Multi-scroll Chaotic Systems Research Articles

Regulation and analysis of multi-scroll attractors in a class of memristive Hopfield neural networks with high Lyapunov exponents

Multi-scroll chaotic systems have gained significant attention in recent years. However, there is little research on the chaotic systems of memristive neural networks with multiple complex structural attractors. A class of multi-scroll chaotic systems are designed in this paper, which are based on Hopfield neural network by using memristors as synapses. Implementing synaptic control with memristors at various coupling positions allows for the creation of multiple chaotic attractors with distinct topological structures. The proposed systems exhibit special and irregular shaped attractors, and can generate a fixed number of scrolls. Through analysis we find that the systems have high Lyapunov exponents, indicating strong sensitivity and randomness. Meanwhile, the systems exhibit extremely rich dynamic behaviors, such as symmetric coexistence phenomena. Especially, we observe signal amplitude control and offset boosting phenomena in the systems. In addition, simulation circuits are designed based on such chaotic systems to verify the physical feasibility of the systems.

Design and FPGA implementation of nested grid multi-scroll chaotic system

Conventional multi-scroll chaotic systems are often constrained by the number of attractors and the complexity of generation, making it challenging to meet the increasing demands of communication and computation. This paper revolves around the modified Chua’s system. By modifying its differential equation and introducing traditional nonlinear functions, such as the step function sequence and sawtooth function sequence. A nested grid multi-scroll chaotic system (NGMSCS) can be established, capable of generating nested grid multi-scroll attractors. In contrast to conventional grid multi-scroll chaotic attractors, scroll-like phenomena can be initiated outside the grid structure, thereby revealing more complex dynamic behavior and topological features. Through the theoretical design and analysis of the equilibrium point of the system and its stability, the number of saddle-focused equilibrium points of index 2 is further expanded, which can generate (2 N+2) × M attractors, and the formation mechanism is elaborated and verified in detail. In addition, the generation of an arbitrary number of equilibrium points in the y-direction is achieved by transforming the x and y variables, which can generate M×(2 N+2) attractors, increasing the complexity of the system. The system’s dynamical properties are discussed in depth via time series plots, Lyapunov exponents, Poincaré cross sections, 0–1 tests, bifurcation diagrams, and attraction basins. The existence of attractors is confirmed through numerical simulations and FPGA-based hardware experiments.

A Piecewise Linear Approach for Implementing Fractional-Order Multi-Scroll Chaotic Systems on ARMs and FPGAs

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 4×4-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.

Constructing a New Multi-Scroll Chaotic System and Its Circuit Design

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.

Expanded multi-scroll attractor system analysis and application for remote sensing image encryption

Exploring special multi-scroll chaotic systems is meaningful work. This paper studies an expanded multi-scroll chaotic system consisting of eight terms with one nonlinearity. It is generated by modifying the nonlinear term of the newly constructed chaotic system by a polynomial function. The unique mathematical model makes the unstable index-2 equilibria increase in four dimensions, which contributes to the number of scrolls expanding in each phase plane unidirectionally. Dynamic analysis finds that the system can yield complex nested multi-scroll attractors and has single-parameter-based synchronization control of amplitude and offset boosting behavior. Moreover, circuit implementation verifies the physical existence of the proposed system. Also, an image encryption algorithm for remote sensing images is established. Permutation and diffusion operations are performed using new pixel coordinates derived from the data of the chaotic matrix, which comes from the same position as the pixel matrix. Relevant tests suggest that the algorithm is secure and able to resist undesirable interference.

Multi-scroll attractor and its broken coexisting attractors in cyclic memristive neural network

This paper proposes a simple-structured memristive neural network, which incorporates self-connections of memristor synapses alongside both unidirectional and bidirectional connections. Different from other multi-scroll chaotic systems, this network structure has a more concise three-neuron structure. This simple memristive neural network can generate a number of multi-scroll attractors in manageable quantities and shows the characteristics of the coexisting attractors and amplitude control. In particular, when the parameters are changed, the coexisting attractors break up around the center of gravity into two centrosymmetric chaotic attractors. Abundant dynamic behaviors are studied through phase portraits, bifurcation diagrams, Lyapunov exponents, and attraction basins. The feasibility of the system is demonstrated by building a circuit realization platform.

Design of pseudorandom number generator based on a controllable multi-double-scroll chaotic system

The generation, implementation, and application of multi-scroll chaotic systems have been extensively studied due to their complex dynamic behaviors. This paper introduces a new controlled multi-double-scroll chaotic system and its corresponding implementation on a digital platform, the Field Programmable Gate Array (FPGA). The proposed system can generate 2N+1 and 2N+2 numbers of double-scroll attractors. In particular, the system has controllable multistability and multi-double-scroll complete amplitude control characteristics. The numerical analyses of Lyapunov exponents, bifurcation diagrams, phase diagrams, and basin of attraction show that the system exhibits complex dynamic behaviors. Furthermore, a new pseudo-random number generator (PRNG) is designed based on the multi-double-scroll chaotic system (MDSCS), in which a post-processing circuit is used to improve the quality of the generator. The PRNG has a high data throughput and uses only a small amount of resources in the target FPGA. The proposed PRNG uses less than 1% of the resources of the target FPGA and delivers a data bit throughput of 7.2 Gbps/s. To our best knowledge, it reaches the highest data throughput of PRNG based on continuous chaotic systems. The evaluation sequences are tested using generic randomness tests such as NIST SP800-22, TestU01, and Histograms. The experimental results show that the random sequences generated by our design have passed all the test items, which means that they will have excellent performance in the application.

A general method for generating multi-scroll and multi-wing chaotic systems and its implementation of attractor reproduction

Abstract In comparison to traditional chaotic systems, the multi-scroll and multi-wing chaotic systems are more complicated. The design and execution of sophisticated multi-scroll or multi-wing chaotic attractors attract a lot of attention. However, these constructed nonlinear functions cannot be applied to extended multi-scroll and multi-wing attractors at the same time. To this end, this paper proposes a new function which can be used to generate multi-scroll and multi-wing chaotic attractors in both double-scroll and double-wing chaotic systems. Using this function, multi-scroll and multi-wing chaotic systems can be constructed directly without relying on whether the chaotic system has some symmetry (odd symmetry or even symmetry). The construction method presented is generally applicable to chaotic systems with multi-scroll and multi-wing self-excited attractors.The main point of this method is as follows: firstly, the piecewise linear (PWL) saturation function is nested within the cosine nonlinearity function , and the resulting nested COS-PWL function. Secondly, to enable the expansion of multi-wing and multi-scroll, the nested COS-PWL function is incorporated into the double-wing and double-scroll systems in different manners. The maximum Lyapunov exponent (MLE) and the bifurcation diagram route for increasing the number of wings and scrolls confirm the feasibility and effectiveness of the method. Finally, the three-element method is used to determine a Sinusoidal function, which can generate attractor self-reproduction in the corresponding dimension by replacing the state variables of the multi-scroll and multi-wing systems, so that an infinite number of coexisting attractors can be obtained by simply changing the initial values of the variables, i.e., multiple stability can be generated.

A family of new generalized multi-scroll Hamiltonian conservative chaotic flows on invariant hypersurfaces and FPGA implementation

A family of new generalized multi-scroll Hamiltonian conservative chaotic systems(HCCSs) are firstly found by investigating Hamiltonian energy of a Hamiltonian conservative chaotic system. And the novelty of the family of new generalized multi-scroll HCCSs is that they are on some invariant hypersurfaces depending on Hamiltonian energy. Subsequently, some numerical analysis are given to show multi-scroll chaotic dynamics and complex characteristics of the family of new generalized multi-scroll HCCSs. Finally, FPGA circuits are employed to implement the family of new generalized multi-scroll HCCSs, which not only physically show the existence of multi-scroll dynamics, but also provide a family of new pseudo-random signal generators for engineering applications.

Grid-scroll memristive chaotic system with application to image encryption

The construction of multi-scroll chaotic systems has been a hot research topic in recent years. In this paper, a four-dimensional multi-scroll chaotic system is established on the basis of a simple three-dimensional chaotic system by adding a flux-controlled non-volatile memristor. After that, a novel system is extended to generate grid-scroll chaotic attractors by replacing the linear term in the multi-scroll chaotic system with a triangular wave function. The evolution of chaos is studied by means of bifurcation diagrams and Lyapunov exponent diagrams. The existence of coexisting attractors is observed, and the attraction basin of coexisting attractors is drawn. In addition, the partial amplitude of the proposed system can be controlled by adjusting the parameters. The results of circuit implementation are consistent with the numerical simulation, verifying the physical existence of the system. Finally, according to the proposed grid-scroll memristive chaotic system, a new image encryption algorithm is designed. It adopts a novel confusion-diffusion structure to scramble and diffuse all pixels of the plain-image. And the security analysis experiments are performed to verify the good encryption performance of the proposed scheme such as information entropy analysis, correlation coefficient analysis, differential attack analysis and robustness analysis.

Multi-Scroll Attractor and Multi-Stable Dynamics of a Three-Dimensional Jerk System

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.

Complex Dynamics, Hardware Implementation and Image Encryption Application of Multiscroll Memeristive Hopfield Neural Network With a Novel Local Active Memeristor

Because of the nonlinearity and memory, memristors are the most suitable electrical component for simulating synapses. A novel local active and nonvolatile memristor is designed. By circuit experiments, its memristive properties are verified. By introducing this memristor, this brief constructs a 4D memristive Hopfield neural network (MHNN) which can perform complex dynamics, such as controllable double-scrolls attractors and controllable initial offset boosting coexistence. Compared with other multiscroll chaotic systems, the autonomy equation of the system is smooth for discarding the sign function. In addition, this MHNN performs well in image encryption applications for the significant complexity of multiscroll. Through safety analysis, the information entropy of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$512 \times 512$ </tex-math></inline-formula> Lena graph is 7.9993, which is very close to the ideal value of 8. Besides, the number of pixels changing rates (NPCR) and the unified averaged changed intensity (UACI) are 99.6097% and 33.4621%, which are almost equal ideal values. Finally, this brief designs the digital circuit of the multiscroll MHNN signal generator and verifies the function with the help of a field programmable gate array (FPGA) and oscilloscope. Besides, by designing a pseudo-random number generation circuit, FPGA can directly encrypt the image and transmit it to the input and output devices.

Stability in the Synchronization of Multi-scroll Chaotic Systems using Hamiltonian Forms

This work contributes to the subject of stability in the synchronization of identical chaotic systems, coupled by means of a control law. Initially, chaotic systems with multiple scrolls are expressed in a Hamiltonian form, this allows us to observe particular characteristics of the system, since the system is divided into a conservative part and a dissipative part. This procedure helps in the design of a control law that allows identical chaotic systems to be synchronized. In particular, the technique of Model-Matching is used for the design of the control law, but an optimization is performed in the design of a control law, based only on the conservative part of the chaotic systems in Hamiltonian form. This gives us a new point of view in the design of the controllers used to achieve synchronization between chaotic systems, which simplifies the calculations that are carried out to obtain a law of control. A stability analysis is performed in this work to ensure that the synchronization of the two systems coupled by the effect of the control law meet the full system synchronization requirements, some numerical simulations are presented.

The Effect of Using Multi-Scroll Chaotic Systems on Chaos-Based Random Number Generators’ Performance

Chaotic and hyper-chaotic systems are used in various engineering applications such as encryption, communication, and artificial intelligence. Also, chaotic systems are widely used in chaos-based random number generator (RNG) designs as chaotic system signals are not periodic and produce different values continuously. Since multi-scroll chaotic systems (MSCSs) produce more than one scroll, the output values can take more different values than chaotic systems. In this study, the effects of different directional values and/or different numbers of scrolls of multi-scroll chaotic systems on chaos-based random number generators’ performance are investigated with NIST 800-22 (National Institute of Standards and Technology) and correlation coefficient tests. As a result of the research, it has been concluded that the use of multi-scroll chaotic systems with different directional values and/or different numbers of scrolls does not always have a direct positive effect on the performance of chaos-based random number generators. Thus, it is necessary to use a special pre-process method that will vary according to the multi-scroll chaotic system to be used for chaos-based RNG designs with good performance.

A Multiring Julia Fractal Chaotic System With Separated-Scroll Attractors

Fractal and chaos are two major phenomena in nonlinear systems. A few combinations of chaos with fractal for generating multiscroll attractors are reported. Thus, based on Chua multiscroll chaotic systems, a novel multiring chaotic system is proposed via Julia multifractal processes. The main purpose of this work is to make up for the deficiency of various fractal processes to further improve the performance of chaotic systems. A new class of multiscroll attractors is generated by combining different fractal processes, including symmetrical multiring attractor, separated multiring attractor, and nested multiring attractor. Furthermore, dynamic behaviors of the fractal chaotic system are analyzed by means of Lyapunov exponents and bifurcation diagram to indicate that the system exhibits chaotic characteristics. Experiments on the spectrum entropy (SE) complexity and image encryption performance demonstrate the good performance of the proposed fractal-based attractors. To verify the feasibility of multiring chaotic system, the various fractal-based attractors generated are implemented by micro controller unit (MCU), and the separated multiring attractors are observed on the digital oscilloscope.

Experimental verification of the multi-scroll chaotic attractors synchronization in PWL arbitrary-order systems using direct coupling and passivity-based control

Because one of the main applications of chaos synchronization is oriented to data encryption and random number generators, fractional-order chaotic systems showing multi-scroll attractors have pointed out as feasible solutions to improve the performance of secure communications schemes. Nowadays, recent works have reported the analysis and control of fractional-order synchronization schemes for multi-scroll chaotic systems. However, the synchronization schemes from those works present mostly theoretical numerical simulations and give circuit simulations employing a simulator software rather than physical implementations. That way, our first contribution is to describe herein the experimental realization of a synchronization scheme for a fractional-order multi-scroll chaotic oscillator with a 4-scroll attractor and fractional-order q=0.9. First, we implemented the multi-scroll systems using an opamp-based analog circuit and an RC-based fractance device to emulate a fractional-order integrator. As a second contribution, we analytically demonstrate the complete synchronization between two identical multi-scroll systems employing two fractional-order controllers and the fractional-order derivative in the Caputo sense. By utilizing the Laplace transform theory, we analyze the one-way coupling method where the response (slave) system’s behavior depends on an identical drive (master) system. As a third contribution, we propose a passivity-based synchronization controller. This synchronization strategy consists of designing a virtual state feedback input and then based on the fractional-order Lyapunov theory, a control law obtains by designing an appropriate storage function to stabilize the system around its stable zero equilibrium. The experimental measures of the implemented synchronization scheme present suitable results compared to analytical and numerical findings.

Active Realization of Fractional-Order Integrators and Their Application in Multiscroll Chaotic Systems

This paper presents the design, simulation, and experimental verification of the fractional-order multiscroll Lü chaotic system. We base them on op-amp-based approximations of fractional-order integrators and saturated series of nonlinear functions. The integrators are first-order active realizations tuned to reduce the inaccuracy of the frequency response. By an exponential curve fitting, we got a convenient design equation for realizing fractional-order integrators of orders from 0.1 to 0.95. The results include simulations in SPICE of the mathematical description and the electronic implementation and experimental measurements that confirm them. Monte Carlo and sensitivity tests revealed a robust realization. Contrary to its passive counterparts, the suggested realizations significantly reduce design and implementation efforts by favoring resistors and capacitors with commercial values and reducing hardware requirements.

Scaling of Attractors of a Multiscroll Memristive Chaotic System and its Generalized Synchronization with Sliding Mode Control

Memristor can greatly enhance the complexity of a chaotic system because of its nonlinear characteristics. In this paper, three different memristor models are introduced to the Yang system. The chaotic attractors with single scroll and double scrolls can be obtained by adjusting the action intensities of three memristors and all the attractors inherit the scaling property of attractors of the Yang system. By employing the complex polynomials transformation method in the chaotic system to expand the number of scrolls of the system, the ring-shaped multiscroll attractors are generated, and the number of scrolls can be changed by adjusting the powers of complex polynomials, which show that the memristive system has flexible scalability. Next, a synchronization method for the multiscroll chaotic system is proposed. The generalized synchronization controller and parameter adaptive law are designed by employing sliding mode control. The sufficient conditions for synchronization are given by Lyapunov stability theory. This method can be applied to the synchronization of multiscroll systems generated by means of changing the state variables of the original system by function transformation and then adding the transformation matrix to the system. Compared with the existing synchronization method, this method has a wider scope of application, and it can synchronize two multiscroll chaotic systems with greater difference. In addition, the conditions to be satisfied in this method are simpler. Finally, the method proposed above is applied to the synchronization between a chaotic system with a ring-shaped eight-scroll attractor and a grid-shaped [Formula: see text]-scroll attractor chaotic system with interference signals. The numerical simulation results verify the effectiveness of the method.

Finite-time synchronization of multi-scroll chaotic systems with sigmoid non-linearity and uncertain terms

In this article, the finite-time synchronization of multi-scroll chaotic systems with sigmoid non-linearity in the presence of uncertain stochastic parameters has been studied. The finite-time synchronization is a popular topic from a practical point of view in engineering, which has many applications in the areas like chaos-based communication, image and voice encryptions. The present chaotic system scrolls (up to five) are incorporated with the help of hyperbolic tangent functions (sigmoid non-linearity), but in real applications, there is no limit of multi-scroll in a chaotic system. Finite-time synchronization is achieved with the help of Lyapunov stability using some lemmas and definitions. The Lyapunov exponents are also calculated for multi-scroll chaotic systems with uncertainties and it confirms that the multi-scroll system is chaotic with uncertainties. A drive has been taken to achieve finite-time synchronization of multi-scroll chaotic systems with uncertain parameters, where the multi-scrolls are generated by nonlinear functions.

Generation of grid multi-wing chaotic attractors and its application in video secure communication system

In order to study the application of chaos in video image encryption, a real-time video secure communication system based on a new grid multi-wing chaotic system is proposed in this paper. First, by introducing sawtooth wave functions to the Lorenz system, a new grid multi-wing butterfly chaotic system with complicated dynamical behaviors is obtained. Compared with the existing multi-scroll and multi-wing chaotic systems, The system structure is simple and more easier to be implemented in a digital system. Then, a chaos-based pseudorandom random number generator is developed by implementing post-processing procedure. The present video secure communication system is designed with the closed-loop feedback scheme. The corresponding hardware implementation is developed by FPGA platforms, and the experimental results are given to verify its feasibility. Furthermore, a series of widely used secure analyses are applied to prove that the system has good security performance.