Abundant Dynamical Behaviors Research Articles

Coexisting and multiple scroll attractors in a Hopfield neural network with a controlled memristor

Abstract In this paper, a method of generating multi-double scroll attractors is proposed based on the memristor Hopfield neural network (HNN) under pulse control. First, the original hyperbolic-type memristor is added to the neural network mathematical model, and the influence of this memristor on the dynamic behavior of the new HNN is analyzed. The numerical results show that after adding the memristor, the abundant dynamic behaviors such as chaos coexistence, period coexistence and chaos period coexistence can be observed when the initial value of the system is changed. Then the logic pulse is added to the external memristor. It is found that the equilibrium point of the HNN can multiply and generate multi-double scroll attractors after the pulse stimulation. When the number of logical pulses is changed, the number of multi-double scroll attractors will also change, so that the pulse can control the generation of multi-double scroll attractors. Finally, the HNN circuit under pulsed stimulation was realized by circuit simulation, and the results verified the correctness of the numerical results.

Bionic firing activities in a dual mem-elements based CNN cell

Firing activities provide the potential possibility for achieving bio-brain functionality with high energy-efficient and high-speed information processing performance. This inspires the design of bionic circuits to generate firing activities and develop brain-like applications. In this paper, a dual mem-elements based cellular neural network (CNN) cell is constructed to produce bionic firing activities, in which a non-ideal memcapacitor and an N-type locally active memristor are employed to emulate the functions of the neuronal membrane. The proposed CNN cell has an excitation-dependent equilibrium trajectory and stability. Numerical analysis shows that the dual mem-elements based CNN cell has abundant dynamical behaviors of forward/reverse period-doubling bifurcation routes, chaos crisis, tangent bifurcation, and bubbles with the change of model parameters of the CNN cell, memcapacitor, and exciting source. As a result, the rich firing patterns’ transition can be observed from the two-dimensional dynamics evolution. The analog circuit of the proposed CNN cell is designed, and then a PCB-based hardware circuit is implemented. The experimental results certify the accuracy of the theoretical and numerical analysis.

Dynamics of memristive Liu system and its DSP implementation

In this paper, a novel four-dimensional memristive system is investigated to generate abundant dynamical behaviors. By combining the Liu chaotic system with an ideal memristor, an enhanced chaotic system is proposed. Dynamical analysis indicates that the new system sustains stable chaotic states and exhibits complex behaviors, with the help of the Lyapunov exponents, bifurcation diagrams, Poincaré section, parametric offset boosting, and SE complexity. The coexistence of attractors is investigated by the variation of parameters. The chaotic performance is enhanced in the proposed system, broadens the range of parameters for chaotical oscillations, and transforms periodic and quasiperiodic states into chaos. The practical applicability and feasibility of the system are validated via NIST testing and DSP implementation. The system exhibits resilient dynamical characteristics that make it highly suitable for deployment in various domains, including secure communication and signal detection.

Collective behaviors of fractional-order FithzHugh–Nagumo network

Brain connection is the mechanism of determining the brain function and cognition, and the small world network is widely investigated among these connections. In this paper, the small world complex brain network is constructed by the fractional-order neurons, and the fractional-order memristor is used to connect these neurons. Compared with the integer-order counterpart, the fractional-order memristor describes precisely the distortion of hysteresis curves, and it exhibits more abundant dynamical behaviors to function as the synapse. In addition, the neurons with different fractional orders represent the individual differences during the cell differentiation. For the two coupled neurons, the phase lock is achieved in the bursting and spiking neurons, and it denotes a time delay between the two firing process, whereas the chaotic neurons cannot synchronize. In small world network, the collective behaviors of neurons tend to synchronize with the increasing of the coupling intensity. These results promote the understanding of information coding and transformation in complex brain network.

The excitation of high-order localized waves in (3+1)-dimensional Kudryashov-Sinelshchikov equation

The aim of this work is to explore the excitation of high-order localized waves in the (3+1)-dimensional Kudryashov-Sinelshchikov equation, which is used to describe the dynamic of liquid with gas bubble. First of all, classical N-soliton solutions are constructed by means of Hirota bilinear form and symbolic calculation. What’s more, the high-order breather waves are derived through the degeneration process of the N-soliton solutions with conjugate parameter. Then, high-order lump waves are constructed by taking long wave limit technique on N-soliton solutions. Finally, the high-order mixed localized waves involving resonant Y-type solitons, high-order breather waves and high-order lump waves are obtained by utilizing some comprehensive methods. Abundant dynamical and evolutionary behaviors of these results are investigated specifically, some figures are presented to shed light on the nonlinear phenomena hidden in the high-order localized waves vividly.

Dynamic behaviors analysis of fraction-order neural network under memristive electromagnetic induction

The dynamic behaviors of coupled neurons with different mathematical representations have received more and more attention in recent years. The coupling among heterogeneous neurons can show richer dynamic phenomena, which is of great significance in understanding the function of the human brain. In this paper, we present a fraction-order heterogeneous network with three neurons, which is built by coupling an FN neuron with two HR neurons. Complex electromagnetic surroundings have meaningful physical influence on the electrical activities of neurons. To imitate the effects of electromagnetic induction on the three-neuron heterogeneous network, we introduce a fraction-order locally active memristor in the neural network. The characteristics of this memristor are carefully analyzed by pinched hysteresis loops and its locally active characteristic is proved by the power-off plot and the DC <i>v-i</i> plot. Then, the parameter-dependent dynamic activities are investigated numerically by using several dynamical analysis methods, such as the phase diagrams, bifurcation diagrams, Lyapunov exponent spectra, and attraction basins. In addition, the network also reveals rich dynamic behaviors, including coexisting activities, anti-monotonicity phenomena, transient chaos and firing patterns, providing support for further investigating the firing patterns of the human brain. In particular, complex dynamics, including coexisting attractors, anti-monotonicity, and firing patterns, can be influenced by the order and strength of electrical synaptic coupling and electromagnetic induction. The control of the bistable state can be realized through the time feedback control method, so that the bistable state can be transformed into an ideal monostable state. The study of the fraction-order memristive neural network may expand the field of view for understanding the collective behaviors of neurons. Finally, based on the ARM platform, we give a digital implementation of the fraction-order memristive neural network, which can verify the consistency with the numerical simulation results. In the future, we will explore more interesting memristive neural networks and study different types of methods to control the firing behaviors of the networks.

Four Novel Dual Discrete Memristor-Coupled Hyperchaotic Maps

Unlike the high-dimensional hyperchaotic system based on a continuous memristor, the low-dimensional map coupled by discrete memristor (DM) and traditional chaotic map can also generate hyperchaos. However, the hyperchaotic map constructed by two DMs has not attracted much attention. To this end, a generalized two-dimensional dual DM-coupled hyperchaotic mapping model is reported in this paper, and four specific maps are provided. The proposed maps have line invariant points, which can be interpreted as allowing arbitrary real values for the initial condition associated with the DM, and the stability is investigated in detail. Furthermore, the coupling strength-dependent and initial condition-dependent complex dynamics of four maps are studied by numerical simulations, and the dynamical performance is evaluated from the perspective of quantitative analysis. It is shown that the considered maps are capable of exhibiting the three characteristic fingerprints of memristors in arbitrary parameter spaces, and this characteristic has gained attention for the first time. In particular, the complete control of the considered maps by variable substitution is performed, which can generate arbitrary switched hyperchaotic behaviors. In addition, four pseudo-random number generators are designed based on the proposed maps, and the randomness is tested by using the NIST SP800-22 software. In general, the proposed maps can not only generate abundant dynamical behaviors, but also enrich the DM circuits and provide a reference for applications based on chaos. Finally, the developed digital hardware circuit implementation platform verifies the results of the numerical method.

Symmetry, Multistability and Antimonotonicity of a Shinriki Oscillator with Dual Memristors

In this paper, a type of modified dual memristive Shinriki oscillator is constructed with a flux-controlled absolute-type memristor and a voltage-controlled generic memristor, and the proposed oscillator with abundant dynamical behaviors, including the multistability and antimonotonicity, is comprehensively studied through dynamical distribution graphs, bifurcation diagrams, Lyapunov exponents and phase portraits. It is found that the passive/active state of memristor, which means different characteristics in the [Formula: see text]–[Formula: see text] domain with positive and negative parameters of the elements, can affect the state of the oscillator. For example, if the memristor is active, the oscillator will change more frequently in the multistable region. Also, it is noted that, for inherent initial-related symmetry and circuit structures with duality, both phenomena have strong symmetric characteristics and opposite evolution trends modulated by values of corresponding components. Especially, the bubbles, which are symmetric about parameters with duality and own complex evolution laws, have rarely been explored in previous works. In addition, the memristive oscillator is modularized based on field programmable gate array (FPGA) technology, and the multiple coexisting attractors are captured, which verifies the accuracy of the numerical results.

A new fractional-order 5D memristive chaotic system with special extreme multistability and its application to image encryption

Introducing memristor into the chaotic system can enrich the dynamic behaviors of the chaotic system. We propose a symbolic function memristor model and introduce it into a chaotic system to construct a fractional-order 5D memristor chaotic system. Through dynamic analysis of equilibrium point, Lyapunov exponents, phase diagram and bifurcation diagram, it is found that the system has abundant dynamic behaviors, for example, the change of equilibrium point type with parameters, transient chaos, offset-boosting and a special kind of extreme multistability. And with the change of parameters, the attractor state and shape will appear rich changes. Then the correctness of the system is verified by circuit simulation. The chaotic system is introduced into the process of image encryption, and an encryption system is constructed, which is composed of Zigzag scrambling, Hilbert curve scrambling, DNA encryption and GF257 diffusion algorithm. Finally, through a variety of security verification, the results show that the encryption system has good security and can resist many kinds of attacks effectively.

Projection Synchronization of Three-Dimensional Chaotic Systems With Active Control Based on DNA Strand Displacement.

The emergence of biological computing based on DNA strand displacement has allowed chaotic systems to have more abundant dynamic behaviors. So far, the synchronization of chaotic systems based on DNA strand displacement has been mainly realized by coupling control and PID control. In this paper, the projection synchronization of chaotic systems based on DNA strand displacement is achieved using an active control method. First, some basic catalytic reaction modules and annihilation reaction modules are constructed based on the theoretical knowledge of DNA strand displacement. Second, the chaotic system and the controller are designed according to the above mentioned modules. On the basis of chaotic dynamics, the complex dynamic behavior of the system is verified by the lyapunov exponents spectrum and the bifurcation diagram. Third, the active controller based on DNA strand displacement is used to realize the projection synchronization between the drive system and the response system, where the projection can be adjusted within a certain range by changing the value of the scale factor. The result of projection synchronization of chaotic system is more flexible, which is realized by active controller. Our control method provides an efficient way to achieve synchronization of chaotic systems based on DNA strand displacement. The designed projection synchronization is verified to have excellent timeliness and robustness by the results Visual DSD simulation.

Multiple modes of bursting phenomena in a vector field of triple Hopf bifurcation with two time scales

For the development of the theoretical framework of slow–fast dynamics, it is essential to study the mechanism of bursting oscillations with high co-dimensional bifurcations in the high-dimensional vector field. This paper is aimed to explore the bursting oscillations in the normal form up to the third order of a vector field with triple Hopf bifurcation at the origin. We establish an eight-dimension system with the slow–fast effect, in which two time scales exist. The fast subsystem is represented as a six-dimensional vector field with triple Hopf bifurcation at the origin, while the slow subsystem is adopted to adjust the variation of parameters in the fast subsystem, which may lead to abundant dynamic behaviors. To better understand the qualitatively different behaviors in the fast subsystem, the trivial solution, single-mode, two-mode, and three-mode movements are introduced based on the projections of trajectory on the three sub-planes. As a result, several types of bursting phenomena, such as Hopf/Hopf bursting oscillations with single-mode, Hopf/Hopf/fold bursting oscillations with single-mode, Hopf/Hopf bursting oscillations with two-mode have been revealed, the mechanism of which is obtained upon the superposition of equilibrium branches together with bifurcations in the fast subsystem as well as transformed phase portrait. In addition, for bursting oscillations on the whole phase space, the amplitude of spiking oscillations may change according to two different limit cycles, which leads to an abrupt change of the oscillating amplitude on the sub-plane, corresponding to different projections of Poincaré. It can be observed the state variables on the different sub-planes alternate between synchronized and non-synchronized states in the fast subsystem, which may be accounted for by the bifurcations.

Bifurcation and Chaotic Behavior of Duffing System with Fractional-Order Derivative and Time Delay

In this paper, the abundant nonlinear dynamical behaviors of a fractional-order time-delayed Duffing system under harmonic excitation are studied. By constructing Melnikov function, the necessary conditions of chaotic motion in horseshoe shape are detected, and the chaos threshold curve is obtained by comparing the results obtained through the Melnikov theory and numerical iterative algorithm. The results show that the trend of change is the same, which confirms the accuracy of the chaos threshold curve. It could be found that when the excitation frequency ω is larger than a certain value, the Melnikov theory is not valid for these values. Furthermore, by numerical simulation, some numerical results are obtained, including phase portraits, the largest Lyapunov exponents, and the bifurcation diagrams, Poincare maps, time histories, and frequency spectrograms at some typical points. These numerical simulation results show that the system exhibits some new complex dynamical behaviors, including entry into the state of chaotic motion from single period to period-doubling bifurcation and chaotic motion and periodic motion alternating under the necessary condition of chaotic occurrence. In addition, the effects of time delay, fractional-order coefficient, fractional order, linear viscous damping coefficient, and linear stiffness coefficient on the chaotic threshold curve are discussed, respectively. Those results reveal that there exist abundant nonlinear dynamic behaviors in this fractional-order system, and by adjusting these parameters reasonably, the system could be transformed from chaotic motion to non-chaotic motion.

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

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

Dynamic Analysis and Fuzzy Fixed-Time Optimal Synchronization Control of Unidirectionally Coupled FO Permanent Magnet Synchronous Generator System

This paper focuses on dynamic analysis and the fuzzy fixed-time optimal synchronization control problem of unidirectionally coupled fractional-order (FO) permanent magnet synchronous generator (PMSG) system. The synchronization model between FO master and slave PMSGs with capacitive and resistive couplings is built. The dynamic analysis fully reveals its abundant dynamical behaviors including chaotic oscillations and gives stability/instability boundaries with the designed numerical method. In controller design, the hierarchical type-2 fuzzy neural network (HT2FNN) with a transformation is designed to approximate unknown functions, the fixed-time command filter matched up with the compensating signal is proposed to achieve precise estimate and fast convergence, and a fixed-time preconfigured performance function integrated with a smooth and invertible function is built to realize fixed-time convergence and performance constraint. Then a fuzzy fixed-time optimal synchronization control scheme fusing with the HT2FNN, filter, performance function and optimal control is developed under the FO backstepping theory. The stability analysis proves that all signals of the closed-loop system are bounded along with the cost function being minimized. Finally, numerical simulation results verify the feasibility and advantages of our scheme.

Abundant soliton solutions and different dynamical behaviors of various waveforms to a new (3+1)-dimensional Schrödinger equation in optical fibers

Abundant soliton solutions and different dynamical behaviors of various waveforms to a new (3+1)-dimensional Schrödinger equation in optical fibers

Global Behavior Analysis of Nonlinear Energy Sink Systems Using the Parallelized Multi-Degrees-of-Freedom Cell Mapping Method

Abstract Nonlinear energy sink (NES) systems, when applied to a physical system with multiple interference sources, exhibit abundant nonlinear dynamic behaviors. However, current research in this respect is limited within the theoretical scope of deterministic systems. According to the theory of cell mapping, this paper introduces a parallel restructured algorithm to improve the performance of cell mapping and cell processing, and a parallelized multidegrees-of-freedom (DOF) cell mapping (PMDCM) method is given. With the method, the global behavior of NES systems is analyzed so that the dynamical behavior of multiple stable attractors within typical parameter intervals can be captured. The research results show that for NES systems, there is the phenomenon of multiple stable attractors coexisting in multiple typical parameter intervals, which occurs between periodic and periodic attractor, periodic and quasi-periodic, periodic and chaotic attractor. While revealing the corresponding relationship between different types of attractors and their basin of attraction, these findings verify that the new cell mapping method has high computational efficiency and accuracy and can provide a theoretical basis for the study of high-dimensional nonlinear systems' global behavior and optimal control.

Hyperchaotic Oscillation and Multistability in a Fourth Order Smooth Chua System with Implementation Using No Analog Multipliers

Abundant dynamical behaviors and succinct realization forms are of great significance for nonlinear circuits and systems. Hyperchaos represents more complex oscillation mode compared with chaos, while multistability means more complex essential features compared with robustness. And the coexistence of the two is rare but interesting. Based on the classic Chua model, a fourth order smooth Chua system is constructed through utilizing linear feedback control method in this article. The introduction of hyperbolic sine nonlinearity remarkably simplifies the physical realization form for the new system. The two interesting nonlinear phenomena of hyperchaotic oscillation and multistability are discovered by adjusting system parameters. Dynamic analyses including bifurcation diagrams and Lyapunov exponent spectra comprehensively reveal the evolution of oscillatory behaviors brought by the variation of system parameters, whereupon more attractors such as periodic attractors, chaotic attractors in different oscillation scales, coexisting quasi-periodic attractors, coexisting multiple attractors and coexisting periodic attractors are found. And the attractor morphologies of this system are rare and significantly different from those in the reported Chua circuits and systems. Hardware circuit experiments without using analog multipliers are performed from a four-layer printed circuit board, which efficiently verify the simulated dynamical behaviors within the acceptable deviation.

Spatial dynamics of a fractional predator-prey system with time delay and Allee effect

In this paper, we investigate the spatial dynamics of a class of spatial fractional predator-prey systems with time delay and Allee effect. Firstly, Hopf bifurcation and Turing bifurcation conditions are obtained by using stability theory and bifurcation theory. Then, the abundant dynamic behaviors of the system are demonstrated by numerical simulation. Finally, numerical results show that time delay, Allee effect, and fractional diffusion can affect the formation 3D Turing patterns such as spherical, tube, spherical and tube coexisting and the constitution of 3D spiral wave patterns.

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.

Modeling Allee Effect in the Leslie-Gower Predator–Prey System Incorporating a Prey Refuge

This paper considers a Leslie-Gower predator–prey system with Allee effect and prey refuge. By considering the prey refuge constant as a parameter, we analyze the stability of the equilibria in the system, and find that there are abundant dynamic behaviors. It is shown that the model can undergo a sequence of bifurcations including saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation of codimension two or three as the parameters vary. Moreover, the model underdoes a degenerate Hopf bifurcation of codimension two and has two limit cycles, where the inner one is stable and the outer one is unstable. Through some numerical simulations, the occurrence of Bogdanov–Takens bifurcation and Hopf bifurcation of codimension two are confirmed.