Search Results (16)

Traditional Hopfield neural networks (HNNs) suffer from limitations in generating controllable chaotic dynamics, which are essential for applications in neuromorphic computing and secure communications. Memristors, with their memory-dependent nonlinear characteristics, provide a promising approach to regulate neuronal activities, yet systematic studies on attractor offset behaviors remain scarce. In this study, we propose a fully memristive electromagnetic radiation neural network by incorporating three distinct memristors as external electromagnetic stimuli into an HNN. The parameters of the memristors were tuned to modulate chaotic oscillations, while variations in initial conditions were employed to explore multistability through bifurcation and basin stability analyses. The results demonstrate that the system enables large-scale amplitude control of chaotic signals via memristor parameter adjustments, allowing arbitrary scaling of attractor amplitudes. Various offset behaviors emerge, including parameter-driven symmetric double-scroll relocations in phase space and initial-condition-induced offset boosting that leads to extreme multistability. These dynamics were experimentally validated using an STM32-based electronic circuit, confirming precise amplitude and offset control. Furthermore, a multi-channel pseudo-random number generator (PRNG) was implemented, leveraging the initial-boosted offset to enhance security entropy. This offers a hardware-efficient chaotic solution for encrypted communication systems, demonstrating strong application potential. Full article

This work is devoted to the hyperbolic sine function (HSF) control-based finite-time bipartite synchronization of fractional-order spatiotemporal networks and its application in image encryption. Initially, the addressed networks adequately take into account the nature of anisotropic diffusion, i.e., the diffusion matrix can be not only non-diagonal but also non-square, without the conservative requirements in plenty of the existing literature. Next, an equation transformation and an inequality estimate for the anisotropic diffusion term are established, which are fundamental for analyzing the diffusion phenomenon in network dynamics. Subsequently, three control laws are devised to offer a detailed discussion for HSF control law’s outstanding performances, including the swifter convergence rate, the tighter bound of the settling time and the suppression of chattering. Following this, by a designed chaotic system with multi-scroll chaotic attractors tested with bifurcation diagrams, Poincaré map, and Turing pattern, several simulations are pvorided to attest the correctness of our developed findings. Finally, a formulated image encryption algorithm, which is evaluated through imperative security tests, reveals the effectiveness and superiority of the obtained results. Full article

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. Full article

The symmetric multi-scroll strange attractor has shown great potential in chaos-based applications due to its high complexity in phase space. Here, the approach of symmetrization is employed for attractor doubling to generate pseudo-multi-scroll attractors in a discrete map, where a carefully selected offset constant is the key to organizing coexisting attractors. By choosing the Hénon map to generate the pseudo-multi-scroll attractor and implementing the digital circuit on a microcontroller, this study fills a significant gap in the research on discrete chaotic systems. The complexity performance is further validated using a pseudo-random number generator, demonstrating substantial academic contributions to the field of chaos theory. Additionally, a pseudo-multi-scroll attractor-based squirrel search algorithm is first developed, showcasing its practical application in mobile robot path planning. This work not only advances the theoretical understanding of chaotic systems but also provides practical methods for implementation in digital systems, offering valuable insights for policy-making in advanced robotic systems and intelligent manufacturing. Full article

This manuscript introduces a piecewise linear decomposition method devoted to a class of fractional-order dynamical systems composed of piecewise linear (PWL) functions. Inspired by the Adomian decomposition method, the proposed technique computes an approximated solution of fractional-order PWL systems using only linear operators and specific constants vectors for each sub-domain of the PWL functions, with no need for the Adomian polynomials. The proposed decomposition method can be applied to fractional-order PWL systems composed of nth PWL functions, where each PWL function may have any number of affine segments. In particular, we demonstrate various examples of how to solve fractional-order systems with 1D 2-scroll, 4-scroll, and $4\times 4$-grid scroll chaotic attractors by applying the proposed approach. From the theoretical and implementation results, we found the proposed approach eliminates the unneeded terms, has a low computational cost, and permits a straightforward physical implementation of multi-scroll chaotic attractors on ARMs and FPGAs digital platforms. Full article

Multi-scroll chaotic systems have complex dynamic behaviors, and the multi-scroll chaotic system design and analysis of their dynamic characteristics is an open research issue. This study explores a new multi-scroll chaotic system derived from an asymptotically stable linear system and designed with a uniformly bounded controller. The main contributions of this paper are given as follows: (1) The controlled system can cause chaotic behavior with an appropriate control position and parameters values, and a new multi-scroll chaotic system is proposed using a bounded sine function controller. Meanwhile, the dynamical characteristics of the controlled system are analyzed through the stability of the equilibrium point, a bifurcation diagram, and Lyapunov exponent spectrum. (2) According to the Poincaré section, the existence of a topological horseshoe is proven using the rigorous computer-aided proof in the controlled system. (3) Numerical results of the multi-scroll chaotic system are shown using Matlab R2020b, and the circuit design is also given to verify the multi-scroll chaotic attractors. Full article

At present, memristive neural networks with various topological structures have been widely studied. However, the memristive neural network with a star structure has not been investigated yet. In order to investigate the dynamic characteristics of neural networks with a star structure, a star memristive neural network (SMNN) model is proposed in this paper. Firstly, an SMNN model is proposed based on a Hopfield neural network and a flux-controlled memristor. Then, its chaotic dynamics are analyzed by using numerical analysis methods including bifurcation diagrams, Lyapunov exponents, phase plots, Poincaré maps, and basins of attraction. The results show that the SMNN can generate complex dynamical behaviors such as chaos, multi-scroll attractors, and initial boosting behavior. The number of multi-scroll attractors can be changed by adjusting the memristor’s control parameters. And the position of the coexisting chaotic attractors can be changed by switching the memristor’s initial values. Meanwhile, the analog circuit of the SMNN is designed and implemented. The theoretical and numerical results are verified through MULTISIM simulation results. Finally, a color image encryption scheme is designed based on the SMNN. Security performance analysis shows that the designed cryptosystem has good security. Full article

In this work, complex dynamics are found in a fractional-order multi-scroll chaotic system based on the extended Gamma function. Firstly, the extended left and right Caputo fractional differential operators are introduced. Then, the basic features of the extended left Caputo fractional differential operator are outlined. The proposed operator is shown to have a new fractional parameter (higher degree of freedom) that increases the system’s ability to display more varieties of complex dynamics than the corresponding case of the Caputo fractional differential operator. Numerical results are performed to show the effectiveness of the proposed fractional operators. Then, rich complex dynamics are obtained such as coexisting one-scroll chaotic attractors, coexisting two-scroll chaotic attractors, or approximate periodic cycles, which are shown to persist in a shorter range as compared with the corresponding states of the integer-order counterpart of the multi-scroll system. The bifurcation diagrams, basin sets of attractions, and Lyapunov spectra are used to confirm the existence of the various scenarios of complex dynamics in the proposed systems. Full article

Since the Lorenz chaotic system was discovered in 1963, the construction of chaotic systems with complex dynamics has been a research hotspot in the field of chaos. Recently, memristive Hopfield neural networks (MHNNs) offer great potential in the design of complex, chaotic systems because of their special network structures, hyperbolic tangent activation function, and memory property. Many chaotic systems based on MHNNs have been proposed and exhibit various complex dynamical behaviors, including hyperchaos, coexisting attractors, multistability, extreme multistability, multi-scroll attractors, multi-structure attractors, and initial-offset coexisting behaviors. A comprehensive review of the MHNN-based chaotic systems has become an urgent requirement. In this review, we first briefly introduce the basic knowledge of the Hopfiled neural network, memristor, and chaotic dynamics. Then, different modeling methods of the MHNN-based chaotic systems are analyzed and discussed. Concurrently, the pioneering works and some recent important papers related to MHNN-based chaotic systems are reviewed in detail. Finally, we survey the progress of MHNN-based chaotic systems for application in various scenarios. Some open problems and visions for the future in this field are presented. We attempt to provide a reference and a resource for both chaos researchers and those outside the field who hope to apply chaotic systems in a particular application. Full article

A multi-scroll attractor reflects the structural diversity of the dynamic system, and multi-stability behavior reflects its state diversity. Multi-scroll and multi-stability chaotic systems can produce complex random sequences, which have important application values in the field of data security. However, current works on multi-scroll–multi-steady behavior have been carried out separately, rather than simultaneously. This paper considers a three-dimensional Jerk system with a sinusoidal nonlinear term. The basic dynamic behaviors, such as the stability of equilibrium points, bifurcation of parameters and initial values, phase diagrams, and basins of attraction, were analyzed. It was found that the system has infinite equilibrium points. Moreover, the system not only generates complex dynamics, such as single-scroll, double-scroll, and multi-scroll but also realizes the self-reproduction of these dynamic characteristics by controlling the initial value of the system. Therefore, by expanding the equilibrium point, the effective controls of the system’s structural diversity and state diversity are realized at the same time, having important theoretical significance and application value. Full article

Fractional-order chaotic systems are widely used in the field of encryption because of its initial value sensitivity and historical memory. In this paper, the fractional-order definition of Caputo is introduced based on a nonideal flux-controlled memristive Hopfield neural network model, when changing the parameters of the fractional-order memristive Hopfield neural network (FMHNN) can generate a different amount of multi-scroll attractors. Some dynamical behaviors are investigated by numerical simulation, especially analyzed coexistence and bifurcation under different orders and different coupling strengths. The results show that the chaotic system of FMHNN has abundant dynamic behaviors. In addition, a chaotic audio encryption scheme under a Message Queueing Telemetry Transport (MQTT) protocol is proposed and implemented by Raspberry Pi; the audio encryption system based on FMHNN has a broad future in intelligent home and other IoT applications. Full article

A three directional (3-D) multi-scroll chaotic attractors based on the Jerk system with nonlinearity of the sine function and sign function is introduced in this paper. The scrolls in the X-direction are generated by the sine function, which is a modified sine function (MSF). In addition, the scrolls in Y and Z directions are generated by the sign function series, which are the superposition of some sign functions with different time-shift values. In the X-direction, the scroll number is adjusted by changing the comparative voltages of the MSF, and the ones in Y and Z directions are regulated by the sign function. The basic dynamics of Lyapunov exponent spectrum, phase diagrams, bifurcation diagram and equilibrium points distribution were studied. Furthermore, the circuits of the chaotic system are designed by Multisim10, and the circuit simulation results indicate the feasibility of the proposed chaotic system for generating chaotic attractors. On the basis of the circuit simulations, the hardware circuits of the system are designed for experimental verification. The experimental results match with the circuit simulation results, this powerfully proves the correctness and feasibility of the proposed system for generating 3-D grid multi-scroll chaotic attractors. Full article

A novel chaotic system for generating multi-scroll attractors based on a Jerk circuit using a special form of a sine function (SFSF) is proposed in this paper, and the SFSF is the product of a sine function and a sign function. Although there are infinite equilibrium points in this system, the scroll number of the generated chaotic attractors is certain under appropriate system parameters. The dynamical properties of the proposed chaotic system are studied through Lyapunov exponents, phase portraits, and bifurcation diagrams. It is found that the scroll number of the chaotic system in the left and right part of the x-y plane can be determined arbitrarily by adjusting the values of the parameters in the SFSF, and the size of attractors is dominated by the frequency of the SFSF. Finally, an electronic circuit of the proposed chaotic system is implemented on Pspice, and the simulation results of electronic circuit are in agreement with the numerical ones. Full article

Based on simplified Lorenz multiwing and Chua multiscroll chaotic systems, a rotation compound chaotic system is presented via transformation. Based on a binary fractal algorithm, a new ternary fractal algorithm is proposed. In the ternary fractal algorithm, the number of input sequences is extended from 2 to 3, which means the chaotic attractor with fractal transformation can be presented in the three-dimensional space. Taking Lorenz system, rotation Lorenz system and compound chaotic system as the seed chaotic systems, the dynamics of the complex chaotic attractors with fractal transformation are analyzed by means of bifurcation diagram, complexity and power spectrum, and the results show that the chaotic sequences with fractal transformation have higher complexity. As the experimental verification, one kind of complex chaotic attractors is implemented by DSP, and the result is consistent with that of the simulation, which verifies the feasibility of digital circuit implement. Full article

Designing chaotic oscillators using complementary metal-oxide-semiconductor (CMOS) integrated circuit technology for generating multi-scroll attractors has been a challenge. That way, we introduce a current-mode piecewise-linear (PWL) function based on CMOS cells that allow programmable generation of 2–7-scroll chaotic attractors. The mathematical model of the chaotic oscillator designed herein has four coefficients and a PWL function, which can be varied to provide a high value of the maximum Lyapunov exponent. The coefficients are implemented electronically by designing operational transconductance amplifiers that allow programmability of their transconductances. Design simulations of the chaotic oscillator are provided for the $0.35\mathsf{\mu }$ m CMOS technology. Post-layout and process–voltage–temperature (PVT) variation simulations demonstrate robustness of the multi-scroll chaotic attractors. Finally, we highlight the synchronization of two seven-scroll attractors in a master–slave topology by generalized Hamiltonian forms and observer approach. Simulation results show that the synchronized CMOS chaotic oscillators are robust to PVT variations and are suitable for chaotic secure communication applications. Full article