Double Scroll Research Articles

Generating a new chaotic system using two chaotic Rossler-Chua coupling systems

This paper presents a new chaotic system by resulting from the coupling two different chaotic systems such as the Rӧssler system and the Chua system, where the x2 dynamic of the Rӧssler system was coupled with z1 dynamic of the Chua system. Some of the basic dynamics behaviors were examined for new chaotic system by using the Matlab program. In this work was noticed a difference in the time series of the Chua system and this in turn led to a difference in the attractor, as the attractor of the Chua system changed from double scroll to single scroll. This new behavior led to change of the bandwidth of the Chua system that mean the Rӧssler system affected on the Chua system and this led to an increase in the possibility of using this system in different applications.

Analysis of a quarter car suspension based on a Kelvin–Voigt viscoelastic model with fractional-order derivative

The dynamical analysis of a quarter car suspension based on the Kelvin–Voigt​ viscoelastic model with fractional order derivative is analytically and numerically studied in this paper. Depending on the non-dimensional quadratic stiffness parameter, the potential configuration associated with the system under study is possible to obtain monostable and, asymmetric bistable configurations. Thanks to the multiple time scale method, the effects of viscoelastic parameters on the vibration amplitude of the quarter car suspension model are analytically investigated and the numerical simulations confirmed the analytical results. Moreover, the vibration amplitude of the quarter car suspension model can display periodic and dissipative chaotic behaviors by varying the non-dimensional quadratic stiffness, the fractional-order, and the road profile parameters. One scroll and double scroll chaotic attractors are generated by varying the non-dimensional quadratic stiffness parameter. Finally, the threshold conditions for the appearance of homoclinic bifurcation (left well and right well) in the case of asymmetric bistable potential are investigated using Melnikov’s criterion. Melnikov’s predictions are validated numerically by using the basin of attraction of initial conditions. It is found that the decrease of fractional order contributes to increasing the regular motion domain up to a critical value of the fractional-order and after it, decreases rather.

Experimental investigation of double scroll cutting pattern by using button cutting tool of raise boring machines

Full-scale linear cutting test is one the most preferred and reliable methods for performance prediction of mechanical miners. Raise Boring Machines (RBMs) are one of these mechanical miners which are commonly used for drilling / excavation of shafts for different purposes in mining and tunneling projects. This study aims at investigating experimentally double spiral (scroll) cutting pattern in raise boring machine. Full-scale linear cutting tests are performed by using button (kerf) cutting tool. The tests are performed at line spacing of 25 mm and varying depths of cut per revolution by using two hard rocks (diabase and granodiorite) that were obtained from the Yusufeli Dam and HEPP project site. The average and maximum normal, rolling and side forces; specific energy; yield; and coarseness index are measured in each cut. The study indicates that optimum ratio of line spacing to depth of cut per revolution are obtained as 8.3 and 4.2 for diabase and granodiorite samples, respectively.

Studying the Chaotic Dynamics Using Rossler-Chua Systems Combined with A Semiconductor Laser

In this paper, two different chaotic dynamic systems are coupled using a semiconductor laser to produce a new chaotic system. These two chaotic systems are Rossler and Chua systems. X-dynamic of Rossler system was coupled optically using optical fiber as a carrier of signal with x, y, and z-dynamics of Chua system. The results were analyzed and the behavior of Chua system was found to be changing in time series which, in turn, changed the attractor. The Chua attractor was converted from double scroll to single scroll. The results obtained from connecting two different systems in chaotic behavior showed a remarkable increase in the bandwidth of Chua system. This increase in bandwidth opens up a wide field for many applications, the most important of which is in the field of secure communications.

Nonlinear dynamics and chaos in Caputo-like discrete fractional Chen system

In this paper, a fractional Caputo-difference form of discrete Chen system is introduced. The considered system is proposed by implementing the fractional Caputo-like difference operator instead of the standard forward difference operator in the linearized form of the Chen system. The dynamics of the suggested discrete system are numerically illustrated for different fractional orders using phase plots, bifurcation diagrams and Lyapunov exponents. Double scroll chaotic attractors for the considered system are displayed. This study confirms the possible existence of chaos in Caputo-like discrete fractional systems.

A two-directional grid multiscroll hidden attractor based on piecewise linear system and its application in pseudo-random bit generator

In this work, we present a methodology to design dynamical systems that display two-directional grid multiscroll hidden attractors. Self-excited attractors based on piecewise linear systems emerging from its equilibria in order to exhibit in one-directional (1D) and two-directional (2D) grid multiscroll hidden attractors. The particularity of this methodology is that the self-excited multiscroll attractor is separated into several double scroll self-excited attractors to allow the appearance of a hidden attractor. Next, the hidden attractor is used as the core of a cryptographic safe pseudo-random bit generator and statistical evidence is provided to ensure its convenience in cryptographic applications.

A Novel Technique for Donor Insertion and Unfolding in Descemet Membrane Endothelial Keratoplasty.

To describe a novel technique for insertion and unfolding of donor Descemet membrane (DM) scroll during DM endothelial keratoplasty (DMEK). DMEK was performed in 20 eyes. The donor DM scroll was loaded into a glass injector, and the fluid column was manipulated to achieve a double scroll configuration. After performing DM stripping and inferior iridectomy, an anterior chamber (AC) maintainer was placed in position. The glass injector was inserted into the AC and the tip rotated to ensure correct orientation of the DM scroll. After disconnecting the IV tubing from the AC maintainer, the donor DM scroll was delivered into the AC. Donor unfolding was performed using surface tapping techniques. An air bubble was injected under the graft to obtain a complete AC air fill. The outcomes of this surgical technique were analyzed. Donor DM insertion and unfolding were performed without any complications in all eyes. Double scroll configuration of donor DM in the injector could be achieved in 17 of 20 eyes (85%). A controlled delivery, maintaining correct scroll orientation, was possible in all eyes. Intraoperative duration from insertion to unfolding was 252.7 ± 45.4 seconds. A clear cornea was achieved in all eyes, with 75% of eyes (15/20) achieving corrected distance visual acuity of 20/20 or better, at average follow-up duration of 13.3 ± 1.3 months. Endothelial cell loss at 6 months follow-up was noted to be 16.7%. Partial donor detachment was noted in 2 eyes (10%), which resolved spontaneously. Our novel technique of donor DM insertion allows easy donor insertion and unfolding, thereby simplifying the DMEK procedure.

Dynamic behavior of fractional-order memristive time-delay system and image encryption application

This paper presents dynamic behavior of a fractional-order memristive time-delay system and its application in image encryption. First, a fractional-order memristive time-delay system is proposed, and the stability and bifurcation behaviors of the system are theoretically analyzed. Some limited conditions for describing the stability interval and switching between different dynamic behaviors are derived. Second, the dynamic characteristics of the system are analyzed through the coexisting attractors, coexisting bifurcation diagrams, the Largest Lyapunov exponents (LLE), the 0-1 test. When parameters change, such as time delay and fractional order, the system transits from steady state to periodic state, single scroll chaotic state, double scroll chaotic state. Furthermore, an image encryption scheme based on the fractional-order memristive time-delay system is introduced, and some statistical features are analyzed. Finally, numerical simulations verify the validity of the theoretical analysis and safety of the image encryption scheme based on the fractional-order delayed memristive chaotic system.

Application of particle swarm optimization with ANFIS model for double scroll chaotic system

The predictions for the original chaos patterns can be used to correct the distorted chaos pattern which has changed due to any changes whether from undesired disturbance or additional information which can hide under chaos pattern. This information can be recovered when the original chaos pattern is predicted. But unpredictability is most features of chaos, and time series prediction can be used based on the collection of past observations of a variable and analysis it to obtain the underlying relationships and then extrapolate future time series. The additional information often prunes away by several techniques. This paper shows how the chaotic time series prediction is difficult and distort even if Neuro-Fuzzy such as Adaptive Neural Fuzzy Inference System (ANFIS) is used under any disturbance. The paper combined particle swarm (PSO) and (ANFIS) to exam the prediction model and predict the original chaos patterns which comes from the double scroll circuit. Changes in the bias of the nonlinear resistor were used as a disturbance. The predicted chaotic data is compared with data from the chaotic circuit.

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.

Construction and Analysis of Variable Wall Thickness Profile of Double Scroll Teeth of Scroll Compressor

Based on the changing trend of the geometric parameters of the scroll profile of the involute of circle with variable radii, a new type of double scroll teeth profile of scroll compressor composed of involute of circle with variable radii is constructed. The construction process of the baseline is discussed, and the baseline equations are deduced. According to the principle of normal equidistance, the geometric model of variable wall thickness double scroll teeth is established, and the suction volume, tooth thickness, and scroll area of double scroll teeth are compared with those of the double scroll teeth composed of ordinary involute of circle. The results show that: compared with ordinary double scroll teeth with constant thickness, the scroll diameter when two scrolls mesh is reduced and the revolutions of orbiting scroll required in the working process are reduced and it is easy to select better scroll profile through adjusting the changing coefficient of the base circle radius k, which provides a reference for scroll machinery using environment-friendly working fluids.

Composite One- to Six-Scroll Hidden Attractors in a Memristor-Based Chaotic System and Their Circuit Implementation

Chaotic systems with hidden multiscroll attractors have received much attention in recent years. However, most parts of hidden multiscroll attractors previously reported were repeated by the same type of attractor, and the composite of different types of attractors appeared rarely. In this paper, a memristor-based chaotic system, which can generate composite attractors with one up to six scrolls, is proposed. These composite attractors have different forms, similar to the Chua’s double scroll and jerk double scroll. Through theoretical analysis, we find that the new system has no fixed point; that is to say, all of the composite multiscroll attractors are hidden attractors. Additionally, some complicated dynamic behaviors including various hidden coexisting attractors, extreme multistability, and transient transition are explored. Moreover, hardware circuit using discrete components is implemented, and its experimental results supported the numerical simulations results.

Initial offset boosting coexisting attractors in memristive multi-double-scroll Hopfield neural network

Memristors are widely considered to be promising candidates to mimic biological synapses. In this paper, by introducing a non-ideal flux-controlled memristor model into a Hopfield neural network (HNN), a novel memristive HNN model with multi-double-scroll attractors is constructed. The parity of the number of double scrolls can be flexibly controlled by the internal parameters of the memristor. Through theoretical analysis and numerical simulation, various coexisting attractors and amplitude control are observed. Particularly, the interesting and rare phenomenon of the memristor initial offset boosting coexisting dynamics is discovered, in which the initial offset boosting coexisting double-scroll attractors with banded attraction basins are distributed in a line along the boosting route with the variation of the memristor initial condition. In addition, it is also found that the number of the initial offset boosting coexisting double-scroll attractors is closely related to the total number of scrolls and ultimately tends to infinity with increasing the total number of scrolls, meaning the emergence of extreme multistability. Then, the random performance of the initial offset boosting coexisting double-scroll attractors is tested by the NIST test suite. Moreover, an encryption scheme based on them is also proposed. The obtained results show that they have excellent randomness and are suitable for image encryption application. Finally, numerical simulation results are well demonstrated by circuit experiments, showing the feasibility of the designed memristive multi-double-scroll HNN model.

Anatomical description of the head in Ossimi Sheep: Sectional anatomy and Computed Tomographic approach

The objective of present investigation was to explore the morphology of the head in Ossimi sheep. Here, the nasal, paranasal sinuses, oral, orbital, cranial, auricular and pharyngeal cavities were examined by the computed tomography images interpreted by the anatomical sections. Also, three-dimensional CT of the head was reconstructed. The Ossimi sheep was shown to possess long, double scroll dorsal and ventral nasal conchae that occupy the nasal cavity rostrally, while its caudal part possessed five small ethmoidal triangular nasal projections, the largest one was the middle nasal concha and the others contained the ethmoidal sinus. The dorsal and middle nasal meatuses were narrow and the ventral meatus was wide and lead directly to the choanae. The nasal septum failed to reach the floor at the caudal part of the nasal cavity thus forming a single channel that continued to the nasopharynx. The vomeronasal organ extended throughout the length of the nasal cavity from the incisive duct caudal to the dental pad to the level of the 3rd upper cheek tooth. The paranasal sinuses identified were the frontal, maxillary, lacrimal, palatine and ethmoidal sinuses. The study recorded the absence of the sphenoid sinus and small palatine sinus in sheep different from that in other ruminants. The obtained results may be useful as a basic anatomical reference for the surgeons and pathologists.

Emergence of a square chaotic attractor through the collision of heteroclinic orbits

In this work we introduce a square chaotic attractor based on the collision of two heteroclinic orbits. Before the collision, the system presents the coexistence of two double scroll attractors that are generated via piecewise linear (PWL) systems that deal with two saddle-foci equilibria of different classes, i.e. the dimensions of the unstable and stable manifolds of one of the equilibrium points are one and two, respectively, and vice versa for the unstable and stable manifolds associated with the second equilibrium point. The new class is constructed based on the existence of a heteroclinic loop by linear affine systems with two saddle-focus equilibrium points of different types. The chaotic behavior of the proposed system is tested by the maximum Lyapunov exponent and the 0–1 chaos test. This new class of PWL dynamical systems exhibits different behaviors and allows the generation of different basins of attraction for the coexistence of two or more attractors. Therefore, a mechanism to establish bistability is provided and how it can be controlled (i.e. bistability annihilation) via varying system’s parameters is demonstrated.

The effects of symmetry breaking on the dynamics of a simple autonomous jerk circuit

We investigate the dynamics of a simple jerk circuit where the symmetry is broken by forcing a dc voltage. The analysis shows that with a zero forcing dc voltage, the system displays a perfect symmetry and develops rich dynamics including period doubling, merging crisis, hysteresis, and coexisting multiple (up to six) symmetric attractors. In the presence of a non-zero forcing dc voltage, several unusual and striking nonlinear phenomena occur such as coexisting bifurcation branches, hysteresis, asymmetric double scroll strange attractors, and multiple coexisting asymmetric attractors for some appropriate sets of system parameters. In the latter case, different combinations of attractors are depicted consisting for instance of two, three, four, or five disconnected periodic and chaotic attractors depending solely on the choice of initial conditions. The investigations are carried out by using standard nonlinear analysis tools such as Lyapunov exponent plots, bifurcation diagrams, basins of attraction, and phase space trajectory plots. The theoretical results are checked experimentally and a very good agreement is found between theory and experiment.

On the Dynamics and Control of a Fractional Form of the Discrete Double Scroll

This paper is concerned with the dynamics and control of the fractional version of the discrete double scroll hyperchaotic map. Using phase portraits and bifurcation diagrams, we show that the general behavior of the proposed map depends on the fractional order. We also present two control schemes for the proposed map, one that adaptively stabilizes the map, and another to achieve the complete synchronization of a pair of maps. Numerical results are presented to illustrate the findings.

Dynamical Effects of Neuron Activation Gradient on Hopfield Neural Network: Numerical Analyses and Hardware Experiments

Hyperbolic tangent function, a bounded monotone differentiable function, is usually taken as a neuron activation function, whose activation gradient, i.e. gain scaling parameter, can reflect the response speed in the neuronal electrical activities. However, the previously published literatures have not yet paid attention to the dynamical effects of the neuron activation gradient on Hopfield neural network (HNN). Taking the neuron activation gradient as an adjustable control parameter, dynamical behaviors with the variation of the control parameter are investigated through stability analyses of the equilibrium states, numerical analyses of the mathematical model, and experimental measurements on a hardware level. The results demonstrate that complex dynamical behaviors associated with the neuron activation gradient emerge in the HNN model, including coexisting limit cycle oscillations, coexisting chaotic spiral attractors, chaotic double scrolls, forward and reverse period-doubling cascades, and crisis scenarios, which are effectively confirmed by neuron activation gradient-dependent local attraction basins and parameter-space plots as well. Additionally, the experimentally measured results have nice consistency to numerical simulations.

Synchronization of Multi-Layer Networks: From Node-to-Node Synchronization to Complete Synchronization

Multi-layer networks, which incorporate multiple subsystems and different kinds of interactions, are recently believed to have a stronger ability in modelling various real-world systems than traditional single-layer complex networks. Motivated by this observation, we try to show how to achieve node-to-node synchronization and complete synchronization in multi-layer networks under directed switching communication topologies, where the nodes have Lur’e type dynamics. First, by using the multiple Lyapunov stability theory, we show that node-to-node synchronization in two-layer networks can be achieved by choosing sufficiently large coupling strength as well as suitable feedback gain matrix if the average dwell time is bounded below by some positive constants. The result is then extended to the case of multiple layers. Furthermore, if the inner communication topologies of each layer have a directed spanning tree with a common root, we show complete synchronization in the considered network can be achieved if the control parameters and the average dwell time are appropriately chosen. Finally, the theoretical results are validated through performing numerical simulations on Chua’s circuit with double scroll attractors.

Effects of symmetric and asymmetric nonlinearity on the dynamics of a novel chaotic jerk circuit: Coexisting multiple attractors, period doubling reversals, crisis, and offset boosting

Abstract A novel self-driven RC chaotic jerk circuit with the singular feature of having a smoothly adjustable nonlinearity and symmetry is proposed and investigated. The novel chaotic circuit is mathematically modeled by a third order system with a single nonlinear term in the form ϕ k ( x ) = 0.5 ( exp ( k x ) − exp ( − x ) ) where parameter k maps a smoothly adjustable control resistor. Obviously for k = 1 , the system is point symmetric with respect to the origin of the system coordinates since the nonlinear term reduces to the hyperbolic sine function. The case k ≠ 1 corresponds to a non-symmetric system. The numerical experiment reveals a plethora of events including period doubling route to chaos, hysteresis, periodic windows, asymmetric double scroll chaos, symmetric double scroll chaos, and coexisting bifurcations branches as well. This latter phenomenon induces multiple coexisting attractors consisting of two, three, four, five, or six disconnected symmetric or asymmetric attractors for the same set of parameter values when monitoring solely the initial conditions. Laboratory experimental measurements are carried out to confirm the theoretical predictions.