Multi-scroll Attractors Research Articles

Privacy Protection of Medical Data Based on Multi-Scroll Memristive Hopfield Neural Network

Memristive Hopfield neural network (MHNN) has complex dynamic behavior, which is suitable for encryption applications. In order to ensure the information security of the medical data transmitted in Internet of Things (IoT), we propose three new MHNN models by using a non-ideal flux-controlled memristor model with multi-piecewise nonlinearity. In these models, there are complex dynamical behaviors such as coexisting attractors, multi-scroll attractors and grid multi-scroll attractors. In terms of hardware, the proposed model is implemented using field programmable gate array (FPGA). In addition, we provide a complete set of medical data sharing solution, which are helpful for the referral patients to receive timely medical treatment. The whole solution is successfully verified on Raspberry Pi, the encrypted Computed Tomography (CT) image is transmitted safely under Message Queuing Telemetry Transport (MQTT) protocol, and the CT image is subjected to basic security analysis. The results show that the ciphertext histogram is evenly distributed, the correlation between adjacent pixels is almost 0, the information entropy reaches 7.9977, and the values of number of pixels change rate (NPCR) and unified average change intensity (UACI) are 99.6078% and 33.4875%. The solution not only performs the exchange of medical data, but also protects the privacy of patients.

A Memristive Synapse Control Method to Generate Diversified Multistructure Chaotic Attractors

Due to the synapse-like nonlinearity and memory characteristics, memristor is often used to construct memristive neural networks with complex dynamical behaviors. However, memristive neural networks with multistructure chaotic attractors have not been found yet. In this article, a novel method for designing multistructure chaotic attractors in memristive neural networks is proposed. By utilizing a multipiecewise memristive synapse control in a Hopfield neural network (HNN), various complex multistructure chaotic attractors can be produced. Theoretical analysis and numerical simulation demonstrate that multiple multistructure chaotic attractors with different topologies can be generated by conducting the memristive synapse-control in different synaptic coupling positions. Differing from traditional multiscroll attractors, the generated multistructure attractors contain multiple irregular shapes instead of simple scrolls. Meanwhile, the number of structures can be easily controlled with the memristor control parameters. Furthermore, we design a module-based analog memristive neural network circuit and the arbitrary number of multistructure attractors can be obtained by selecting corresponding control voltages. Finally, based on the memristive HNNs, a novel image encryption cryptosystem with a permutation-diffusion structure is designed and evaluated, exhibiting its excellent encryption performances, especially the extremely high key sensitivity.

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.

Symmetric multi-scroll attractors in magnetized Hopfield neural network under pulse controlled memristor and pulse current stimulation

In this paper, a new method for generating symmetric multi-scroll attractors is proposed based on a magnetized Hopfield neural network (HNN). Firstly, the mathematical model of magnetized HNN by introducing an original memristor to describe the effect of electromagnetic radiation is established, and its dynamic characteristics are analyzed. Then, utilizing two types of multi-level-logic pulse functions, the complex attractor structure changes of the magnetized HNN under external pulse controlled memristor, pulse current stimulation and their combined effects are discussed respectively. The numerical simulation results show that the magnetized HNN under pulse controlled memristor or/and pulse current stimulation can produce multi-scroll chaotic and multi-scroll periodic strange attractors with symmetric structure. The number of attractors can be changed by setting the different series of multi-level-logic pulse functions. Finally, the magnetized HNN circuit under pulse controlled memristor or/and pulse current stimulation is realized by using discrete analog electronic devices. The experimental results verify the validity of the numerical simulation results.

Generation of Multistability through Unstable Systems

In this work, we propose an approach to generate multistability based on a class of unstable systems that have all their roots in the right complex half-plane. Multistability is the coexistence of multiple stable states for a set of system parameters. The approach is realized by using linear third order differential equations that consists of two parameters. The first bifurcation parameter transforms the unstable system with all its roots in the right complex half-plane into an unstable system with one root in the left complex half-plane and two roots remaining in the right complex half-plane. With this first transformation, the system is capable of generating attractors by means of a piecewise linear function and the system presents monostability. We then use the another bifurcation parameter to switch from a monostable multiscroll attractor to several multistable states showing a single-scroll attractor.

CHUA’S CIRCUITS DYNAMIC WITH NON-IDEAL COILS AND HYSTERESIS

A complete dynamic study (theoretical, experimental and numerical) of a chaotic Chua-type circuit was carried out using a non-ideal coil with non-negligible values of its internal resistance. This set up means that the circuit does not show chaotic behavior. In order to observe chaoticity, a modification that introduces hysteresis to the Chua diode is proposed. Using spaces of up to three control parameters it is shown how this modification adapts to a wide range of non-ideal inductors presenting more extensive chaotic regions than the classical Chua circuit. These simulations are tested with a physical circuit and demonstrate how the modifications result in a simpler experimental design of the inductor. By simplifying the experimental design the difficulties caused by the electronic component is eliminated and results in the absence of Rössler type attractors and the asymmetry of the double-scroll attractors.

Multiscroll in Bidirectionally Coupled Jerk Oscillators: Theoretical Analysis and PSpice Verification

It is known that the unidirectional coupling between a double-scroll system (drive) in the oscillation regime with another system (response) in the steady-state, gives rise to more complex dynamics such as multiscroll attractors [Dana et al., 2008]. This paper shows the way to move from n-scroll to 2n-scroll in a bidirectional coupling scheme between two jerk oscillators. The coupling considered can be interpreted as a perturbation to each oscillator’s amplitude through a signal proportional to the amplitude of the other. Indeed, starting from a situation of two oscillators which can generate double-scroll dynamics individually, we perform a bidirectional coupling of both oscillators which results in more complex dynamics such as multistability and 4-scroll chaotic attractor in the coupled system (not observed in an isolated oscillator). The underlined coupling scheme provides an efficient method for generating higher-order multiscroll attractors. The route to the 4-scroll chaotic attractor is explained with the help of bifurcation diagrams, plots of several phase portraits, and basins of attractions. Indirect validation of the theoretical analysis is carried out by using the PSpice simulation software.

A novel type of chaotic attractor with a multiunit structure: from multiscroll attractors to multi-bond orbital attractors

A novel type of chaotic attractor with a multiunit structure: from multiscroll attractors to multi-bond orbital attractors

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.

High-complex chaotic system based on new nonlinear function and OTA-based circuit realization

This research proposes a chaotic system using a new modified rectangular nonlinear function based on the principle of the third-order jerk concept. The dynamic behaviors can be realized by manipulating only one parameter from periodic, non-periodic, single-scroll, and double-scroll attractors. The proposed chaotic system can achieve relatively high-complexity chaotic behavior, as evident in the comparatively high Kaplan-Yorke dimension (DKY = 2.43) with the existing jerk systems. The performances of the proposed chaotic system are also computed and simulated through the MATLAB numerical analysis, including scenarios of bifurcations branches, Lyapunov exponents, route to chaos, phase portraits, 0–1 test charts, and Poincaré map. The proposed system analyzes the coexistence of asymmetric or symmetric attractors confirmed with basins of attraction to highlight the presence of multiple attractors, which influence initial conditions. In addition, the system's offset boosting and amplitude control are included. Furthermore, a double-scroll attractor and a four-scroll attractor can also be achieved by placing nonlinear functions at different positions in the third-order system., The circuits of the proposed chaotic systems can be conveniently realized using commercially available operational transconductance amplifiers (OTAs) and capacitors. The performances of the circuit are also proven through the experimental results compared with the numerical results. The experimental results exhibit the ease of construction based on the new nonlinear function and are suitable for applying in the data encryptions.

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.

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.

Chaos and Dynamics Induced by the Amplitude of Load Torque: Analysis and Control

In this contribution, the dynamics and chaos control of a permanent magnet reluctance synchronous motor (PMSM) with constant, sinusoidal, and square load torque taking the reference rotor speed and the stator voltage on d-q-axis are analyzed. The analysis shows that under some conditions, varying amplitude of the load torque, PMSM presents period doubling route to chaos with one and double-scroll attractors and bursting oscillations. We use the self-feedback delay controller to stabilize the output trajectory by forcing to the periodic solution. Furthermore, an analog circuit is designed and implemented in OrCAD-PSpice software to verify the numerical results and good agreement is observed with numerical results.

Dynamic Analysis and Audio Encryption Application in IoT of a Multi-Scroll Fractional-Order Memristive Hopfield Neural Network

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

Segmented disc dynamo with symmetric multidirectional patterns of multiscroll chaotic attractors

Rather than adopting the common approach that the multiscroll attractors are obtained through reconstruction of nonlinear functions, this work presents a systematic mechanism for designing multidirectional symmetric multiscroll patterns of chaotic attractors. The present scheme is based on the multilevel signal that work with the nonlinearities and state variables of chaotic system to generate symmetric multidirectional patterns of multiscroll chaotic attractors (MPMCA). To illustrate the mechanism of this systematic approach, the coefficients of quadratic terms of segmented disc dynamo are modified and designed a chaotic system that generates a chaotic attractor consisting of two pairs of symmetrical attractors. The state variables under the multilevel signal are transformed and then combined the symmetric attractors to generate a complete family of symmetric MPMCA in multi directions (including one directional (1D), two directional (2D) and three directional (3D) symmetric patterns of multiscroll chaotic attractors). Chaotic dynamical characteristics of symmetric MPMCA are analysed​ by means of equilibrium points distribution and phase portraits. Matlab simulations results obtained in this work are consistent well with our theoretical analysis verifying the effectiveness of the presented scheme.

Dynamic analysis and application in medical digital image watermarking of a new multi-scroll neural network with quartic nonlinear memristor.

Memristor is widely used in various neural bionic models because of its excellent characteristics in biological neural activity simulation. In this paper, a piecewise nonlinear function is used to transform the quartic memristor, which is introduced into the ternary Hopfield neural network (HNN) with self-feedback, and a piecewise quartic memristive chaotic neural network model with multi-scroll is constructed. Through simulation analysis, the number of scroll layers changes with memristor parameters and has significant coexistence of multi-scroll attractors and high initial value sensitivity has been found. Using its excellent unpredictability, a digital watermarking algorithm based on wavelet transform is improved and used in the protection of personal medical data. The results show that it not only improves the confidentiality and convenience, but also ensures its robustness and has good encryption effect.

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.

Three-Dimensional Sine Chaotic System With Multistability and Multi-Scroll Attractor

In this brief a simple three-dimensional sine chaotic system formed by six terms is presented. The only one nonlinear term, i.e., the sinusoidal function makes the system have infinitely many equilibria. This prominent feature lead to infinitely many coexisting self-excited and hidden attractors with the same or different shapes, also namely homogenous multistability or heterogenous multistability. Furthermore, the multiple equilibria make the system successfully generate multi-scroll attractors. These complicated dynamical behaviors are numerically studied and further experimentally demonstrated by microcontroller-based hardware platform.

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.

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.