Number Of Scrolls Research Articles

Hidden multi-scroll and coexisting self-excited attractors in optical injection semiconductor laser system: Its electronic control.

In this paper, we investigate hidden and coexisting self-excited multi-scroll attractors by using a modified rate equations model of semiconductor lasers (REM-SCLs) subjected to optical injection by exploring various quantifying analytical and numerical methods. The multi-leveled dynamics sticks out the existence of several sets of equilibria that asymptotically attract trajectories originating outside of them. Chaos topology based on the impact of equilibria allows the describing of the so-called stable or unstable multi-scroll chaotic attractors. Shaping of the new coexisting self-excited multi-scroll attractor, whose source is from coupling of equilibria, is analyzed, as well as its structural dynamics along with the dynamical emergence of the hidden multi-scroll attractor in the restricted interval, defined by an additional decisive parameter. Additionally, specific 3D plots with embedded contour plots obtained by harnessing two-parameter bifurcation analysis clarify structural dynamics of such a multi-scroll attractor and accurately circumscribe stretching of its fractal-like basin of attraction. Strange metamorphoses undergone by the fractal-like basin of attraction of the studied multi-scroll attractor are stepwisely parsed in the map of two-codimension bifurcation as its scroll number evolves. At last, an electronic circuit of equivalent REM-SCLs is designed and simulated in the PSpice environment alongside a tailored electronic controller. The achieved results align with the ones of numerical analysis; besides, temporal controlling of optical waves pertaining thereto is also fulfilled.

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.

Parametric controllable planar multi-scroll chaotic attractors in a 3-D memristive tabu learning single neuron model

In this paper, a three-dimensional (3D) autonomous tabu learning single neuron model is proposed, which is achieved by using a sinusoidal activation function and introducing a memristor synapse. This model exhibits the remarkable capability to produce a series of planar multi-scroll chaotic attractors, and its unique feature lies in the ability to control the number of scrolls. The investigation of the planar multi-scroll chaotic attractors and its dynamical behaviors is conducted through the analysis of phase plane portraits, bifurcation diagrams, and spectral entropies. The numerical simulations unveil a compelling relationship between the number of chaotic scrolls and specific control parameters governing the model. To further validate the findings, a 3D autonomous tabu learning single neuron model is implemented on a digital hardware platform. In an effort to extend the practical significance of this research, the multi-scroll chaotic phenomenon generated by the proposed model is deployed for image encryption. The fusion of mathematical modeling, digital hardware implementation, and practical application underscores the universality and significance of the proposed single neuron model in the fields of chaotic systems and engineering applications.

Generation multi-scroll chaotic attractors using composite sine function and its application in image encryption

In this paper, a composite sine function is proposed and applied in a chaotic system, which is capable of generating definite number of chaotic attractors. The proposed composite sine function possesses infinite breakpoints, but it can produce a fixed number of scrolls by adjusting its parameters. Compared to other chaotic systems with multiple scrolls chaotic attractors, the realization circuit of the chaotic system with compound sine function allows obtaining different numbers of scrolls by adjusting only one resistance value. As a result, the circuit structure remains unchanged despite variations in the number of scrolls. Various analytical methods are applied to study the dynamical behaviors of the proposed chaotic system, including Lyapunov exponent, equilibrium point, bifurcation diagram, phase diagram, spectral entropy and C0-algorithm. Furthermore, based on the analysis of dynamical characteristics, the electronic circuits of the proposed system are given on Multisim circuit simulation software, and the multi-scroll chaotic attractors exhibit consistency with the numerical simulation results. Finally, we incorporated the proposed chaotic system into a Deoxyribonucleic acid coding algorithm for image encryption, and this method exhibits excellent encryption efficiency and high level of security.

Superconductivity coupling of harmonic resonant oscillators: Homogeneous and heterogeneous extreme multistability with multi-scrolls.

Understanding and characterizing multistabilities, whether homogeneous or heterogeneous, is crucial in various fields as it helps to unveil complex system behaviors and provides insights into the resilience and adaptability of these systems when faced with perturbations or changes. Homogeneous and heterogeneous multistabilities refer, respectively, to situation in which various multiple stable states within a system are qualitatively similar or distinct. Generating such complex phenomena with multi-scrolls from inherent circuits is less reported. This paper aims to investigate extreme multistability dynamics with homogeneous and heterogeneous multi-scrolls in two coupled resonant oscillators through a shunted Josephson junction. Analysis of equilibrium points revealed that the system supports both hidden and self-excited attractors. Various dynamical tools, including bifurcation diagrams, spectrum of Lyapunov exponents, and phase portraits, are exploited to establish the connection between the system parameters and various complicated dynamical features of the system. By tuning both system parameters and initial conditions, some striking phenomena, such as homogeneous and heterogeneous extreme multistability, along with the emergence of multi-scrolls, are illustrated. Furthermore, it is observed that one can readily control the number of scrolls purely by varying the initial conditions of the investigated system. A multi-metastable phenomenon is also captured in the system and confirmed using the finite-time Lyapunov exponents. Finally, the microcontroller implementation of the system demonstrates strong alignment with the numerical investigations.

A Novel Conservative Chaotic Circuit with a Cubic Root

A new conservative chaotic system with a cubic root is constructed in this paper. The system has no equilibria. An offset parameter is introduced into the system. By modulating different offset parameters and corresponding initial conditions, multiple coexisting conservative chaotic attractors can be generated consequently. By replacing the offset parameter with an appropriate sign function, multiscroll conservative chaotic attractors can be generated effectively by connecting multiple coexisting chaotic attractors based on offset boosting. Moreover, the number of scrolls tend to be infinite. Finally, a novel chaotic circuit with a cubic root is designed successfully by connecting two multipliers in opposite directions. The hardware implementation of the chaotic circuit is also completed to further verify the existence of conservative chaos and the consistency with the numerical simulations.

Multi-scroll and coexisting attractors in a Hopfield neural network under electromagnetic induction and external stimuli

The application of external stimuli to biological neurons is a valuable tool for investigating neuronal properties, understanding neural circuitry, and developing therapeutic interventions for neurological disorders. In this article, we propose a discrete fractional Hopfield neural network model consisting of four neurons to explore the influence of external stimuli in the presence of electromagnetic induction and radiation. To incorporate the electromagnetic induction between connected neurons, we construct and employ a discrete fractional sine memristor. Additionally, we introduce a multi-level pulse function to the sine memristor element to examine the chaotic dynamics of the neural network model. The qualitative behavior of the network model is demonstrated through stability analysis and bifurcation diagrams showcasing chaos. The study also focuses on understanding the coexisting behavior of the neural network model in the presence and absence of external stimuli. Moreover, we investigate the generation of multi-scroll attractors by varying the level of the pulse function, which is introduced to electromagnetic induction. Numerical simulations reveal that increasing the level of the multi-pulse function doubles the number of scrolls in the attractors when external stimuli are present. The findings presented in this article contribute to our understanding of discrete fractional memristors and shed light on the dynamical behavior of neurons and their electrical activity in the brain. Innovation within the discrete fractional-order Hopfield neural networks realm entails the creation and utilization of fresh ideas, methodologies, and strategies that harness fractional-order dynamics to confront diverse hurdles and enhance the effectiveness of Hopfield networks. Discrete fractional-order Hopfield neural networks have the capacity to propel an array of applications forward, spanning artificial intelligence, machine learning, control systems, and optimization, showcasing their potential for substantial progress in various domains.

A physical memristor-based chaotic system and its application in colour image encryption scheme

This work proposes a physical memristor (TaOx) based new 4D chaotic system with 3D multi-scroll, no equilibrium point, spiking behaviour, coexistence bursting oscillation and multistability. Using this physical memristor-based chaotic system, a novel and efficient colour image encryption algorithm has been developed using a unique box scrambling method and bit-wise XOR operations. Many interesting and new dynamics of a material-based memristive chaotic system are reported here, like 3D multi-scroll chaotic attractors, bursting characteristics, multistability, a neuronal system like spiking behaviours etc using Lyapunov spectrum and bifurcation plots. It is observed that the number of scrolls is changed with the total simulation time. This novel memristive chaotic system has limit cycles with controllable spikes and bursting oscillation. In addition, the system shows chaotic bursting oscillation under a different set of parameters and initial conditions. The coexistence of the bursting phenomena is studied here. The bursting and spiking characteristic is important for material-based memristors in neuromorphic applications. 3D Chaotic multi-scroll and multistability properties make the image encryption method more efficient and secure. Such characteristics are rare in physical memristor-based chaotic systems and using this, the image encryption algorithm is also rare in recent findings. Therefore, a new secure image encryption algorithm for colour images is proposed here, based on the unique box scrambling method, bitwise XOR operation and pseudo-random number generation using the proposed memristive chaotic system. Various tests like NPCR, UACI, histogram analysis, correlation study, information entropy analysis, robustness against external noise, etc have been performed to check the algorithm’s robustness and efficiency and test the capability to resist statistical and differential attacks.

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 chaotic attractors with multi-wing via oscillatory potential wells

The oscillatory potential well can cause the mass points in it to move chaotically, which can be considered as a physical mechanism of chaos generation. Based on this physical mechanism, the oscillatory multi-well potential with multi-concave is used to generate controllable multi-scroll chaotic attractors with multi-wing. These attractors have two kinds of topological units: scroll and wing. The topological structure of the attractor depends on the shape of the potential well, that is, the wells and the concaves correspond to the scrolls and wings of the attractor. By constructing potential wells with different shapes, we can obtain attractors with different topological structure. The number of scrolls and wings of the chaotic attractor can be controlled by the oscillation amplitude of the potential well and damping coefficient, that is, stronger oscillation or smaller damping allows the mass points to enter the higher potential wells or concaves, resulting in attractors with more scrolls or wings. As the number of scrolls or wings increases, the attractors become more complex, as expressed by Poincaré maps and quantified by the Kaplan-Yorke dimensions. Kaplan-Yorke dimensions of attractors can be adjusted continuously from 2 to 3 by the two control parameters. Finally, the results are confirmed by Multisim simulation circuit and actual analog circuit.

Two-dimensional non-autonomous neuron model with parameter-controlled multi-scroll chaotic attractors

This work presents a two-dimensional (2-D) non-autonomous tabu learning single neuron (TLSN) model based on sinusoidal activation function (SAF), which can generate a class of multi-scroll chaotic attractors with parameters controlling the number of scrolls. The SAF-based TLSN (SAF-TLSN) model has a periodically alterable equilibrium point closely related to the model parameters. Its equilibrium trajectories with different stability types can be distributed in the phase plane, resulting in the generation of multi-scroll chaotic attractors. The multi-scroll chaotic attractors and their dynamical behaviors are investigated by phase plane portrait, maximal Lyapunov exponent, bifurcation diagram, and cross section. The numerical results demonstrate that each scroll of multi-scroll chaotic attractors is associated with the equilibrium trajectories composed of infinitely many unstable equilibrium points, and the scroll distributions of multi-scroll chaotic attractors are confined to the distribution ranges of equilibrium trajectories that are controlled by the model parameters. Besides, using a digital hardware platform, the SAF-TLSN model is implemented and the multi-scroll chaotic attractors with different scrolls are obtained experimentally to verify the numerical ones.

Generating n-Scroll Chaotic Attractors From a Memristor-Based Magnetized Hopfield Neural Network

This brief presents a novel method to generate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> -scroll chaotic attractors. First, a magnetized Hopfield neural network (HNN) with three neurons is modeled by introducing an improved multi-piecewise memristor to describe the effect of electromagnetic induction. Theoretical analysis and numerical simulation show that the memristor-based magnetized HNN can generate multi-scroll chaotic attractors with arbitrary number of scrolls. The number of scrolls can be easily changed by adjusting the memristor control parameters. Besides, complex initial offset boosting behavior is revealed from the magnetized HNN. Finally, a magnetized HNN circuit is designed and various typical attractors are verified.

Dynamics of a class of Chua’s oscillator with a smooth periodic nonlinearity: Occurrence of infinitely many attractors

In this article, a class of Chua’s system with smooth periodic nonlinear term is considered. This kind of systems exhibits strange dynamical behavior. Except for the existence of infinitely many equilibrium points, period-doubling bifurcation, and double-scroll attractors, this type of systems has strange dynamics different from the classical Chua’s system: (i) the coexistence of (infinitely) many non-chaotic strange attractors; (ii) the coexistence of (infinitely) many spiral chaotic attractors; (iii) the coexistence of multi-scroll attractors; (iv) “growing”-scroll attractors, where the number of scrolls is an increasing function with respect to the time variable.

Multistability route in a PWL multi-scroll system through fractional-order derivatives

Recently, it was discovered that the use of fractional derivatives induces the occurrence of multistable states in PWL systems with multiple scrolls. In this paper, we show the emergence of multistable states in a PWL system and study the route of the system to transition from monostable to multistable behavior by reducing the order of the fractional derivative. Using bifurcation diagrams that describe the evolution of the attractor as a consequence of the change in derivative order and basins of attraction, we show that the system has a path to multistability that reveals the mechanism for the occurrence of this behavior. The resulting dynamics shows the coexistence of up to n + 2 attractors, where n is the number of scrolls generated by the integer-order system, which has been confirmed for two different n-multi-scrolls systems. Moreover, the existence and uniqueness of the dynamics is proved by Poincaré sections, which show that the behaviors exist, and coexist, and are not a section of the same solution.

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.

Complex Network Dynamics of Multiscroll Chaotic Attractors and Their Output-Feedback Pinning Synchronization

The saturated function series have been successfully used to generate multiscroll chaotic attractors. In this paper, we revisit multiscroll chaotic attractors via saturated function series. We find that with a small constant drift acting on the saturated function, the number of scrolls will greatly decrease. This phenomenon brings extra difficulty in pinning synchronization of networked multiscroll chaotic attractors. A new output-feedback pinning controller is designed based on the unknown-input distributed observer to estimate the synchronization error, which has the advantage of saving the communication cost as the transmission of the observer information is not needed.

Multi-scroll fractional-order chaotic system and finite-time synchronization

The definition of fractional calculus is introduced into the 5D chaotic system, and the 5D fractional-order chaotic system is obtained. The new 5D fractional-order chaotic system has no equilibrium, multi-scroll hidden attractor and multi-stability. By analyzing the time-domain waveform, phase diagram, bifurcation diagram and complexity, it is found that the system has no equilibrium but is very sensitive to parameters and initial values. With the variation of different parameters, the system can produce attractors of different scroll types accompanied by bursting oscillation. Secondly, the multi-stability of the hidden attractor is studied. Different initial values lead to the coexistence of attractors of different scroll number, which shows the advantages of the system. The correctness and realizability of the fractional-order chaotic system are proved by analog circuit and physical implement. Finally, because of the high security of multi-scroll attractor and hidden attractor, finite-time synchronization based on the fractional-order chaotic system is studied, which has a good application prospect in the field of secure communication.

Design and FPGA implementation of a memristor-based multi-scroll hyperchaotic system

A novel memristor-based multi-scroll hyperchaotic system is proposed. Based on a voltage-controlled memristor and a modulating sine nonlinear function, a novel method is proposed to generate the multi-scroll hyperchaotic attractors. Firstly, a multi-scroll chaotic system is constructed from a three-dimensional chaotic system by designing a modulating sine nonlinear function. Then, a voltage-controlled memristor is introduced into the above-designed multi-scroll chaotic system. Thus, a memristor-based multi-scroll hyperchaotic system is generated, and this hyperchaotic system can produce various coexisting hyperchaotic attractors with different topological structures. Moreover, different number of scrolls and different topological attractors can be obtained by varying the initial conditions of this system without changing the system parameters. The Lyapunov exponents, bifurcation diagrams and basins of attraction are given to analyze the dynamical characteristics of the multi-scroll hyperchaotic system. Besides, the field programmable gate array (FPGA) based digital implementation of the memristor-based multi-scroll hyperchaotic system is carried out. The experimental results of the FPGA-based digital circuit are displayed on the oscilloscope.

Generating grid chaotic sea from system without equilibrium point

The dynamical system without equilibrium point is considered having hidden dynamics. It is relatively difficult to locate the attractor in the state space as its attraction basin has nothing to do with the equilibrium point. Especially, generation of multi-scroll chaos from no-equilibrium system is a challenging task. In this paper, using sine function, a modified Sprott-A system without equilibrium point but with perpetual points is presented. In particular, this system has the conservative property of zero-sum Lyapunov exponents and thus can generate chaotic sea rather than an attractor. The locations of the scrolls of chaotic sea are found having potential relevance to the sine nonlinearities and perpetual points. Different number of scrolls can be extended only adjusting the system parameters. Three cases of five-term Sprott-A system variants with single-direction multi-scroll/multi-double-scroll chaotic sea and two-direction grid chaotic sea are demonstrated. Besides, hidden tori are found coexisting with the chaotic sea. Numerical simulations and hardware experiments both confirm the complex dynamics of the system.

Compressor Profile Optimization Based on Hybrid Intelligent Algorithm

In order to improve the working characteristics of the scroll compressor, according to the scroll profile of the compressor, the energy efficiency ratio (EER) of the scroll compressor is taken as the objective function, and the number of scroll turns N and knots are determined based on the genetic annealing algorithm. The distance p, the height of the scroll body h, and the thickness of the scroll profile t are optimized. In the optimized solution set, three sets of optimized profile and initial profile are selected for theoretical calculation of thermodynamic characteristics and volume characteristics, and the specific influence of scroll compressor profile parameters on compressor characteristics is explored in detail, and compared with the unoptimized scroll. The initial parameters of the rotary compressor are compared with the theoretical performance. The results show that the pitch p has a significant effect on the energy efficiency ratio and discharge volume of the scroll compressor, and the number of scroll turns N has a significant effect on the characteristic of suction volume. Three kinds of optimized scroll profile parameters S2, S3, S4 are selected in the optimal solution set. Compared with the initial value S1, the working characteristics are improved. The energy efficiency ratio was increased by 38.10%, 42.58%, and 50.26%; the suction volume was increased by 66.1%, 82.3%, and 73.9%; the exhaust volume was increased by 21.1%, 29.6%, and 50%; the internal volume ratio was increased by 36.4%. 40.9%, 27.3%. It is proved that the use of genetic annealing algorithm achieves the purpose of improving the compressor's operating characteristics.