are analyzed, showcasing their potential for secure communication and encryption applications. The fractional-order approach enhances the system's flexibility and adaptability, making it suitable for a wide range of practical uses, including secure data transmission, image encryption, and complex signal processing. Finally, based on the proposed fractional-order system, we designed a simple and efficient medical image encryption scheme and analyzed its security performance. Experimental results validate the theoretical findings, confirming the system's robustness and effectiveness in generating complex chaotic behaviors. [ABSTRACT FROM AUTHOR]

Chaotic systems, characterized by complex nonlinear dynamical behaviors, are suitable for generating abundant key streams for digital image encryption. Nevertheless, some encryption systems suffer from weak key space and inadequate key randomness due to a narrow chaotic parameter range and small Lyapunov exponents. Additionally, we must address the crucial issue of keys lacking or weakly relating to plaintext, as this can hinder the algorithm's ability to effectively defend against differential attacks. This paper proposes a hybrid one-dimensional and two-dimensional cross-feedback hyperchaotic system framework to enhance key security. We combine the system with a diffusive DNA coding operation to enhance the statistical relationship between plaintext and ciphertext. Firstly, our method associates the current pixel of plaintext images with hyperchaotic sequences to determine the DNA encoding rule for the next pixel. Secondly, we transform ordinary images into DNA matrices through DNA-coding operations with diffusivity. Thirdly, we rearrange the hyperchaotic sequence to change the position of the corresponding DNA matrix elements during scrambled diffusion. Finally, we XOR it with the DNA-encoded hyperchaotic sequence to generate the ciphertext image. The experimental results demonstrate that the performance of our proposed algorithm has significantly improved in terms of information entropy, correlation, histogram, and other analytical parameters. Additionally, the algorithm exhibits a notably robust resistance to differential attacks. [ABSTRACT FROM AUTHOR]

Neuromorphic computing draws inspiration from the brain to design energy-efficient hardware for information processing, enabling highly complex tasks. In neuromorphic computing, chaotic phenomena describe the nonlinear interactions and dynamic behaviors. Chaotic behavior can be utilized in neuromorphic computing to accomplish complex information processing tasks; therefore, studying chaos is crucial. Today, more and more color images are appearing online. However, the generation of numerous images has also brought about a series of security issues. Ensuring the security of images is crucial. We propose a novel fourth-direction hyperchaotic system in this paper. In comparison to low-dimensional chaotic systems, the proposed hyperchaotic system exhibits a higher degree of unpredictability and various dynamic behaviors. The dynamic behaviors include fourth-direction hyperchaos, third-direction hyperchaos, and second-direction hyperchaos. The hyperchaotic system generates chaotic sequences. These chaotic sequences are the foundation of the encryption scheme discussed in this paper. Images are altered by employing methods such as row and column scrambling as well as diffusion. These operations will alter both the pixel values and positions. The proposed encryption scheme has been analyzed through security and application scenario analyses. We perform a security analysis to evaluate the robustness and weaknesses of the encryption scheme. Moreover, we conduct an application scenario analysis to help determine the practical usability and effectiveness of the encryption scheme in real-world situations. These analyses demonstrate the efficiency of the encryption scheme. [ABSTRACT FROM AUTHOR]

This paper aims to propose a material-independent novel locally-active memristor (LAM) model, which is applied to the fourth-order autonomous chaotic oscillation circuit to investigate the phenomena and mechanisms of chaos and hyperchaos. Under different parameter configurations, the LAM-based system can exhibit rich dynamic behaviours and multistability, such as multi-equilibrium points, period, chaos and hyperchaos attractors. The phase diagram, bifurcation diagram, Lyapunov exponential spectrum, basins of attraction and dynamics map are used to analyse the complex dynamics of the system. In addition, we find many types of coexisting attractors, including periodic attractors with multiple different periods, hyperchaotic attractors with multiple different scrolls, a double-scroll chaotic attractor with a shock wave orbit and so on. This work fills the gap by theoretical analysis and numerical simulation. And proves that the local activity and non-volatility of memristors are two important reasons for the generation of complex and coexisting attractors. [ABSTRACT FROM AUTHOR]

Inspired by basic circuit connection methods, memristors can also be utilized in the construction of complex discrete chaotic systems. To investigate the dynamical effects of hybrid memristors, we propose two hybrid tri-memristor hyperchaotic (HTMH) mapping structures based on the hybrid parallel/cascade and cascade/parallel operations, respectively. Taking the HTMH mapping structure with hybrid parallel/cascade operation as an example, this map possesses a spatial invariant set whose stability is closely related to the initial states of the memristors. Dynamics distributions and bifurcation behaviours dependent on the control parameters are explored with numerical tools. Specifically, the memristor initial offset-boosting mechanism is theoretically demonstrated, and memristor initial offset-boosting behaviours are numerically verified. The results clarify that the HTMH map can exhibit hyperchaotic behaviours and extreme multistability with homogeneous coexisting infinite attractors. In addition, an FPGA hardware platform is fabricated to implement the HTMH map and generate pseudorandom numbers (PRNs) with high randomness. Notably, the generated PRNs can be applied in Wasserstein generative adversarial nets (WGANs) to enhance training stability and generation capability. [ABSTRACT FROM AUTHOR]

Memristor is a nonlinear electronic component with good plasticity, and it is widely used to function as the synapse. Significantly, the inter-neuronal communication is a discrete process via neurotransmitter, and the discrete memristor is suitable to emulate its dynamics. With the introduction of discrete memristors, the discrete neural models coupled with memristors have received widespread attention. In this paper, we present a one-dimensional (1D) Chialvo map, and a discrete memristor-coupled Chialvo neuron (DMCCN) map is designed by bidirectionally coupling two 1D Chialvo maps through a discrete memristor. The DMCCN map exhibits a linear fixed point set dependent on the initial condition of the memristor, and the stability is discussed. The dynamical behaviors are investigated through Lyapunov exponents (LEs), bifurcation diagrams, phase portraits, and firing patterns. The results indicate that the DMCCN map generates complex dynamical behaviors, such as chaos, hyperchaos and coexisting firing patterns. Specifically, these behaviors are highly dependent on the initial conditions, leading to initial-induced heterogeneous multistability and initial-boosted homogeneous multistability. Furthermore, a hardware circuit based on the DSP platform is designed to implement the DMCCN map, verifying its application potential. [ABSTRACT FROM AUTHOR]

Fractional calculus is an effective method used to analyze the dynamics of nonlinear systems and provide more precise results. In this study, firstly, the 4-dimensional Pang system is introduced and its dynamic analyses demonstrating the hyperchaotic structure are given. Then, fractional-order calculations of the system are presented and the dynamics of the system for different fraction orders are investigated. At this point, according to the results obtained from Lyapunov exponents and phase-space representation, the Pang system exhibits periodic, chaotic, and hyperchaotic behaviors in different fractional orders. The results obtained at the end of this study present that the system is hyperchaotic for the fractional order of 3.52 and it is also confirmed that more accurate results are obtained than the integer-order analysis. In the next part of the study, adaptive synchronization of the fractional-order system is performed. Three different cases are examined and it is demonstrated that synchronization is achieved in all cases.

We consider the rich dynamics and bifurcations exhibited by a three-dimensional quadratic map. Torus doubling bifurcations are central to bifurcation theory. Such bifurcations can only occur in a higher dimensional map of dimensions greater than or equal to three. After subsequent doublings, formation of Shilnikov attractors and hyperchaotic attractors are usually observed. It is shown that the map under consideration shows both resonant and ergodic torus doubling bifurcation. We show that the system exhibits resonant torus doubling bifurcation in which the doubled mode-locked periodic orbits lies on a Möbius strip. Additionally, we also illustrate the doubling of ergodic tori and analyze the bifurcation via the use of the second Poincaré section and the method of Lyapunov bundles. The analysis involves the techniques of construction of one-dimensional manifolds, a one-parameter continuation of saddle periodic orbits, and multi-dimensional Newton–Raphson method to locate the saddle periodic orbits. [ABSTRACT FROM AUTHOR]

This paper studies the dynamics and integrability of a variable-length coupled pendulum system. The complexity of the model is presented by joining various numerical methods, such as the Poincaré cross sections, phase-parametric diagrams, and Lyapunov exponents spectra. We show that the presented model is hyperchaotic, which ensures its nonintegrability. We gave analytical proof of this fact analyzing properties of the differential Galois group of variational equations along certain particular solutions of the system. We employ the Kovacic algorithm and its extension to dimension four to analyze the differential Galois group. Amazingly enough, in the absence of the gravitational potential and for certain values of the parameters, the system can exhibit chaotic, integrable, as well as superintegrable dynamics. To the best of our knowledge, this is the first attempt to use the method of Lyapunov exponents in the systematic search for the first integrals of the system. We show how to effectively apply the Lyapunov exponents as an indicator of integrable dynamics. The explicit forms of integrable and superintegrable systems are given. [ABSTRACT FROM AUTHOR]

A new 4-D two-scroll hyperchaotic system with three quadratic nonlinearities is detailed in this research paper. It is shown that the new hyperchaotic system has a unique saddle-point rest point at the origin and it exhibits self-excited attractor. A detailed stability analysis is carried out for the new hyperchaotic system.It is also proved that the new hyperchaotic system has multistability with coexisting attractors. Next, we invoke integral sliding mode control for the global self-synchronization of the new hyperchaotic system taken as leader-follower systems.An electronic circuit design of the new two-scroll hyperchaotic system is developed in MultiSim, which is beneficial for engineering applications. [ABSTRACT FROM AUTHOR]

Firstly, a novel six-dimensional (6D) hyperchaotic particle motion system is formulated. The equilibrium points and their characteristics, Poincaré sections, Lyapunov exponents, bifurcations and multi-periodic windows are studied. Secondly, we present two nonholonomic constrained systems. In order to analyze the particle motion trajectories under constraints, the explicit equations for constrained systems are given. Based on Lyapunov exponents, Poincaré maps and bifurcations, we can see that the different hyperchaotic phenomena of the particle motion can be generated by introducing nonholonomic constraints. Finally, the stability control of the 6D hyperchaotic particle motion system is realized by separately using constraint control method and linear feedback control method. Numerical simulations of the dynamical behaviors of the six-dimensional hyperchaotic particle motion system are carried out in order to illustrate the complex phenomena of the systems and verify the analysis results. [ABSTRACT FROM AUTHOR]

Comparing with dissipative systems, conservative systems have distinguished advantages in information processing and secure communication. It is of practical importance and theoretical significance to design conservative chaotic systems based on memristors due to their special features of complexity and flexibility. In this paper, a novel conservative hyperchaotic memristor system is proposed. The rich dynamics of the system, including extreme multistability and hyperchaos, are analyzed by using phase portraits, time series, bifurcation diagrams and Lyapunov exponents, and confirming the system is conservative. Based on the Hamiltonian theory, a specific energy function of the system is constructed and the generation mechanism of the extreme multistability is revealed and analyzed. Interestingly, a special heart-shaped attractor is found from the system. Finally, the theoretical results are verified and demonstrated through physical circuit implementation, demonstrating its potential for future applications. [ABSTRACT FROM AUTHOR]

Image encryption is essential for protecting sensitive visual data during transmission and storage. This article introduces a novel image encryption technique that utilizes three hyperchaotic systems: a memristor system, a hyperchaotic 7D system, and an Erbium-doped fiber laser-based system. The algorithm takes advantage of the complex dynamics and random-like behavior of these systems to implement a robust three-layer encryption process. In each layer, a pseudo-random number generator is used to construct an S-box and an encryption key, which are subsequently applied to the image data. The proposed encryption scheme is implemented in software using Wolfram Mathematica® version 13.1 and in hardware using an AMD Virtex-7 FPGA VC707, achieving encryption rates of 9.7 Mbps and 0.67 Gbps, respectively. Experimental results demonstrate the scheme’s effectiveness in terms of encryption quality, resistance to various cryptanalysis attacks, and a high computational efficiency. The proposed method shows significant potential for secure image transmission and storage applications.

This paper is focused on nonlinear behavior associated with fourth-order autonomous lumped circuit known as the Reinartz oscillator. It is shown that this naturally sinusoidal oscillator with three-winding transformer can exhibit robust chaotic behavior. Essential scalar polynomial nonlinearity is not intrinsic to transformer, but to active two-port element described by impedance parameters. The existence of complex solutions is proved on numerical as well as experimental basis. In the first case, conventional routines for qualitative analysis of dynamical flows were applied, based on either prescribed set of differential equations or generated data sequence. In detail, a pair of the largest Lyapunov exponents are visualized with respect to key system parameters, attractor dimensions, recurrence plots and bifurcation diagrams showing interesting routes-to-chaos scenarios are provided to illustrate complexity of observed steady state motion. For the second case, electronic circuits dynamically equivalent to investigated mathematical model will be designed, constructed, and experimentally measured. Chaotic signals captured as oscilloscope screenshots will be compared to theoretical counterparts.

We present a hyperchaotic oscillator with two linear terms and seven nonlinear terms that displays special algebraic properties. Notably, the introduced oscillator features distinct equilibrium types: single-point, line, and spherical equilibria. The introduced oscillator exhibits attractive dynamics like hyperchaos with two wing attractors. To gain a better understanding, we provide the bifurcation and Lyapunov exponents. The Kolmogorov–Sinai entropy is applied to show the complexity of the oscillator. A microcontroller realization confirms the reliability of the oscillator. The proposed oscillator is successfully applied for RNG.

We report the appearance of supertransient chaos in a single and two-coupled Liénard system with the influence of external periodic force. The existence of transient dynamics in a model is significantly long before it settles into the asymptotic steady state of periodic dynamics understood as supertransient chaos. The two diffusively coupled forced Liénard systems exhibit extremely long transient dynamics when their frequencies of the external forcing are slightly mismatched. Additionally, the coupled system signifies supertransient hyperchaotic dynamics for a specific set of system parameters. This study involves different numerical characterizations, statistical analysis, and hardware implementation using an analog electronic circuit.

Stabilization by a periodic pulsed force of trajectories running away to infinity in the three-dimensional Rössler system at a threshold of a saddle-node bifurcation, birth of equilibrium states is studied. It is shown that the external pulsed action stabilizes dynamical regimes in a fairly wide range of external signal parameters. Stabilized regimes can be periodic, quasi-periodic, or chaotic. Three types of chaotic oscillations are revealed depending on the spectrum of Lyapunov exponents: the simplest (classical) and multi-dimensional (hyperchaos and chaos with additional zero Lyapunov exponent). Scenarios for the development of multi-dimensional chaos have been studied in detail. The universality of the observed behavior when changing the direction of the external force is investigated. It is shown that the effects are universal, when force acts in a plane corresponding to focal behavior of the trajectories, stabilization is not observed if the direction of force is perpendicular to the plane. The universality of the obtained picture is studied when the autonomous dynamics of the model change, it is shown that for small periods of external action, the picture is determined by a transient processes of autonomous model and remains characteristic for stabilization. With an increase in the period of external force, the properties of an autonomous model appear. [ABSTRACT FROM AUTHOR]

Unlike the high-dimensional hyperchaotic system based on a continuous memristor, the low-dimensional map coupled by discrete memristor (DM) and traditional chaotic map can also generate hyperchaos. However, the hyperchaotic map constructed by two DMs has not attracted much attention. To this end, a generalized two-dimensional dual DM-coupled hyperchaotic mapping model is reported in this paper, and four specific maps are provided. The proposed maps have line invariant points, which can be interpreted as allowing arbitrary real values for the initial condition associated with the DM, and the stability is investigated in detail. Furthermore, the coupling strength-dependent and initial condition-dependent complex dynamics of four maps are studied by numerical simulations, and the dynamical performance is evaluated from the perspective of quantitative analysis. It is shown that the considered maps are capable of exhibiting the three characteristic fingerprints of memristors in arbitrary parameter spaces, and this characteristic has gained attention for the first time. In particular, the complete control of the considered maps by variable substitution is performed, which can generate arbitrary switched hyperchaotic behaviors. In addition, four pseudo-random number generators are designed based on the proposed maps, and the randomness is tested by using the NIST SP800-22 software. In general, the proposed maps can not only generate abundant dynamical behaviors, but also enrich the DM circuits and provide a reference for applications based on chaos. Finally, the developed digital hardware circuit implementation platform verifies the results of the numerical method. [ABSTRACT FROM AUTHOR]

Fractional calculus is an effective method used to analyze the dynamics of nonlinear systems and provide more precise results. In this study, firstly, the 4-dimensional Pang system is introduced and its dynamic analyses demonstrating the hyperchaotic structure are given. Then, fractional-order calculations of the system are presented and the dynamics of the system for different fraction orders are investigated. At this point, according to the results obtained from Lyapunov exponents and phase-space representation, the Pang system exhibits periodic, chaotic, and hyperchaotic behaviors in different fractional orders. The results obtained at the end of this study present that the system is hyperchaotic for the fractional order of 3.52 and it is also confirmed that more accurate results are obtained than the integer-order analysis. In the next part of the study, adaptive synchronization of the fractional-order system is performed. Three different cases are examined and it is demonstrated that synchronization is achieved in all cases. [ABSTRACT FROM AUTHOR]

The dynamics of two coupled neuron models, the Hindmarsh – Rose systems, are studied. Their interaction is simulated via a chemical coupling that is implemented with a sigmoid function. It is shown that the model may exhibit complex behavior: quasi-periodic, chaotic and hyperchaotic oscillations. A phenomenological scenario for the formation of hyperchaos associated with the appearance of a discrete Shilnikov attractor is described. It is shown that the formation of these attractors leads to the appearance of in-phase bursting oscillations. [ABSTRACT FROM AUTHOR]

With the combined call for increased network throughput and security comes the need for high-bandwidth, unconditionally secure systems. Through the combination of true random number generators (TRNGs) for unique seed values, and four-dimensional Lorenz hyperchaotic systems implemented on a Stratix 10 Intel FPGA, we are able to implement 60 MB/s encryption/decryption schemes with 0% data loss on an unconditionally secure system with the NIST standard using less than 400 mW. Further, the TRNG implementation allows for unique encryption outputs for similar images while still enabling proper decryption. Histogram and adjacent pixel analysis on sample images demonstrate that without the key, it is not possible to extract the plain text from the encrypted image. This encryption scheme was implemented via PCIe for testing and analysis. [ABSTRACT FROM AUTHOR]

We study the complex dynamics of a discrete analogue of the classical flow dynamical system — Rössler oscillator. Minimal ensembles of two and three coupled discrete oscillators with different topologies are considered. As the main research tool we used the method of Lyapunov exponents charts. For coupled systems, the possibility of two-, three- and four-frequency quasi-periodicity is revealed. Illustrations in the form of Fourier spectra are presented. Doublings of invariant curves, two- and three-dimensional tori are found. The transition from two-dimensional tori to three-dimensional ones occurs through a quasi-periodic saddle-node bifurcation of invariant tori or through a quasi-periodic Hopf bifurcation. A discrete version of the hyperchaotic Rössler oscillator is also discussed. It exhibits dynamical behavior close to a flow system in some measure. [ABSTRACT FROM AUTHOR]

This paper contributes to the problem of chaos and hyperchaos localization in the fundamental structure of analog building blocks dedicated to single-tone harmonic signal generation. This time, the known Reinartz sinusoidal oscillator is addressed, considering its conventional topology, both via numerical analysis and experiments using a flow-equivalent lumped electronic circuit. It is shown that physically reasonable values of circuit parameters can result in robust dynamical behavior characterized by a pair of positive Lyapunov exponents. Mandatory numerical results prove that discovered strange attractors exhibit all necessary fingerprints of structurally stable chaos. The new "chaotic" parameters are closely related to the standard operation of the investigated analog functional block. A few interestingly shaped, strange attractors have been captured as oscilloscope screenshots. [ABSTRACT FROM AUTHOR]

This research study proposes a novel hyperchaotic finance system with profit margin and then utilizes the Adomian Decomposition Method (ADM) to tackle the solution of the associated Caputo-derivative based fractional-order hyperchaotic finance system with profit margin. Numerical simulations and analyses, such as the evaluation of Lyapunov exponents (LE), the creation of bifurcation diagrams (BD), the complexity analysis (CA), and the 0-1 test, are used to get a full picture of the system. The results of our study demonstrate the occurrence of periodic, chaotic, and hyperchaotic dynamics in the system. Furthermore, the overall criteria for the existence and the uniqueness of the exact solutions for Caputo fractional-order models are presented. In addition, we present a control methodology for the fractional-order hyperchaotic financial system utilizing the Laplace transform and linear feedback control. Significantly, our research showcases a noteworthy correlation between the analytical results and numerical simulations, emphasizing the soundness and effectiveness of our methodology.

The design of chaotic systems with complex dynamic behaviors has always been a key aspect of chaos theory in engineering applications. This study introduces a novel fractional-order system characterized by hidden dynamics, hyperchaotic behavior, and multi-scroll attractors. By employing fractional calculus, the system’s order is extended beyond integer values, providing a richer dynamic behavior. The system’s hidden dynamics are revealed through detailed numerical simulations and theoretical analysis, demonstrating complex attractors and bifurcations. The hyperchaotic nature of the system is verified through Lyapunov exponents and phase portraits, showing multiple positive exponents that indicate a higher degree of unpredictability and complexity. Additionally, the system’s multi-scroll attractors are analyzed, showcasing their potential for secure communication and encryption applications. The fractional-order approach enhances the system’s flexibility and adaptability, making it suitable for a wide range of practical uses, including secure data transmission, image encryption, and complex signal processing. Finally, based on the proposed fractional-order system, we designed a simple and efficient medical image encryption scheme and analyzed its security performance. Experimental results validate the theoretical findings, confirming the system’s robustness and effectiveness in generating complex chaotic behaviors.

In this work, we present new results for a two-scroll 4-D hyperchaotic system with a unique saddle point equilibrium at the origin. The bifurcation and multi-stability analysis for the new hyperchaotic system are discussed in detail. As a control application, we develop a feedback control based on integral sliding mode control (ISMC) for the complete synchronization of a pair of two-scroll hyperchaotic systems developed in this work. Numerical simulations using Matlab are provided to illustrate the hyperchaotic phase portraits, bifurcation diagrams and synchronization results. Finally, as an electronic application, we simulate the new hyperchaotic system using Multisim for real-world implementations.

Recently, the application of memristors in chaotic systems has been extensively studied. Unfortunately, there is limited literature on the introduction of discrete memristors into chaotic maps, especially into non-classical multidimensional maps. For this reason, this paper establishes a new three-dimensional parabolic chaotic map model; in order to improve the complexity and randomness of the map, it is coupled with a square-charge-controlled discrete memristor to design a new four-dimensional memristive hyperchaotic map. Firstly, the stability of the two maps is discussed. And their dynamical properties are compared using Lyapunov exponential spectra and bifurcation diagrams. Then, the phase diagram and iteration sequence of the 4D memristive hyperchaotic map are obtained. Meanwhile, we investigate the hyperchaotic states, the transient chaos, state transfer and attractor coexistence phenomena of the four-dimensional memristive map. In particular, the special state transfer phenomenon of switching from a periodic attractor to a quasi-periodic attractor and the special coexistence phenomenon of a quasi-periodic attractor coexisting with a quasi-periodic attractor around fixed points are found, which have not been observed in other systems. Finally, the phase-track diagrams and iterative sequence diagrams of the four-dimensional memristive map are verified on a digital experimental platform, revealing its potential for practical applications. [ABSTRACT FROM AUTHOR]

With the advancement in information and communication technologies (ICTs), the widespread dissemination and sharing of digital images has raised concerns regarding privacy and security. Traditional methods of encrypting images often suffer from limitations such as a small key space and vulnerability to brute-force attacks. To address these issues, this paper proposes a novel eighth-order hyperchaotic system. This hyperchaotic system exhibits various dynamic behaviors, including hyperchaos, sub-hyperchaos, and chaos. The encryption scheme based on this system offers a key space larger than 22338. Through a comprehensive analysis involving histogram analysis, key space analysis, correlation analysis, entropy analysis, key sensitivity analysis, differential attack analysis, and cropping attack analysis, it is demonstrated that the proposed system is capable of resisting statistical attacks, brute force attacks, differential attacks, and cropping attacks, thereby providing excellent security performance. [ABSTRACT FROM AUTHOR]

In this paper, a modified hyperchaotic memristor-based Chua's circuit and its generalized discrete model are reported. First, the dynamics of the continuous system are investigated using different tools, including stability theory, phase portraits, Lyapunov exponents, and bifurcation diagrams, showing that the proposed circuit model possesses a line of equilibrium and exhibits rich dynamics, including the coexistence of regular and chaotic attractors, and mixed-mode oscillations. Lastly, a generalized discrete version of the proposed system is formulated and its dynamics are explored. Using the Jury criterion, we constructed a useful stability test to localize some stable regions in three parameter planes. The presence of an exact periodic solution is numerically highlighted, and the occurrence of chaotic behavior is confirmed using the largest Lyapunov exponent and the 0−1 test. [ABSTRACT FROM AUTHOR]

We generalize the two-dimensional Lozi map in order to systematically obtain piecewise continuous maps in three and higher dimensions. Similar to higher dimensional generalizations of the related Hénon map, these higher dimensional Lozi maps support hyperchaotic dynamics. We carry out a bifurcation analysis and investigate the dynamics through both numerical and analytical means. The analysis is extended to a sequence of approximations that smooth the discontinuity of the derivatives in the Lozi map. [ABSTRACT FROM AUTHOR]

In this research, we examine the solution of ordinary fractional differential equations using Atangana’s beta derivative. Our approach is divided into two parts. First, we establish conditions under which the fractional differential equation has a unique solution. Next, we develop a numerical technique based on the Adams–Bashforth method and demonstrate its convergence. We then apply the numerical method to a test example to evaluate its efficacy. Finally, we analyze three nonlinear chaotic and hyperchaotic fractional dynamical systems, calculating the strange attractors for various fractional orders. We analyzed the fractional chaotic systems and showed that by changing the order of the fractional derivative, some properties of the system change, such as the area of the chaos attracting region.

The dual-memristor hyperchaotic map has not yet received much attention and application, and its complexity and flexibility deserve further improvement. To this end, a novel two-dimensional hybrid dual-memristor (HDM) hyperchaotic map with complexity enhancement is constructed by connecting two different discrete memristors through self-feedback and sinusoidal transformation. The considered map has line invariant points related to the initial conditions of the discrete memristor, and it is Lyapunov stable or unstable. Taking the coupling strengths, memristor parameters, and initial conditions as tunable parameters, complex dynamical behaviors are investigated by numerical methods, including special bifurcation modes, a plethora of strange attractors, hyperchaotic attractors, and multi-stability behaviors. In particular, the performance evaluation of strange attractors with different topological structures is performed, and the complexity enhancement characteristic of the hyperchaotic attractor is confirmed by excellent dynamical performance indicators. In addition, without affecting the dynamical performance, the flexibility of the HDM map is effectively improved by switching control. Afterward, the pseudo-random number generator and image encryption strategy are proposed based on the hyperchaotic sequences generated by the HDM map to improve its practical application value. The proposed encryption strategy fully utilizes the high randomness, complexity, and flexibility of the hyperchaotic sequences, exhibiting remarkable applicability and robustness in encrypting both typical color and non-typical grayscale images. Comprehensive analyses, including histogram, correlation, and information entropy analyses, as well as differential, data loss, and noise attacks, are conducted to thoroughly evaluate the performance of the encryption strategy. The results indicate that the strategy achieves superior encryption effectiveness and high security. Finally, a microcontroller-based digital circuit experiment platform is developed for verifying the numerical results. [ABSTRACT FROM AUTHOR]

The research on dissipative-chaotic systems has exhibited plenty of remarkable results for past few decades. But unfortunately, the attackers can quickly rebuild the attractors in dissipative systems, causing multiple information security issues. Compared to the dissipative system, the conservative system can efficiently prevent such external attacks due to the presence of flows without any attractors. Therefore, the study of conservative chaotic systems is essential for secure communications and engineering applications. Presently, a few studies are there for non-autonomous conservative systems based on simple memristor. For this reason, a circuit-based new simple memristor is designed and using this, a non-autonomous four-dimensional hyperchaotic conservative system without an equilibrium point is proposed in this paper. The hyperchaotic behavior and multiple offset boosting are explored by phase portraits, bifurcation diagrams, and Lyapunov exponent plots. It is found that due to the presence of applied stimuli, the proposed system also produces time-varying hyperchaos and offset boosting. Furthermore, different torus structures, triggered by initial conditions and control parameters are observed here as the prominent property of the conservative system. An analog circuit of the chaotic system is developed and executed experimentally to verify the correctness of the numerical model. Concurrently, a new algorithm for image encryption using the proposed hyperchaotic map is stated and the efficiency of this algorithm is confirmed by several security tests. [ABSTRACT FROM AUTHOR]

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We propose a high-security hyperchaotic orthogonal frequency division multiplexing (OFDM) encryption scheme based on the orbital angular momentum system, and our research focuses on the security and high-order modulation of the orbital angular momentum (OAM)system. In order to effectively prevent violent decryption by eavesdroppers, we adopt a high security Zhan's hyperchaotic model. In the experiments, the high security performance of the proposed encryption scheme is verified by the OAM optical transmission platform, the proposed encryption scheme has a large key space of 10191, and it is demonstrated that the high-order spatial optical modulation has a small impact on the bit error rate (BER) of the OAM system within the forward error correction (FEC) threshold. The experimental results show that the proposed encryption scheme significantly improves the security performance of OAM spatial optical transmission, and is a promising candidate for the next-generation high-security OAM high-order multiplexed transmission system.

In this paper, we present a 4D hyperchaotic Rabinovich system which obtained by adding a linear controller to 3D Rabinovich system. Based on theoretical analysis and numerical simulations, the rich dynamical phenomena such as boundedness, dissipativity and invariance, equilibria and their stability, chaos and hyperchaos are studied. In addition, the Hopf bifurcation at the zero equilibrium point of the 4D Rabinovich system is investigated. The numerical simulations, including phase diagrams, Lyapunov exponent spectrum, bifurcations, power spectrum and Poincaré maps, are carried out in order to analyze and verify the complex phenomena of the 4D Rabinovich system.

This paper presents a new four-dimensional (4D) hyperchaotic system by introducing a linear controller to 3D chaotic Qi system. Based on theoretical analysis and numerical simulations, the dynamical behaviors of the new system are studied including dissipativity and invariance, equilibria and their stability, quasi-periodic orbits, chaotic and hyperchaotic attractors. In addition, the Hopf bifurcation at the zero equilibrium point and hyperchaos control of the system are investigated. The numerical simulations, including phase diagram, Lyapunov exponent spectrum, bifurcations and Poincaré maps are carried out in order to analyze and verify the complex phenomena of the 4D hyperchaotic system.

In this paper, a closed-loop supply chain is modeled based on hyperchaotic dynamics. Then, synchronization of the two hyperchaotic closed loop supply chains is performed with a proportional integral (PI) sliding mode controller design method. Using Lyapunov stability theory, it has been proved that the PI sliding mode controller can converge the synchronization error to zero in a limited time. The most important issue in the design of control strategies is the behavior of the control signal. In other words, it affects the cost of design and implementation. Numerical simulation results show that the control signal has low amplitude and fluctuations. so, the PI sliding mode control method can be implemented in the real world. Based on the numerical simulation results, the use of two controllers is proposed to reduce design costs.

Introduction. The article considers the theme of chaos, traditional for philosophy, in the perspective of materiality and spatiality, associated with manifestations of WEB 2.0: the stages of development of the global Web, where interactive content and content from the network users themselves prevail. Theoretical analysis. Here, an attempt is made to comprehend the question of chaos, using methods and models adequate for the digital sphere, as well as thematizing the concept of cyberchaos in order to answer the question of its relevance for modern Internet research. Understanding Internet subject, cyberchaos and the Internet as a chaotic system at the stage of WEB 2.0, where disorder and over-intensity of content predominate, shows it to be both a creative beginning and a method of catastrophic influence on the world, emerging from digital, but at the same time available for control and suppression. Conclusion. The current consideration makes it possible to take into account the scale of Deleuzian nomadology in the WEB 2.0 space. Its result is a series of proposals for the conceptualization of chaos and segments of modern life that relate to the Internet and WEB 2.0: social network subjectivity, information chaos and instant messengers as a driver of chaotic socio-political research.