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In this paper, we introduce a nonlinear vibrations analysis and dynamic responses of a vertical conveyor system under multi excitation forces. By adding the nonlinear proportional derivative controller (NPD) to the vibrating motion of the vertical vibration conveyor, the energy was transferred between uncontrolled and controlled system. We calculate the approximate solutions of the vibrating system utilizing the method of multiple scales. In addition, we investigate the stability at worst resonance cases using phase plane technique, equations of frequency response, and averaging method. The vertical vibration conveyor behavior was studied numerically at the values of its different parameters. The results exhibit the efficiency of NPD control unit to avoid the oscillations of the vertical conveyor system. Numerical simulations have been carried out using MAPLE and MATLAB software's to ensure the fidelity of our results. Comparisons are made between analytical and numerical results. Also, findings of the present work are discussed in details and compared with published works.

We recently derived a method, local orthogonal rectification (LOR), that provides a natural and useful geometric frame for analyzing dynamics relative to a base curve in the phase plane for two-dimensional systems of ODEs (Letson and Rubin, SIAM J. Appl. Dyn. Syst., 2018). This work extends LOR to apply to any embedded base manifold in a system of ODEs of arbitrary dimension and establishes a corresponding system of LOR equations for analyzing dynamics within the LOR frame, which maps naturally back to the original phase space. The LOR equations encode geometric properties of the underlying flow and remain valid, in general, beyond a local neighborhood of the embedded manifold. In addition to developing a general theory for LOR that makes use of a given normal frame, we show how to construct a normal frame that conveniently simplifies the computations involved in LOR. Finally, we illustrate the utility of LOR by showing that a blow-up transformation on the LOR equations provides a useful decomposition for studying trajectories' behavior relative to the embedded base manifold and by using LOR to identify canard behavior near a fold of a critical manifold in a two-timescale system.

The discontinuous dynamical behavior of a modified Duffing-Rayleigh system with a piecewise quadratic function is studied. The necessary and sufficient conditions for motion switchability at the velocity boundary are investigated through the theory of discontinuous dynamical systems. Various motions through the boundary are demonstrated by the parameter maps and coexisting bifurcation diagrams. The coexistence of an example system under different initial conditions is illustrated by attraction basins, and the trajectories in phase plane. The periodic and chaotic motions with different mapping structures are analyzed for a better understanding of the motion switching mechanism. Through Multisim, the circuit experiment proves the effectiveness of the theoretical analysis.

For the nonlinear Schrodinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity, by using the method of dynamical systems, we investigate the bifurcations and exact traveling wave solutions. Because obtained traveling wave system is an integrable singular traveling wave system having a singular straight line and the origin in the phase plane is a high-order equilibrium point. We need to use the theory of singular systems to analyze the dynamics and bifurcation behavior of solutions of system. For \begin{document}$ m>1 $\end{document} and \begin{document}$ 0 , corresponding to the level curves given by \begin{document}$ H(\psi, y) = 0 $\end{document} , the exact explicit bounded traveling wave solutions can be given. For \begin{document}$ m = 1 $\end{document} , corresponding all bounded phase orbits and depending on the changes of system's parameters, all exact traveling wave solutions of the equation can be obtain.

Time-optimal (TO) control is promising in many fields since it ensures to accomplish a task in minimum time. However, its direct application may cause oscillations or even limit cycles due to the restriction of the sampling period, state measurement accuracy and so on. To solve these problems, a compound proximate time-optimal (PTO) control law is proposed by considering system bandwidth and saturation in this paper. With the help of the phase plane, the switching zone and linear zone of the PTO control law are constructed. When the system state lies outside of both switching and linear zones, the bang-bang optimal scheme is adopted to achieve rapidity. And when the system state enters the switching zone, the bang-bang suboptimal scheme is developed to avoid oscillations or even limit cycles. Proportional-derivative (PD) scheme is applied in the linear zone to obtain local asymptotic stability around the origin. Numerical simulations are carried out to corroborate the advantages of the PTO control law.

In this paper authors present the results of a research that had a purpose to develop a method of constructing a neuroregulator model for the case of optimization of the control structure of a technological cycle. The method's implementation is based upon the automation of a production process when a physical controller, that operates the technological process according to a given program, is present. In order to achieve this goal, the artificial neural network approaches were implemented to create a mathematical model of the neuroregulator. The mathematical model of the neuroregulator is based on a physical prototype, and the procedure of a real-time control synthesis (adaptive control) is based on recurrent neural network training. The neural network architecture includes LSTM blocks, which are capable of storing information for long periods of time. A method is proposed for constructing a neuroregulator model for control of a production cycle when solving the task of the optimal trajectory finding on the phase plane of the technological cycle states. In the considered task of the optimal trajectory finding the mathematical model of the neuroregulator receives at each moment of time information about the current system state, the adjacent system states and the movement direction on the phase plane of states. Movement direction is determined by the given control optimization criteria. Based on the research results it was found that recurrent networks with LSTM modules can be used successfully as an approximator for the agent's Q-function to solve the given problem when the partially observed region of system states has a complex structure. The choice of the method of adaptation to the control actions and the external environmental disturbances proposed in the paper satisfies the requirements for the adatation process performance, as well as the requierments for the control processes quality, when there is lack of information about the nature of random control disturbances. The experimental environment, as well as the neural network models was implemented using the Python programming language with TensorFlow library.

Бесперебойное электроснабжение нефтяных и газовых месторождений остается важнейшей задачей мировой экономики. Одним из важнейших факторов, влияющих на отключение электроснабжения месторождений, является нарушение устойчивой работы генераторов. Устойчивая работа генераторов может быть нарушена при возникновении переходных процессов, вызванных короткими замыканиями или импульсными воздействиями на линии электропередач. При этом в электроэнергетической системе могут возникнуть уединенные волны – солитоны, характеризующиеся большой амплитудой и высокой скоростью распространения волны. В данной работе описываются причины возникновения таких волн. Приводится решение волнового уравнения электромеханической системы электроэнергетики, описывающего распространение уединенных волн. Решение рассматривается в фазовой плоскости, приводится численный пример расчета солитонного решения. Цель: найти солитонные решения в переходных процессах электромеханических систем и объяснить причины их возникновения, дать объяснения этого физического явления, определить, какую роль это явление играет в оценке устойчивости работы генераторов и предложить мероприятия по устранению нарушения устойчивости при наличии солитонной волны. Методы: метод фазовой плоскости, численное решение дифференциального уравнения методом Рунге– Кутта, метод пространства состояния. Результаты. Обнаружено, что при приближении решения уравнения турбина–генератор к границе динамической устойчивости возникают солитоны – одиночные всплески величины угла генератора. Выводы. При распространении эти волны ведут себя как частицы, что позволяет производить анализ обмена энергиями (потоками мощности) так же как анализ обмена энергиями механических частиц. При нарушении устойчивости возникают гармонические колебания, которые преобразуются в группу солитонов, распространение которых можно рассматривать как распространение частиц. The paper introduces the solution of the wave equation of the electromechanical system of electric power engineering in the form of a soliton - solitary wave. The causes of such waves are described. The solutions are considered in the phase plane, a numerical example of the soliton solution was given. The main aim of the research is to find the soliton solutions in transients of electromechanical systems, and explain the causes of their occurrence, explain this physical phenomenon, determine the role this phenomenon plays in the sustainability assessment, and propose the measures to eliminate the violation of stability in the presence of a soliton wave. Methods: phase plane method, numerical differential equation by the Runge-Kutt method, state space method. Results. It was found out that when the turbine-generator equation solution is approaching to the dynamic stability boundary, the solitons - solitary waves of the generator angle - appear. Conclusions. These waves behave like particles at propagation, that allows analyzing the exchange of energies (power flows) as the exchange in mechanical particles with energies. When stability is violated, harmonic oscillations arise, which are transformed into a group of solitons which propagation can be considered as the propagation of particles.

Life sciences are devoted to the research of all aspects of living organisms. As the basic unit of a living organism, biological cells are diverse and different. These differences are a direct reflection of their corresponding biological characteristics, so "seeing the cell more clearly" has been the goal of scientists at home and abroad. Based on the principle of coherent imaging, digital holographic quantitative phase microscopy imaging technology adopts digital recording and numerical reconstruction to realize phase retrieval and three-dimensional reconstruction. It has become one of the most promising methods in cellular research with better imaging efficiency, reconstruction accuracy, and stability. However, in the band-pass filtering process, tilt aberrations are often introduced due to manual frame selection of the +1 order spectrum, which is detrimental to the imaging quality of the system. To address this problem, this paper proposes a method to automatically determine the spectral center based on the surface fitting of a two-dimensional Hann window function, which improves the phase retrieval accuracy and imaging quality without reducing the computational speed. A more accurate spectral center can be obtained by conducting the sub-pixel spectral center of surface fitting. In addition, the accuracy is sub-pixel level and requires sub-pixel displacement correction of the tilted phase plane to obtain better quantitative phase retrieval results and imaging quality.

1AbstractTechnical advances in artificial manipulation of neural activity have precipitated a surge in studying the causal contribution of brain circuits to cognition and behavior. However, complexities of neural circuits challenge interpretation of experimental results, necessitating theoretical frameworks for system-atic explorations. Here, we take a step in this direction, using, as a testbed, recurrent neural networks trained to perform a perceptual decision. We show that understanding the computations implemented by network dynamics enables predicting the magnitude of perturbation effects based on changes in the network’s phase plane. Inactivation effects are weaker for distributed network architectures, are more easily discovered with non-discrete behavioral readouts (e.g., reaction times), and vary considerably across multiple tasks implemented by the same circuit. Finally, networks that can “learn” during inactivation recover function quickly, often much faster than the original training time. Our framework explains past empirical observations by clarifying how complex circuits compensate and adapt to perturbations.

In order to detect arc fault in line, an arc fault identification algorithm is proposed. The current waveform of common household and office appliances in normal operation and the current waveform of arc fault and point contact arc fault in its branch are tested. The information dimension of current phase plane diagram is calculated based on fractal theory, and the arc fault identification model. Phase space reconstruction of current waveform, The Information Dimensions of Current Phase Plane Based on Fractal Theory is considered as one of the characteristics of arc fault identification, then the current zero-off time is taken as the second characteristic, the arc fault identification model based on SVM is established. The simulation result shows that: the proposed algorithm can effectively identify arc fault and provide a reference for the development of a new type of AFCI with fault location function.

The unrestrained free surface oscillation of liquid hydrogen in a rectangular tank due to external forces are analyzed using a mapping technology called sigma transformation. Here the governing equation and boundary conditions were developed using potential flow theory for irrotational and incompressible fluid flow. The function sigma is used for transforming the non-linear physical domain into a fixed rectangular domain and thereby reducing the complexity of analysis by avoiding the re-meshing of the liquid domain in each time step. The numerical investigation is done for medium steepness non-breaking inviscid flow conditions under horizontal excitations due to turning and lane change accelerations. For shallow waves, the viscous effect becomes more evident. Sloshing behaviour of the liquid hydrogen was analyzed by generating wave profiles, elevation, phase plane and sloshing force diagram. The non-linearity of waves was also investigated using a fast Fourier transform. This method seems to be more simple and accurate at medium steepness, for higher steepness waves non-linear effect become more significant, this is due to the development of low energy level frequencies, which can be understood from the spectrum analysis. At high steepness phase plane diagram changes its circular shape to oval.

The hair bundles of inner hair cells in the auditory nervous exhibit spontaneous oscillations, which is the prerequisite for an important auditory function to enhance the sensitivity of inner ear to weak sounds, otoacoustic emission. In the present paper, the dynamics of spontaneous oscillations and relationships to steady state are acquired in a two-dimensional model with fast variable X (displacement of hair bundles) and slow variable $$X_a$$ . The spontaneous oscillations are derived from negative stiffness modulated by two biological factors (S and D) and are identified to appear in multiple two-dimensional parameter planes. In (S, D) plane, comprehensive bifurcations including 4 types of codimension-2 bifurcation and 5 types of codimension-1 bifurcation related to the spontaneous oscillations are acquired. The spontaneous oscillations are surrounded by supercritical and subcritical Hopf bifurcation curves, and outside of the curves are two cases of steady state. Case-1 and Case-2 steady states exhibit Z-shaped (coexistence of X) and N-shaped (coexistence of $$X_a$$ ) X-nullclines, respectively. In (S, D) plane, left and right to the spontaneous oscillations are two subcases of Case-1, which exhibit the stable equilibrium point locating on the upper and lower branches of X-nullcline, respectively, resembling that of the neuron. Lower to the spontaneous oscillations are 3 subcases of Case-2 from left to right, which manifest stable equilibrium point locating on left, middle, and right branches of X-nullcline, respectively, differing from that of the neuron. The phase plane for steady state is divided into four parts by nullclines, which manifest different vector fields. The phase trajectory of transient behavior beginning from a phase point in the four regions to the stable equilibrium point exhibits different dynamics determined by the vector fields, which is the basis to identify dynamical mechanism of complex forced oscillations induced by external signal. The results present comprehensive viewpoint and deep understanding for dynamics of the spontaneous oscillations and steady states of hair bundles, which can be used to well explain the experimental observations and to modulate functions of spontaneous oscillations.