The electric activities of neurons are dependent on the complex electrophysiological condition in neuronal system, and it indicates that the complex distribution of electromagnetic field could be detected in the neuronal system. According to the Maxwell electromagnetic induction theorem, the dynamical behavior in electric activity in each neuron can be changed due to the effect of internal bioelectricity of nervous system (e.g., fluctuation of ion concentration inside and outside of cell). As a result, internal fluctuation of electromagnetic field is established and the effect of magnetic flux across the membrane should be considered during the emergence of collective electrical activities and signals propagation among a large set of neurons. In this paper, the variable for magnetic flow is proposed to improve the previous Hindmarsh–Rose neuron model; thus, a four-variable neuron model is designed to describe the effect of electromagnetic induction on neuronal activities. Within the new neuron model, the effect of magnetic flow on membrane potential is described by imposing additive memristive current on the membrane variable, and the memristive current is dependent on the variation of magnetic flow. The dynamics of this modified model is discussed, and multiple modes of electric activities can be observed by changing the initial state, which indicates memory effect of neuronal system. Furthermore, a practical circuit is designed for this improved neuron model, and this model could be suitable for further investigation of electromagnetic radiation on biological neuronal system.
This work is devoted to the investigation of mathematical models of drilling systems described by ordinary differential equations. Here, we continue the study done by the researchers from Eindhoven where the two-mass mathematical model of a drilling system has been investigated (Mihajlovic et al. J. Dyn. Syst. Meas. Control 126(4): 709–720, 2004; de Bruin et al. Automatica 45(2): 405–415, 2009). The modified version of this model, which takes into account a full description of an induction motor, is studied. It is shown that such complex effects as hidden oscillations may appear in these kinds of systems. These effects may lead to drill string failures and breakdowns.
Based on generalized bilinear forms, lump solutions, rationally localized in all directions in the space, to dimensionally reduced p-gKP and p-gBKP equations in (2+1)-dimensions are computed through symbolic computation with Maple. The sufficient and necessary conditions to guarantee analyticity and rational localization of the solutions are presented. The resulting lump solutions contain six parameters, two of which are totally free, due to the translation invariance, and the other four of which only need to satisfy the presented sufficient and necessary conditions. Their three-dimensional plots with particular choices of the involved parameters are made to show energy distribution.
This paper is concerned with the adaptive synchronization problem of fractional-order memristor-based neural networks with time delay. By combining the adaptive control, linear delay feedback control, and a fractional-order inequality, sufficient conditions are derived which ensure the drive–response systems to achieve synchronization. Finally, two numerical examples are given to demonstrate the effectiveness of the obtained results.
This paper obtains soliton solutions to optical couplers by two methods. These are sine-cosine function method and Bernoulli's equation approach. There are four laws that are touched upon in this paper. These are Kerr law, power law, parabolic law and dual-power law. The first integration scheme is applicable to Kerr and power laws only where bright soliton solutions are retrievable. The second tool is applicable to parabolic and dual-power laws only that leads to dark and singular solitons for these two nonlinear media.
The cable equation plays a central role in many areas of electrophysiology and in modeling neuronal dynamics. This paper reports an accurate spectral collocation method for solving one- and two-dimensional variable-order fractional nonlinear cable equations. The proposed method is based on shifted Jacobi collocation procedure in conjunction with the shifted Jacobi operational matrix for variable-order fractional derivatives, described in the sense of Caputo. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations. In addition, the method reduces the variable-order fractional nonlinear cable equation to a simpler problem that consists of solving a system of algebraic equations. The validity and effectiveness of the method are demonstrated by solving three numerical examples. The convergence of the method is graphically analyzed. The results demonstrate that the proposed method is a powerful algorithm with high accuracy for solving the variable-order nonlinear partial differential equations.
With symbolic computation, two classes of lump solutions to the dimensionally reduced equations in (2+1)-dimensions are derived, respectively, by searching for positive quadratic function solutions to the associated bilinear equations. To guarantee analyticity and rational localization of the lumps, two sets of sufficient and necessary conditions are presented on the parameters involved in the solutions. Localized characteristics and energy distribution of the lump solutions are also analyzed and illustrated.
The biological Hodgkin–Huxley model and its simplified versions have confirmed its effectiveness for recognizing and understanding the electrical activities in neurons, and bifurcation analysis is often used to detect the mode transition in neuronal activities. Within the collective behaviors of neurons, neuronal network with different topology is designed to study the synchronization behavior and spatial pattern formation. In this review, the authors give careful comments for the presented neuron models and present some open problems in this field, nonlinear analysis could be effective to further discuss these problems and some results could be helpful to give possible guidance in the field of neurodynamics.
This paper presents adaptive dynamic surface control for the flexible model of hypersonic flight vehicle in the presence of unknown dynamics and input nonlinearity. By modeling the flexible coupling as disturbance of rigid body, based on the functional decomposition, the dynamics is divided into attitude subsystem and velocity subsystem. Flight path angle, pitch angle, and pitching rate are involved in the attitude subsystem. To eliminate the inherent problem of “explosion of complexity” in back-stepping, the dynamic surface control is investigated to construct the controller. Furthermore, direct neural control with robust design is proposed without estimating the control gain function and in this way the singularity problem could be avoided. In the last step of dynamic surface design, through the use of Nussbaum-type function, stable adaptive control is presented for the unknown dynamics with time- varying control gain function. The uniform ultimate boundedness stability of the closed-loop system is guaranteed. Simulation result shows the feasibility of the proposed method.
In this paper, we study an issue of input-to-state stability analysis for a class of impulsive stochastic Cohen-Grossberg neural networks with mixed delays. The mixed delays consist of varying delays and continuously distributed delays. To the best of our knowledge, the input-to-state stability problem for this class of stochastic system has still not been solved, despite its practical importance. The main aim of this paper is to fill the gap. By constricting several novel Lyapunov-Krasovskii functionals and using some techniques such as the It formula, Dynkin formula, impulse theory, stochastic analysis theory, and the mathematical induction, we obtain some new sufficient conditions to ensure that the considered system with/without impulse control is mean-square exponentially input-to-state stable. Moreover, the obtained results are illustrated well with two numerical examples and their simulations.
This paper presents an inductor-free memristive circuit, which is implemented by linearly coupling an active band-pass filter (BPF) with a parallel memristor and capacitor filter. Mathematical model is established, and numerical simulations are performed. The results verified by hardware experiments show that the active BPF-based memristive circuit exhibits the dynamical behaviors of point, period, chaos, and period-doubling bifurcation route. Most important of all, the newly proposed memristive circuit has a line equilibrium and its stability closely relies on memristor initial condition, which results in the emergence of extreme multistability. Stability distribution related to memristor initial condition is numerically estimated and the coexistence of infinitely many attractors is intuitively captured by numerical simulations and PSIM circuit simulations.
A discrete fractional logistic map is proposed in the left Caputo discrete delta’s sense. The new model holds discrete memory. The bifurcation diagrams are given and the chaotic behaviors are numerically illustrated.
Spatial patterns are ubiquitous in nature, which have been identified as important factors in dynamics of ecosystems. However, how pattern structures have influence on persistence of populations is far from well being understood. Particularly, whether some characters of spatial pattern can be indicators for ecosystems collapse is not well studied. As a result, we presented a predator–prey system with spatial motion and found that isolation degree (average distance between patterns with high density) of spatial patterns plays an important role in the persistence of populations: If isolation degree is much smaller, then the population will persist; if isolation degree is much larger, then the population density will decrease with increasing space size and run a high risk of extinction as space size is large enough. Our results highlight the relationship between pattern structures and ecosystems collapse.
Allee effect that refers to a positive relationship between individual fitness and population density provides an important conceptual framework in conservation biology. While declining Allee effect causes reduction in extinction risk in low-density population, it provides a benefit in limiting establishment success or spread of invading species. Population models that incorporated Allee effect confer the fundamental role which plays for shaping the population dynamics. In particular, non-spatial predator–prey and invasion models have shown the influence of Allee effects on population dynamics, and spatial models have illustrated its critical roles for pattern formation and the manifestation of traveling wave fronts. We highlight all such no-spatial and spatial population models and their contributions in deeper understanding of population dynamics. In addition, we briefly outline the trends for future research on Allee effect which we think are interesting and widely open.
Offshore platforms are widely used to explore, drill, produce, storage, and transport ocean resources and are usually subject to environmental loading, such as waves, winds, ice, and currents, which may lead to failure of deck facilities, fatigue failure of platforms, inefficiency of operation, and even discomfort of crews. In order to ensure reliability and safety of offshore platforms, it is of great significance to explore a proper way of suppressing vibration of offshore platforms. There are mainly three types of control schemes, i.e., passive control schemes, semi-active control schemes, and active control schemes, to deal with vibration of offshore platforms. This paper provides an overview of these schemes. Firstly, passive control schemes and several semi-active control schemes are briefly summarized. Secondly, some classical active control approaches, such as optimal control, robust control, and intelligent control, are briefly reviewed. Thirdly, recent advances of active control schemes with delayed feedback control, sliding model control, sampled-data control, and network-based control are deeply analyzed. Finally, some challenging issues are provided to guide future research directions.
Under investigation in this paper is a coherently coupled nonlinear Schrödinger system which describes the propagation of polarized optical waves in an isotropic medium. By virtue of the Darboux transformation, some new solutions have been generated on the vanishing and non-vanishing backgrounds, including multi-solitons, bound solitons, one-breathers, bound breathers, two-breathers, first-order and higher-order rogue waves. Dynamic behaviors of those solitons, breathers and rogue waves have been discussed through graphic simulation.
In this paper, an adaptive sliding mode technique based on a fractional-order (FO) switching-type control law is designed to guarantee robust stability for uncertain 3D FO nonlinear systems. A novel FO switching-type control law is proposed to ensure the existence of the sliding motion in finite time. Appropriate adaptive laws are shown to tackle the uncertainty and external disturbance. The calculation formula of the reaching time is analyzed and computed. The reachability analysis is visualized to show how to obtain a shorter reaching time. A stability criterion of the FO sliding mode dynamics is derived based on indirect approach to Lyapunov stability. Advantages of the proposed control scheme are illustrated through numerical simulations.
This study is aimed at examining and comparing several friction force models dealing with different friction phenomena in the context of multibody system dynamics. For this purpose, a comprehensive review of present literature in this field of investigation is first presented. In this process, the main aspects related to friction are discussed, with particular emphasis on the pure dry sliding friction, stick–slip effect, viscous friction and Stribeck effect. In a simple and general way, the friction force models can be classified into two main groups, namely the static friction approaches and the dynamic friction models. The former group mainly describes the steady-state behavior of friction force, while the latter allows capturing more properties by using extra state variables. In the present study, a total of 21 different friction force models are described and their fundamental physical and computational characteristics are discussed and compared in details. The application of those friction models in multibody system dynamic modeling and simulation is then investigated. Two multibody mechanical systems are utilized as demonstrative application examples with the purpose of illustrating the influence of the various frictional approaches on the dynamic response of the systems. From the results obtained, it can be stated that both the choice of the friction force model and friction parameters involved can significantly affect the simulated/modeled dynamic response of mechanical systems with friction.
The aim of the current study is to examine the in-plane and out-of-plane nonlinear size-dependent dynamics of a microplate resting on an elastic foundation, constrained by distributed rotational springs at boundaries. Employing the von Kármán plate theory as well as Kirchhoff’s hypotheses, the equations of motion for the in-plane and out-of-plane directions are derived by means of the Lagrange equations, based on the modified couple stress theory. The potential energies stored in a Winkler-type elastic foundation and the rotational springs at the edges of the microplate are taken into account. The set of second-order nonlinear ordinary differential equations, obtained via the Lagrange scheme, is recast into a double-dimensional set of first-order nonlinear ordinary differential equations with coupled terms by means of a change of variables. The linear natural frequencies of the system are obtained through use of an eigenvalue analysis upon the linear terms of the equations of motion. The nonlinear response, on the other hand, is obtained by means of the pseudo-arclength continuation method. The dynamical characteristics of the system are examined via plotting the frequency–response and force–response curves. The effect of the stiffness of the rotational and translational springs on the nonlinear size-dependent behaviour is also examined. Finally, the effect of employing the modified couple stress theory, rather than the classical theory, on the response is discussed.
This paper presents a novel fifth-order two-memristor-based Chua's hyperchaotic circuit, which is synthesized from an active band pass filter-based Chua's circuit through replacing a nonlinear resistor and a linear resistor with two different memristors. This physical circuit has a plane equilibrium and therefore emerges complex phenomenon of extreme multistability. Based on the mathematical model, stability distributions of three nonzero eigenvalues in the equilibrium plane are exhibited, from which it is observed that four different stability regions with unstable saddle-focus, and stable and unstable node-focus are distributed, thereby leading to coexisting phenomenon of infinitely many attractors. Furthermore, extreme multistability depending on two-memristor initial conditions is investigated by bifurcation diagrams and Lyapunov exponent spectra and coexisting infinitely many attractors' behavior is revealed by phase portraits and attraction basins. At last, a hardware circuit is fabricated and some experimental observations are captured to verify that extreme multistability indeed exists in the two-memristor-based Chua's hyperchaotic circuit.