Double Scroll Research Articles

A novel approach to the convergence analysis of chaotic dynamics in fractional order Chua’s attractor model employing fixed points

This article uses a fixed point method to connect with Chua’s attractor model, incorporating the Atangana–Baleanu derivative using a two-step Lagrange polynomial. We introduce novel fixed point theorems using some special contractions followed by graphical representations of the convergence behavior of the iterative process. The uniqueness and existence of fixed points are demonstrated through theoretical proofs and numerical simulations. This approach of its first kind substantiates the existence and uniqueness of the model, enriching the understanding of its chaotic dynamics. Furthermore, numerical investigations explore distinct scenarios of Chua’s model, encompassing the classic Chua’s attractor, the double scroll attractor, chaos in Chua’s circuit, and behaviors such as cycle behavior and higher parametric values. Each case is analyzed graphically, providing insights into the system’s complex dynamics and validating theoretical predictions.

Dynamic behaviors of far and near memristive electromagnetic induction in spoon neural network.

In this paper, a special spoon neural network is proposed, which is composed of four neurons with direct connection and indirect connection. On this basis, the far induction network and the near induction network (NINN) are constructed by using hyperbolic tangent memristors to explore the influence of electromagnetic induction between neurons at different positions on the dynamic behavior of attractors. NINN exhibits more complex attractor structures and wider chaotic parameters, and also displays a heterogeneous coexisting attractor of limit cycles and chaos under network parameter control. By varying the parameters, coexisting chaotic attractors can be synthesized into a double scrolls attractor, and their oscillation amplitude can be controlled without changing the chaotic characteristics. The type of attractors in human brain determines the clarity of memory. These complex dynamic behaviors demonstrate that near induction has a more pronounced effect on the forgetting and disappearance of memory compared to far induction. Finally, a circuit using switches to change the type of electromagnetic induction is constructed and the results are verified.

A Novel Hypothesis for Migraine Disease Mechanism: The Creation of a New Attractor Responsible for Migraine Disease Symptoms

Migraine Disease (MD) is one of the common primary headaches that can prevent patients from their everyday life. Despite the high prevalence, the pathophysiology of the disease has not been clearly understood yet. Here, the brain is considered as a dynamical system. The Chua’s circuit with a chaotic attractor is the proposed model. This attractor has a one-scroll mode representing a healthy brain and a double-scroll mode representing a migraine sufferer brain. We believe that MD and Chua’s systems have certain behavioral similarities. The boundaries of the attractor are the sensitive brain areas in which any small trigger can start the ictal phase of the migraine. The transition from the inter-ictal phase to the ictal phase in migraine patients occurs due to a decrease in serotonin levels when the brain is within the boundaries of the first attractor. Here, this is the results of the increase of system parameters. In addition, the transition from the ictal phase to the inter-ictal phase in a migraine sufferer brain is caused by a disruption of coordination in the brain’s structures and this lasts for a certain period for every migraine patient. The structures which are the result of the Migraine Generator Network (MGN) and Cortical Spreading Depression (CSD). This explanation may propose newer methods for preventing or curing MD. To better understand MD to control it and shrink the areas involved in this disease, it is better to know the dynamic systems better. It may help prevent the formation of migraine ictal attractor or even make the migraine ictal phase attractor smaller even after it has been formed.

An Integrated Reservoir Predictor Based on Spatiotemporal Information Transformation

Multistep prediction of high-dimensional time series is an essential and challenging task. In this study, we propose an integrated reservoir predictor for making accurate and robust multistep-ahead forecasts based on short-term high-dimensional time series. Initially, a conjugated pair of Spatiotemporal Information (STI) equations is derived using Takens’ embedding theory to transform the spatial information of high-dimensional variables into one-dimensional temporal information of the target variable and vice versa. Next, by exploiting reservoir networks, reservoir-based STI equations are established to efficiently capture nonlinear dynamics of the target system with only linear optimization. Then, through an integration phase, the integrated reservoir predictor can output precise and robust predictions of the multistep-ahead states of any target variable. The integrated reservoir predictor outperforms some other prediction methods (including reservoir computing, long-short-term-memory network, convolutional neural network and support vector regression), when applied to classical dynamic systems (e.g. 60D double scroll model, 40D Lorenz 96 model, and 60D Rössler model) and real-world datasets (solar generation data and PM2.5 concentration records), as indicated by evaluation metrics such as Pearson correlation coefficients exceeding 0.9 and root-mean-square errors below 0.3, even in the presence of noise in training data.

Design and Analysis of Multiscroll Memristive Hopfield Neural Network With Adjustable Memductance and Application to Image Encryption.

Memristor is an ideal electronic device used as an artificial nerve synapse due to its unique memory function. This article presents a design of a new Hopfield neural network (HNN) that can generate multiscroll attractors by utilizing a new memristor as a synapse in the HNN. Differing from the others, this memristor is constructed with hyperbolic tangent functions. Taking the memristor as a self-feedback synapse of a neuron in the HNN, the memristive HNN can yield multidouble-scroll attractors, and its parameters can be used to effectively control the number of double scrolls contained in an attractor. Interestingly, the generation of multidouble-scroll attractors is independent of the memductance function but depends only on the internal state equation. Thus, the memductance function can be adjusted to yield various complex dynamical behaviors. Moreover, amplitude control effects and quantitatively controllable multistability are revealed by numerical analysis. The accurate reproduction of some dynamical behaviors by a designed circuit verifies the correctness of the numerical analysis. Finally, based on the proposed memristive HNN, a novel image encryption scheme in the 3-D setting is designed and evaluated, demonstrating its good encryption performances.

“Double Scroll” sign in a man aged 60 years after closed globe injury

“Double Scroll” sign in a man aged 60 years after closed globe injury

Generating Grid Multi-Scroll Attractors in Memristive Neural Networks

Memristors are well suited as artificial nerve synapses owing to its unique memory function. This paper establishes a novel flux-controlled memristor model using hyperbolic function series. By taking the memristor as synapses in a Hopfield neural network (HNN), three memristive HNNs are constructed. These memristive HNNs can generate multi-double-scroll chaotic attractors or grid multi-double-scroll chaotic attractors. The number of double scrolls in the attractors is controlled by the memristor. Equilibrium points analysis further reveals the generation mechanism of grid multi-double-scroll chaotic attractors. Moreover, numerical simulations indicate the existence of complex dynamics in the memristive HNNs, including extreme multistability and amplitude control. An approach to physically realize grid multi-double-scroll chaotic attractors is also given. Finally, an encryption scheme based on the proposed memristive HNN is designed to demonstrate application potential of the attractors.

Experimental observation of chaotic hysteresis in Chua's circuit driven by slow voltage forcing

The Chua's circuit has been considered as a paradigm for the investigation of chaos, but it still presents many unexplored dynamics. Chaotic hysteresis is an effect observed when the hysteresis phenomenon and chaotic dynamics phenomenon act simultaneously. It is an interesting phenomenon since it is observed in a variety of disciplines, and it has been utilized in explaining the mechanism of bursting oscillations, rich dynamics in thermal convection, and economic financial crisis, to mention a few. In this paper, we report the experimental observation of chaotic hysteresis in a Chua's electronic circuit driven by both a dc and a slow triangular voltage source. One and two hysteresis loops were observed in single and double scroll chaotic regimes, respectively. We present variations of the attractor that are induced by voltage changes, as well as the dependence of the hysteresis effect on both the driving frequency and the distance from the threshold passing from single to double-scroll chaotic regimes. Our results help in expanding our knowledge on unexplored dynamics of Chua's circuit as well as contribute to chaos control theory with a potential for applications.

Cupolets in a chaotic neuron model.

This paper reports the first finding of cupolets in a chaotic Hindmarsh-Rose neural model. Cupolets (chaotic, unstable, periodic, orbit-lets) are unstable periodic orbits that have been stabilized through a particular control scheme by applying a binary control sequence. We demonstrate different neural dynamics (periodic or chaotic) of the Hindmarsh-Rose model through a bifurcation diagram where the external input current, I, is the bifurcation parameter. We select a region in the chaotic parameter space and provide the results of numerical simulations. In this chosen parameter space, a control scheme is applied when the trajectory intersects with either of the two control planes. The type of the control is determined by a bit in a binary control sequence. The control is either a small microcontrol (0) or a large macrocontrol (1) that adjusts the future dynamics of the trajectory by a perturbation determined by the coding function r ( x ). We report the discovery of many cupolets with corresponding control sequences and comment on the differences with previously reported cupolets in the double scroll system. We provide some examples of the generated cupolets and conclude by discussing potential implications for biological neurons.

Double-Scroll Formation by Fluid Column Manipulation in Preloaded DMEK Grafts Prepared From Younger and Older Donor Tissue.

The purpose of this study was to determine whether manipulation of preloaded single-scroll Descemet membrane endothelial keratoplasty (DMEK) grafts within the fluid column of an injector can safely and reliably result in formation of double-scroll DMEK grafts and whether there are differential effects on younger versus older donor tissue. Pairs of DMEK grafts prepared from older (65-80 years) and younger (48-64 years) donors were preloaded into a Straiko modified Jones tube. One member of the pair was manipulated within the fluid column to form a double-scroll graft, and the other remained unmanipulated. Outcomes measured include success rate for double-scroll formation, endothelial cell loss (ECL), and relative scroll width. Older donor grafts formed double scrolls with a 100% success rate. ECL of older donor manipulated grafts was statistically higher than that of unmanipulated mate grafts (17.4% ± 3.5% vs. 13.0% ± 4.2%, P = 0.03), but was still within the acceptable range for transplant. Younger donor grafts were successfully manipulated into double scrolls with a 67% success rate, and there was no difference in the ECL of manipulated and unmanipulated grafts (15.5% ± 4.4% vs. 13.0% ± 4.5%, P = 0.24). For all grafts and conformations, there was a significant relationship between relative scroll width and ECL ( P < 0.01). Fluid column manipulation can be used reliably to form double-scroll DMEK grafts. For younger donor grafts, manipulation yields a double scroll without increasing ECL. For older donor grafts, manipulation results in a minimal, acceptable increase in ECL. Surgeons should weigh the advantage of an easily opened graft against the risk of increased ECL when considering this technique.

Coupled non-oscillatory Duffing oscillators: Multistability, multiscroll chaos generation and circuit realization

We consider the dynamics of two coupled double-well non-oscillatory cubic Duffing oscillators. The coupling is realized by perturbing each one’s amplitude with a signal proportional to the amplitude of the other one. We prove that the coupling induces in the resulting fourth-order self-driven nonlinear system up to nine equilibrium points and engendered extremely complex dynamic features amongst which the coexistence of several bifurcation branches, multiple Hopf bifurcations (when a single parameter is considered), multistable behaviors (e.g. a pair of coexisting double scroll chaos, four coexisting mono-scroll chaos), and four-scroll chaotic attractor. A detailed investigation of these latter features is carried out by resorting to both analytical (i.e. Hopf bifurcation theorem, Routh–Hurwitz criterion) and numerical tools (e.g. Lyapunov exponent diagrams, 1D bifurcation diagrams, frequency spectrum, phase space trajectory plots, time series and attraction basins). An analog circuit implementing the coupled Duffing oscillators is performed by using simple electronic components (i.e. capacitors, resistors, quadruple operational amplifiers and analog multipliers chips). Experimentally captured phase portraits support the conclusions of the theoretical study. One of the important conclusions of this work is that the coupling of non-oscillatory Duffing type oscillators can be considered as an alternative approach to the generation of multiscroll chaos.

Resistorless Chua's diode implementation for chaotic oscillation employing single EXCCCII

A passive resistor-free Chua’s Diode with smooth nonlinearity is introduced using a single Extra X Current Controlled Current Conveyors (EX-CCCII). The proposed Chua’s Diode can generate all fundamental Chua’s chaotic responses like Rossler type attractor, double scroll attractor, and large limit cycle. Chua’s chaotic response can be easily observed in two ways: (i) by tuning the bridging resistor between two capacitors present in a Chua's circuit and (ii) by adjusting the bias current of EX-CCCII present in the proposed Chua’s diode. The proposed Chua’s model is verified with three possible combinations: Chua’s oscillation with a passive inductor, tunable Chua’s oscillation by varying the bias current present in EX-CCCII based negative resistance, and inductor less Chua’s circuit using active inductor based on EX-CCCII. Moreover, the effect of non-idealities present in EXCCCII on negative resistance and dynamical investigation of theproposed Chua’s circuit is described for better understanding. The functionality of the proposed Chua’s circuit is well verified through PSPICE simulation using 0.25 μm TSMC CMOS technology parameters and Current Feedback operational amplifier (CFOA) as IC AD844 based EX-CCCII. In addition to computer simulation, an experimental investigation is also performed by CFOA as IC AD844 to realize EX-CCCII. Finally, a comparison between the existing Chua’s circuit and the proposed one is illustrated for a fair comparison.

Graft Size and Double Scroll Formation Rate in Descemet Membrane Endothelial Keratoplasty

Purpose This study aimed to evaluate the usefulness of intentional double scroll formation of donor Descemet membrane (DM) inside a glass tube inserter (the Fogla technique) in DM endothelial keratoplasty (DMEK) for controlled insertion and unfolding of grafts. Methods Eleven consecutive patients who underwent DMEK were included in this study. We sought to specify graft characteristics in which double scroll configuration was successfully formed using the Fogla technique. We compared donor age, graft size, surgical time, unfolding time, and visual outcomes between patients with and without double scroll configuration. The ability to form double scroll formation of DM grafts of various diameters and unfolding time of DM grafts was evaluated using total seven eye-bank eyes in ex vivo experiments. Results A double scroll configuration inside a glass tube was successfully obtained in six DMEK grafts (54.5%). When comparing clinical features between those with and without double scroll configuration, only graft size was significantly larger in those with double scroll configuration (7.9 ± 0.2 mm) than in those without (7.4 ± 0.4, P = 0.03). There were no significant differences in other features and clinical outcomes, although unfolding-time was shorter in eyes with double scroll configuration (4.6 ± 2.0 min) compared to those without (8.6 ± 8.1, P = 0.21). Ex vivo experiments showed that unfolding time was significantly shorter in double scroll configuration (2.71 ± 0.49 min) than in single scroll (5.02 ± 0.79, P = 0.01). Conclusions A double scroll configuration within a glass tube can be obtained more frequently in larger DMEK grafts (8.0 mm in diameter), which may result in easier and faster DMEK procedures.

Examination of a Modified Graft Preparation Technique to Induce Double-Scroll Formation and Promote the Use of Younger Descemet Membrane Endothelial Keratoplasty Donor Tissue.

The purpose of this study was to determine whether controlled balanced salt solution (BSS) bursts during graft preparation can safely promote formation of a double-scrolled Descemet membrane endothelial keratoplasty (DMEK) graft in younger donor tissue. DMEK grafts prepared from young donor tissue (average age, 55 years; range, 39-66 years) were floated in BSS to spontaneously form scrolls (N = 10 pairs). Controlled BSS bursts were used to promote double-scroll (DS) formation in 1 member of each pair. Grafts were stained, preloaded, and shipped before cell viability analysis. After appropriate training, a less experienced technician performed this technique on 10 additional corneas. Outcomes measured for both technicians include the success rate for obtaining a DS, scroll conformation after shipping, and endothelial cell loss (ECL). There was no difference in ECL between grafts subjected to additional manipulation compared with unmanipulated mate grafts (observer 1: 15.2% ± 3.3% vs. 15.2% ± 4.4%, P = 0.99; observer 2: 16.3% ± 2.9% vs. 15.9% ± 4.5%, P = 0.8). A technician experienced with this technique had a 90% success rate, whereas a less experienced technician had a 70% success rate. The mean ECL of the 10 grafts manipulated by the less experienced technician was not significantly different from results obtained from the experienced technician (observer 1: 18.5% ± 6.0% vs. 15.2% ± 3.3%, P = 0.15; observer 2: 18.1% ± 5.6% vs. 16.3% ± 2.9%, P = 0.34). Scrolls maintained their conformation during shipping events. Double-scroll graft formation using controlled BSS bursts is a reliable technique that can be performed without causing additional damage to DMEK grafts. This technique may make graft unscrolling easier and can promote the use of younger donor tissue for DMEK.

Generating Any Number of Initial Offset-Boosted Coexisting Chua’s Double-Scroll Attractors via Piecewise-Nonlinear Memristor

Due to the natural nonlinearity and unique memory characteristics, memristors are promising candidates for the construction of multiscroll attractors having better application potential in the field of information encryption than the traditional double-scroll attractors. This article proposes a novel memristive multidouble-scroll Chua’s system (MMDSCS) via coupling a nonideal flux-controlled memristor with multipiecewise-linear memductance function in Chua’s system directly. Specially, any number of multidouble-scroll chaotic attractors can be generated through adjusting the internal parameters of the memristor conveniently and without changing the original system’s nonlinearity. Moreover, the amount of double scrolls is also closely related to the strength of the memristive coupling. Another striking highlight is that infinite initial offset-boosted coexisting Chua’s double-scroll attractors with the same shape are produced with the variation of the memristor initial conditions, indicating the emergence of an intriguing phenomenon of homogeneous extreme multistability. This unique property and its formation mechanism are investigated in detail using phase portraits, bifurcation diagrams, Lyapunov exponents, time series, and attraction basins. Furthermore, hardware experiments based on the field-programmable gate array are carried out to confirm the numerical simulations. Finally, an image encryption scheme is designed based on the memristor initial offset boosting dynamics from a perspective of engineering application. In comparison with the existing memristive Chua’s systems, the proposed MMDSCS has many merits, such as multidouble-scroll attractors, memristor initial-controlled chaotic sequences with controllability, good robustness, and high security performance, which is more practical in applications involving information confidential communication.

Impact by double scroll on variable geometry turbine performance

Abstract This paper describes the numerical study for the development of a double scroll(DS) turbine stage combined with a variable geometry system (VGS). Our conventional VGS turbine stage is composed of a mono-scroll housing, variable nozzles and a turbine wheel. A potential to improve engine performance by applying a DS with VGS turbine stage (DS-VGS) was shown by the study. In addition, a DS-VGS has some advantage in the layout for V type engine. Therefore, this study has focused on clarifying the characteristics of aerodynamic performance of a DS-VGS, since the understanding of these fundamental characteristics is important for product development. The differences of performance between a mono-scroll with VGS and a DS with VGS were evaluated as the starting point. In the development for a vaneless DS turbine so far, it has been clarified that the distance from tongue to turbine wheel is one of the important parameters for performance. Therefore, numerical simulations were performed with different tongue-to-nozzle distances. Further, from reliability point of view, the influences of a DS on nozzle moments and aerodynamic forces on nozzle have been examined. The interesting characteristics of a DS-VGS compared to a mono-scroll with VGS are shown.

Rocard’s 1941 Chaotic Relaxation Econometric Oscillator

In the beginning of the Second World War, the French physicist, Yves Rocard, published a book entitled Théorie des Oscillateurs (Theory of Oscillators). In Chapter V, he designed a mathematical model consisting of a set of three nonlinear differential equations and allowing to account for economic cycles. Numerical integration of his model has highlighted a chaotic attractor. Its analysis with classical tools such as bifurcation diagram and Lyapunov Characteristic Exponents has confirmed the chaotic features of its solution. It follows that Rocard’s 1941 chaotic econometric model has thus most likely preceded Lorenz’ butterfly of twenty-two years. Moreover, apart from this historical discovery which upsets historiography, it is also established that this “new old” three-dimensional autonomous dynamical system is a new jerk system whose solution exhibits a chaotic attractor the topology of which varies, from a double scroll attractor to a Möbius-strip and then to a toroidal attractor, according to the values of a control parameter.

Limit cycles and chaos induced by a nonlinearity with memory

Inspired by the observation of a distributed time delay in the nonlinear response of an optical resonator, we investigate the effects of a similar delay on a noise-driven mechanical oscillator. For a delay time that is commensurate with the inverse dissipation rate, we find stable limit cycles. For longer delays, we discover a regime of chaotic dynamics associated with a double scroll attractor. We also analyze the effects of time delay on the spectrum and oscillation amplitude of the oscillator. Our results point to new opportunities for nonlinear energy harvesting, provided that a nonlinearity with distributed time delay can be implemented in mechanical systems.

Multistable systems with nested hidden and self-excited double scroll attractors

Multistable systems with nested hidden and self-excited double scroll attractors

Mathematical models of CBSC over wireless channels and their analysis by using the LeNN-WOA-NM algorithm

In recent studies, several chaos-based secure communication (CBSC) systems are developed and are assumed to be effective and promising tools for balancing communications in wireless applications. The reliability of a wireless communication system is dependent on the synchronization between receiving and transmitting nodes. Therefore, it is crucial to analyze the mathematical models for different enhancements through CBSC and avoid the difficulty in synchronization. We have designed a novel computational paradigm that uses weighted Legendre polynomials to construct series solutions for mathematical models of the Lorenz Chaotic Attractor (LCA) and Double Scroll Attractor (DSA) (with filter and without filter) by using Chua’s circuits. We design fitness functions for each system of differential equations. Two optimization techniques, the Whale Optimization Algorithm (WOA) and the Nelder–Mead (NM) algorithm are adopted to optimize the objective functions and calculate the best set of unknown weights. We name our soft computing paradigm, the LeNN-WOA-NM algorithm. We have compared our results with state-of-the-art, and we establish that the LeNN-WOA-NM algorithm is fast, accurate, and reliable. We present graphical and statistical analysis to further elaborate on the performance of our algorithm.