Narrow Results

Introducing the memcapacitor into the Cellular Neural Network (CNN), the Memcapacitor-Cellular Neural Network (MC-CNN) model with infinitely many equilibrium points is constructed. A series of dynamical behaviors of the MC-CNN are investigated by various nonlinear system analysis means. It is shown that the system has a large maximum Lyapunov exponent in a specific parameter range. And with the variation of parameters, the system is able to produce many different phase trajectories of the attractor. Multistability is also found in the system. The pseudo-randomness of the MC-CNN is calculated by Spectral Entropy (SE) complexity algorithm. The final hardware results proves the physical realizability of the system. The MC-CNN model is intended to provide guidance for neural networks and cryptographic strategies based on the memcapacitor.

This paper presents an investigation into the dynamics of a Memristor-Coupled Heterogeneous Tabu Learning Neuronal Network (MCTLNN) comprising distinct types of Tabu learning neurons. The network adopts both monotone and nonmonotone composite hyperbolic tangent functions as activation mechanisms. A sinusoidal function-based memristor interconnects the neurons, leading to intriguing dynamical interactions. A theoretical analysis reveals the existence of infinitely many equilibria in the MCTLNN. Utilizing multiple numerical simulation methods, including phase portraits, Lyapunov exponents, bifurcation diagrams, and local attraction basin analyses, we systematically investigate the system’s complex dynamics. This study discovers the coexistence of diverse attractors, such as heterogeneous, symmetric, and initial offset-boosted attractors. Notably, we explore the dependence of the system’s dynamics on the initial conditions, specifically highlighting the initially offset boosting phenomenon through extensive simulations. Analog circuit-based hardware implementation not only confirms the model’s theoretical predictions, but also provides a platform for brain-inspired computing applications.

Mechanical metamaterials are highly versatile structures capable of achieving unconventional mechanical properties. Of particular interest are mechanical metamaterials with the ability to tune bandgaps, offering potential applications in vibration control tailored to specific requirements. In this study, a new type of metamaterial is introduced to exhibit a distinctive two-step deformation mode under compressive displacement. Through a combination of numerical simulation and experimental verification, the band structure and vibration characteristics of the novel metamaterial in different stable states are thoroughly examined under compressive displacement. The results demonstrate that, in comparison to the initial structure, the novel mechanical metamaterial effectively addresses the issue of a significant reduction in plateau stress relative to buckling stress following buckling in the second deformation stage. The novel structure showcases notable controllable deformation capabilities, yielding distinct band structures in various stable states, and facilitating bandgap tuning. Furthermore, the study explores the influence of geometrical parameters and stable state transitions on bandgap evolution, supported by frequency response analyses and experimental validation. This research offers a fresh perspective on the utilization of Multistable Multi-Step Deformation Metamaterial (MMDM) for energy absorption and vibration damping applications.

The stochastic dynamical ${\rho }^4$ equation is utilized as a robust framework for modeling the behavior of complex systems characterized by randomness and nonlinearity, with applications spanning various scientific fields. The aim of this paper is to employ an analytical method to identify stochastic traveling wave solutions of the dynamical ${\rho }^4$ equation. Novel hyperbolic and rational functions are investigated through this method. A Galilean transformation is applied to reformulate the model into a planar dynamical system, which enables a comprehensive qualitative analysis. Additionally, the emergence of chaotic and quasi-periodic patterns following the introduction of a perturbation term is addressed. Simulation results indicate that significant changes in the systems’ dynamic behavior are caused by adjusting the amplitude and frequency parameters. Our findings indicate the impact of the method on system dynamics and its efficacy in analyzing solitons and phase behavior in nonlinear models. These discoveries provide fresh perspectives on how the suggested method can lead to notable shifts in the systems’ dynamic behavior. The effectiveness and practicality of the proposed methodology in scrutinizing soliton solutions and phase visualizations across diverse nonlinear models are underscored by these revelations.

In this paper, we design and explore a novel 3D autonomous nonlinear system with logarithmic nonlinearity. It is framed from the Lorenz model by replacing the variable $y$ in the second equation by the nonlinear logarithmic term $ln|y|$. The bi-parametric dynamics of the modeled system is presented. Studies reveal the emergence of shrimp-shaped periodic structures in chaotic regime with period-adding phenomenon, and coexisting periodic-chaotic attractors, which do not occur in the Lorenz model. The stability, bifurcation patterns, and chaotic behaviors of the system are explored with the aid of bifurcation diagrams, phase portraits and Lyapunov exponents. The complexity of the basin sets for the coexisting attractors is also studied.

We investigate the effects of hole doping in the charge-density wave (CDW) state that has the strong electron-phonon (e-p) coupling, using the two-dimensional molecular crystal model. In calculations, we use the mean-filed theory for the interelectronic interactions and the adiabatic approximation for phonons. On the basis of this theory, we calculate e-p states of doped ground states for various values of the doping concentration of holes. From the calculated results, it is found that a multistable nature appears in the doped e-p states just before the CDW-metal phase transition. In order to see the effects of the photoexcitation in the hole-doped states, we also investigate the exciton states taking into account the electron-hole correlation. Results calculated here indicate that the separation of electron and hole occurs in the photoexcited states as a consequence of the energy relaxation of excitons.

In this paper, multistability is discussed for delayed recurrent neural networks with ring structure and multi-step piecewise linear activation functions. Sufficient criteria are obtained to check the existence of multiple equilibria. A lemma is proposed to explore the number and the cross-direction of purely imaginary roots for the characteristic equation, which corresponds to the neural network model. Stability of all of equilibria is investigated. The work improves and extends the existing stability results in the literature. Finally, two examples are given to illustrate the effectiveness of the obtained results.

The axially accelerating beam systems are widely used in engineering fields, which have abundant dynamical phenomena. In this work, we studied the periodic orbits and the phenomena of multistability of the system. The necessary and sufficient conditions for the existence of periodic orbits are derived by Schauder’s fixed point theorem and the averaging method. The theoretical results are in good agreement with the numerical analysis. Multistability is discussed in some set of parameter values. To predict the dynamical behaviors of the system in the long-term run, the effects of external excitation amplitude on the structures of basins of attraction are studied. The results reveal the influences of the physical parameters and initial values on the complex dynamical behaviors of the system, which can offer some guidance for selecting parameter values in engineering application of axially accelerating beams.

In this paper, a novel four-wing chaotic system is constructed based on the Sprott-A system. The novel system contains three nonlinearly quadratic terms, which have rich dynamics such as four-wing hidden attractors, hidden extreme multistability and transient transitions. It is found that this system has high complexity by spectral entropy analysis. In addition, a corresponding hardware analog circuit is designed based on this system by operational amplifiers and multipliers. The experimental results agree with those of the theoretical analysis and confirm that this system is practically feasible. This system will have a good application value in secure communication and cryptography.

The purpose of this work is to introduce a novel 4D chaotic system and investigate its multistability. The novel system has an unstable origin and two stable symmetrical hyperbolic equilibria. When the parameter increases across a critical value, the equilibria lose their stability and double Hopf bifurcations occur with the appearance of limit cycles. A pair of point, periodic, chaotic attractors are observed in the system from different initial values for given parameters. The chaos of the system is yielded via period-doubling bifurcation. A double-scroll chaotic attractor is numerically observed as well. By using the electronic circuit, the chaotic attractor of the system is realized. The control problem of the system is reported. An effective controller is designed to stabilize the system.

This paper presents the universality of the coexistence of complete synchronization with the recently new proposed inverse π-lag synchronization, in the case of mutually coupled systems of Chua's circuit family. The phenomenon of multistability and the nature of this circuit family systems are the key-points which lead to the explanation of this coexistence. For the need of this work, the most representative circuit of this circuits' family, the Chua oscillator, is used. Simulation results confirm this universality in the class of Chua's circuit family.

Recently, a series of typical three-dimensional dissipative chaotic flows where all but one of the nonlinearities are quadratic are studied. Based on this research, a novel chaotic model with only one single linearity is proposed by introducing cubic terms and four new chaotic systems with various characteristics are found. Besides, a chaotic family with a single linearity is constructed with those four chaotic systems and 12 existing systems SL₁–SL${}_{12}$ of the chaotic flows. Exploiting the new systems, basic dynamic behaviors are analyzed, including the strange attractors, equilibrium points, Lyapunov exponents as well as the property of multistability. In addition, the corresponding simulation results are illustrated to show those properties expressly. In realizing the chaotic circuit, we utilize the field programmable gate array (FPGA), which is of considerable flexibility, good programmability and stability, instead of analog devices that are easily affected by surroundings. More importantly, the circuit of the proposed chaotic family is realized on a single FPGA over register transfer level (RTL) using 32-bit fixed-point operation. Finally, an experimental FPGA-based circuit is constructed, and the output results are shown on oscilloscope, which agree well with the numerical simulations.

Chaotic systems with a curve of equilibria have attracted considerable interest in theoretical researches and engineering applications because they are categorized as systems with hidden attractors. In this paper, we introduce a new three-dimensional autonomous system with cubic equilibrium. Fundamental dynamical properties and complex dynamics of the system have been investigated. Of particular interest is the coexistence of chaotic attractors in the proposed system. Furthermore, we have designed and implemented an electronic circuit to verify the feasibility of such a system with cubic equilibrium.

The activation function is crucial in the Hopfield neural network (HNN) to restrict the input–output relation of each neuron. The physical realizability and simplicity of the hardware circuit of activation function are beneficial to promote the practical engineering application of the HNN. However, the HNN commonly used hyperbolic tangent activation function involves a complex hardware circuit implementation. This paper discusses a piecewise-linear activation function (PWL-AF) with simplified circuit implementation and a tri-neuron small-world HNN is built as a paradigm. The hardware implementation circuit of the HNN is greatly simplified, benefited from the PWL-AF with a simple analog circuit. Meanwhile, the dynamics related to the PWL-AF and initial conditions are numerically explored. The numerical results demonstrate that the PWL-AF-based HNN can produce dynamical behaviors like the HNN based on the hyperbolic tangent activation function. Nevertheless, the multistability with up to six kinds of coexisting multiple attractors emerged because of the PWL-AF breakpoint. This can give more flexible and potential aspects in multistability-based engineering applications. Especially, the PWL-AF breakpoint value simultaneously acts as the offset booster and amplitude controller in regulating the offset boosting and amplitude rescaling of neuron states. Afterwards, an analog circuit with three straightforward operational amplifiers (op-amp)-based circuit modules is designed for the PWL-AF, and a PCB-based analog circuit is thereby implemented for the tri-neuron small-world HNN. The hardware experiments agree with the numerical simulations, implying the feasibility of the PWL-AF simplification for the HNN.

One of the usual ways to build up mathematical models corresponding to a wide class of DC–DC converters is by means of piecewise linear differential equations. These models belong to a class of dynamical systems called Variable Structure Systems (VSS). From a classical design point of view, it is of interest to know the dynamical behavior of the system when some parameters are varied. Usually, Pulse Width Modulation (PWM) is adopted to control a DC–DC converter. When this kind of control is used, the resulting mathematical model is nonautonomous and periodic. In this case, the global Poincaré map (stroboscopic map) gives all the information about the system.

The classical design in these electronic circuits is based on a stable periodic orbit which has some desired characteristics. In this paper, the main bifurcations which may undergo this orbit, when the parameters of the circuit change, are described. Moreover, it will be shown that in the three basic power electronic converters Buck, Boost and Buck–Boost, very similar scenarios are obtained. Also, some kinds of secondary bifurcations which are of interest for the global dynamical behavior are presented. From a dynamical systems point of view, VSS analyzed in this work present some kinds of bifurcations which are typical in nonsmooth systems and it is impossible to find them in smooth systems.

We consider the dynamics of the first vibrational mode of a suspension bridge, resulting from the coupling between its roadbed (elastic beam) and the hangers, supposed to be one-sided springs which respond only to stretching. The external forcing is due to time-periodic vortices produced by impinging wind on the bridge structure. We have studied some relevant dynamical phenomena in such a system, like periodic and quasiperiodic responses, chaotic motion, and boundary crises. In the weak dissipative limit the dynamics is mainly multistable, presenting a variety of coexisting attractors, both periodic and chaotic, with a highly involved basin of attraction structure.

We studied multistable oscillatory states of a small neural network model and switching of an oscillatory mode. In the present neural network model, two pacemaker neurons are reciprocally inhibited with conduction delay; one pacemaker neuron inhibits the other via an inhibitory nonpacemaker interneuron, and vice versa. The small network model shows bifurcations from quasi-periodic oscillation to chaos via period 3 with increase in the synaptic weight of the reciprocal inhibition. The route to chaos in the network model is different from that in the single pacemaker neuron. The network model exhibits several multistable states. In a regime of a weak inhibitory connection, in-phase beat, out-of-phase beat (period 3), and chaotic oscillation coexist at the multistable state. We can switch an oscillatory mode by an excitatory synaptic input to one of the pacemaker neurons through an afferent path. In a strong inhibitory connection regime, in-phase beat and out-of-phase beat (period 4) coexist at the multistable state. An excitatory synaptic input through the afferent path leads to the transition from the in-phase beat to the out-of-phase beat. The transition from the out-of-phase beat to the in-phase beat is induced by an inhibitory synaptic input via interneurons. A conduction delay, furthermore, causes the spontaneous transition from the in-phase beat to the out-of-phase beat. These transitions can be explained by phase response curves.

We show that a discrete dynamical system with delayed arguments that describe the computational performance of a neural network with a loop structure can exhibit the coexistence of huge number of periodic solutions, and we describe the domains of attraction for the stable periodic orbits. This demonstrates a great potential of discrete time delayed neural nets with a loop structure for the purpose of storage and retrieval of periodic patterns.

A particularly simple and mathematically elegant example of chaos in a three-dimensional flow is examined in detail. It has the property of cyclic symmetry with respect to interchange of the three orthogonal axes, a single bifurcation parameter that governs the damping and the attractor dimension over most of the range 2 to 3 (as well as 0 and 1) and whose limiting value b = 0 gives Hamiltonian chaos, three-dimensional deterministic fractional Brownian motion, and an interesting symbolic dynamic.

The coexistence of several stable states for a given set of parameters has been observed in many natural and experimental systems as well as in theoretical models. This paper gives an overview over the wide range of applications in different disciplines of science. Furthermore, different system classes possessing multistability are analyzed in terms of the appearance of coexisting attractors and their basins of attraction. It is shown that multistable systems are very sensitive to perturbations leading to a noise-induced hopping process between attractors. The role of chaotic saddles in the escape from attractors in multistable systems is discussed.