Phase Trajectories Research Articles

Applying spline-based phase analysis to macroeconomic dynamics

Abstract The article uses spline-based phase analysis to study the dynamics of a time series of low-frequency data on the values of a certain economic indicator. The approach includes two stages. At the first stage, the original series is approximated by a smooth twice-differentiable function. Natural cubic splines are used as an approximating function y y . Such splines have the smallest curvature over the observation interval compared to other possible functions that satisfy the choice criterion. At the second stage, a phase trajectory is constructed in ( t , y , y ′ ) \left(t,y,y^{\prime} ) -space, corresponding to the original time series, and a phase shadow as a projection of the phase trajectory onto the ( y , y ′ ) (y,y^{\prime} ) -plane. The approach is applied to the values of GDP indicators for the G7 countries. The interrelation between phase shadow loops and cycles of economic indicators evolution is shown. The study also discusses the features, limitations and prospects for the use of spline-based phase analysis.

Circuit design and application for weak signal detection based on Duffing oscillator

A design scheme of Duffing Chaotic circuit module is proposed based on Multisimin study. The operational amplifier is used to construct integrating circuit. The Duffing chaotic circuit is also designed based on the integrating circuit. The results can be shown in a virtual oscilloscope by phase trajectory. The bifurcation results of Duffing oscillator solved by the circuit module are consistent with that obtained by Matlab. The Duffing equation in critical state is selected as Chaos Inspection System (CIS), which is used to identify weak signal with strong noise. Simulation examples show that the chaotic circuit designed in this study can successfully identify many kinds of weak signals, such as sine wave, triangle wave, and square wave etc. This work provides a theoretical basis for the hardware of CIS of weak signal with strong noise.

Fractional-order circuit design with hybrid controlled memristors and FPGA implementation

A hybrid memristor circuit consisting of a magnetic-controlled and a charge-controlled is designed. In order to construct a relatively complex dynamical system, an absolute value and a square root algorithm are embedded in the magnetic-controlled model specially. This novel five-dimensional system is first analyzed with integer order for the existence of chaotic phenomena, and then implemented the fractional-order memristor system by the Grunwald–Letnikov algorithm. Five fractional order values of qi(i=1,2,3,4,5) are taken as identical and different in the numerical simulation, respectively. The bifurcation diagram, 0-1 test, SALI algorithm, and complexity analysis reveal the existence of nonlinear dynamic characteristics to the hybrid fractional-order memristor system. Finally, the commensurate and non-commensurate fractional-order systems based on the frequency domain method are implemented by FPGA technology, their accuracy can reach to 14 decimal places. The results of numerical calculation and circuit simulation have the same phase trajectory, which can verify that the hybrid memristors have practical values in the future.

The global behavior evolution of non-orthogonal face gear-bearing transmission system

The nonlinear global behavior evolution process of the non-orthogonal face gear-bearing system was studied in this paper. Firstly, a multi-degree-of-freedom coupled vibration model of the non-orthogonal face gear-bearing system was established, which took into account factors such as time-varying meshing stiffness, tooth backlash, bearing clearance, and transmission error. Then the cell mapping method was used to calculate the phase trajectory of each cell on the Poincaré section to 1000 steps, and observed the influence of the initial state of the system on the dynamic behavior of the system after a long time history. This paper used the global bifurcation diagram, the global behavior diagram obtained by the cell mapping method, the multi-initial value phase diagram and the Poincaré section to illustrate the evolution process of the global behavior of the system with the excitation frequency, meshing damping and load fluctuation. The research results show that with the change of control parameters, the system gradually changed from periodic response to chaos after a long time history, and presented complex phenomena such as the coexistence of multiple periodic responses and the coexistence of periodic responses and chaotic responses in the evolution process.

Effect of Deposited Charge on Vacuum DC Surface Flashover Under Electron Beam Irradiation

Spacecraft operating environment is rather complex, including high vacuum and energetic electron radiation. Vacuum-insulation interface is usually the weakest link of the insulation system since surface flashover usually occurs at the interface. There are several theories for surface flashover, above which the secondary electron emission avalanche (SEEA) and the electron triggered polarization relaxation (ETPR) are the mainstream. Since researchers have usually assumed that there has been a preexisting positive charge on the insulation surface, when the insulation surface is negatively charged by electron beam irradiation, these theories cannot be directly applied. In this article, the particle tracking method is used to simulate the initial and secondary electron trajectories in the initial phase of surface flashover, and to explain the influence of negative surface charge on surface flashover. The results show that: on the one hand, the negative deposited charge weakens the electric field at the cathode triple junction (CTJ), inhibits the initial electron emission, and hinders the initiation of flashover; on the other hand, the negative deposited charge repels the secondary electrons away from the insulation surface and hinders the multiplication of the secondary electrons and the development of flashover.

Application of the Explicit Euler Method for Numerical Analysis of a Nonlinear Fractional Oscillation Equation

In this paper, a numerical analysis of the oscillation equation with a derivative of a fractional variable Riemann–Liouville order in the dissipative term, which is responsible for viscous friction, is carried out. Using the theory of finite-difference schemes, an explicit finite-difference scheme (Euler’s method) was constructed on a uniform computational grid. For the first time, the issues of approximation, stability and convergence of the proposed explicit finite-difference scheme are considered. To compare the results, the Adams–Bashford–Moulton scheme was constructed as an experimental method. The theoretical results were confirmed using test examples, the computational accuracy of the method was evaluated, which is consistent with the theoretical one, and the simulation results were visualized. Using the example of a fractional Duffing oscillator, waveforms and phase trajectories, as well as its amplitude–frequency characteristics, were constructed using a finite-difference scheme. To identify chaotic regimes, the spectra of maximum Lyapunov exponents and Poincaré points were constructed. It is shown that an explicit finite-difference scheme can be acceptable under the condition of a step of the computational grid.

Chen system-based chaotic transceiver for frequency output quartz transducers

The application of unidirectional synchronization of two coupled Chen systems is exhibited in this work. In spite of the high dependence on initial conditions, which means that two initially close phase trajectories with time become uncorrelated, it is possible to synchronize two dynamic systems to make them evolve identically. Data transmission using chaos requires mixing an information signal with a chaotic carrier. This procedure performs data encryption and spreads the spectrum of an information signal, which increases information security and reliability. Thus, the prospect of using devices with chaotic dynamics in modern telecommunication and telemetry applications is due to several factors, including high information capacity, various frequencies, and confidentiality of messages. The proposed scheme is considered to be used in a measuring transducer design that requires sensors to operate at a long distance from the rest of the scheme. We propose an application of a chaotic oscillator as a transceiver module for a quarts sensor transducer, which could be used in a telemetry application. The process of producing non-periodic but determined oscillations by the non-linear Chen system and signal transmission application, based on it, are the subject of the research. The complete synchronization of two unidirectionally connected Chen systems and its signal transmission application are considered. The goal is to develop a transceiver extension for the quartz measuring transducer scheme to ensure the stable operation of sensors at a long distance from the rest of the scheme. The result of the research: a chaos synchronization scheme was applied to transmit a frequency-modulated signal, obtained from a difference-frequency block of the quartz sensor transducer. Additionally, the mathematical model and numerical modeling of the Chen dynamical system has been done. The numerical solution of the system's differential equations was obtained using Matlab software. To study the change in the dynamic regime depending on the parameters of the model, the spectrum of Lyapunov exponents was calculated and bifurcation diagrams were constructed. The circuit design of the Chen oscillator was built using Multisim software, which uses the PSpice model to simulate electrical components. A model of an analog signal transmission system with chaotic mixing of a frequency output signal with a chaotic carrier has been proposed as an extension of the use of quartz transducers in measuring devices.

Dynamic modeling and analysis of multistage planetary gear system considering tooth crack fault

Due to the influence of signal modulation and multi-stage coupling, the fault diagnosis of multi-stage planetary gear system is difficult. In this paper, a dynamic model of multi-stage planetary gear system containing sun gear crack is proposed from the perspective of nonlinear dynamics to study the failure mechanism and dynamic characteristics of tooth root crack. The mesh stiffness with root crack is calculated by using the modified energy method. By means of global bifurcation diagram, RMS curve, phase diagram, Poincare map, time series diagram and FFT spectrum, the dynamic responses of the system with and without fault are analyzed and compared. The results show that there are differences in the impact of crack failure on the dynamic characteristics at different rotational speeds. The crack disturbance will cause an abrupt change in the vibration amplitude of the system, which can excite the system into chaos from a quasi-periodic state. The system tends to be unstable due to crack failure, which is manifested by irregular expansion of the attractor, divergence of the phase trajectory, periodic pulses in the time series diagram, and sideband frequencies with the sun gear fault characteristic frequency as interval appearing around the meshing frequency. The chaotic motion range should be avoided during crack fault diagnosis.

The influence of the preload on the nonlinear dynamic performance of the rod-fastened Jeffcott rotor system

For a rod-fastened Jeffcott rotor (RFJR) system, this paper establishes a nonlinear dynamic model considering contact interface, preload, unbalanced mass, nonlinear oil film force, and gives the dynamic equation of the rotor system. Then the fourth-order Runge–Kutta method is applied to study the nonlinear dynamics and bifurcation characteristics of the RFJR system. The bifurcation diagram, time-domain vibration waveform, frequency spectrum, phase trajectory, and Poincare section are applied to investigate the influence of the preload on the nonlinear dynamic performance of the rotor system. The results show that the preload affects the instability speed of the synchronous periodic solution of the RFJR system, the rotor system’s first critical speed, and the rotor system’s vibration amplitude passing the first critical speed. Whether the preload of the rotor is saturated or not has a very different effect on the vibration characteristics of the RFJR system. Assuming the fastening force of the tie rods is uneven, it will make the combined rotor have a certain initial bending deformation. At this time, the detuning rate of the tie rod will affect the rotor system’s nonlinear dynamic performance. In addition, the phase angle between the initial bending deformation and the mass eccentricity of the rotor also has a certain impact on the nonlinear dynamic characteristics of the combined rotor. The results obtained in this paper will provide a reference for understanding the nonlinear dynamics of the combined rotor and further research.

Aberrant Phase Precession of Lateral Septal Cells in a Maternal Immune Activation Model of Schizophrenia Risk May Disrupt the Integration of Location with Reward.

Spatial memory and reward processing are known to be disrupted in schizophrenia. Since the lateral septum (LS) may play an important role in the integration of location and reward, we examined the effect of maternal immune activation (MIA), a known schizophrenia risk factor, on spatial representation in the rat LS. In support of a previous study, we found that spatial location is represented as a phase code in the rostral LS of adult male rats, so that LS cell spiking shifts systematically against the phase of the hippocampal, theta-frequency, local field potential as an animal moves along a track toward a reward (phase precession). Whereas shallow precession slopes were observed in control group cells, they were steeper in the MIA animals, such that firing frequently precessed across several theta cycles as the animal moved along the length of the apparatus, with subsequent ambiguity in the phase representation of location. Furthermore, an analysis of the phase trajectories of the control group cells revealed that the population tended to converge toward a common firing phase as the animal approached the reward location. This suggested that phase coding in these cells might signal both reward location and the distance to reward. By comparison, the degree of phase convergence in the MIA-group cells was weak, and the region of peak convergence was distal to the reward location. These findings suggest that a schizophrenia risk factor disrupts the phase-based encoding of location-reward relationships in the LS, potentially smearing reward representations across space.SIGNIFICANCE STATEMENT It is unclear how spatial or contextual information generated by hippocampal cells is converted to a code that can be used to signal reward location in regions, such as the VTA. Here we provide evidence that the firing phase of cells in the lateral septum, a region that links the two areas, may code reward location in the firing phase of cells. This phase coding is disrupted in a maternal immune activation model of schizophrenia risk such that representations of reward may be smeared across space in maternal immune activation animals. This could potentially underlie erroneous reward processing and misattribution of salience in schizophrenia.

Design and circuit implementation of a novel 5D memristive CNN hyperchaotic system

In order to enrich the dynamic characteristics of cellular neural network (CNN) and reveal the influence of memristor nonlinearity on the dynamic behavior of the network, a novel five-dimension memristive cellular neural network hyperchaotic system is designed by replacing the linear resistors in the output module of cellular neural network with two flux-controlled memristors, and the hardware circuit design of the system is completed. Based on Lyapunov exponent spectrum and attractor phase trajectories, the effects of system parameters and initial value on the dynamic characteristics of the memristive cellular neural network model are studied, and the generation conditions of different chaotic attractors and coexisting chaotic attractors are explained. Kolmogorov entropy is used to measure the chaotic degree of the system when the parameters are in different intervals. In this way, the criteria of parameter selection for system application are given. In particular, the effect of Gaussian white noise on the dynamic behavior of memristive CNN chaotic system is studied. The hardware circuit design and characteristic analysis of memristive cellular neural network are completed by the circuit simulation software, and the physical realizability of chaotic characteristics of the memristive cellular neural network model is verified. Furthermore, a secure communication application example based on the hyperchaotic system is given.

Maximally chaotic dynamical systems of Anosov–Kolmogorov and fundamental interactions

In this paper, we give a general review on the application of ergodic theory to the investigation of dynamics of the Yang–Mills gauge fields and of the gravitational systems, as well as its application in the Monte Carlo method and fluid dynamics. In ergodic theory the maximally chaotic dynamical systems (MCDS) can be defined as dynamical systems that have nonzero Kolmogorov entropy. The hyperbolic dynamical systems that fulfill the Anosov C-condition belong to the MCDS insofar as they have exponential instability of their phase trajectories and positive Kolmogorov entropy. It follows that the C-condition defines a rich class of MCDS that span over an open set in the space of all dynamical systems. The large class of Anosov–Kolmogorov MCDS is realized on Riemannian manifolds of negative sectional curvatures and on high-dimensional tori. The interest in MCDS is rooted in the attempts to understand the relaxation phenomena, the foundations of the statistical mechanics, the appearance of turbulence in fluid dynamics, the nonlinear dynamics of Yang–Mills field and gravitating [Formula: see text]-body systems as well as black hole thermodynamics. Our aim is to investigate classical- and quantum-mechanical properties of MCDS and their role in the theory of fundamental interactions.

STUDY ON CHARACTERIZATION METHOD OF PHASE-POINT SATURATION BASED ON THE CAPACITY DIMENSION

In this study, the characterization method and evolution law of phase-point saturation based on the capacity dimension in fractal theory are studied to investigate the running-in process and state identification of a system. Tests of the running-in process under different working conditions were conducted on a ring-disc tribometer, and the friction signals were collected. The phase trajectories in different stages were then constructed and the fractal dimensions were calculated. The phase-point saturation parameter is proposed to reflect the degree of saturation of a phase space filled with phase points, and its reliability, noise-resistance, correlation with the fractal dimension, and effectiveness of the characterization of the running-in process were thoroughly investigated. The results demonstrate that the proposed phase-point saturation parameter is reliable and resistant to noise and can be used to identify changes in the system state. Furthermore, there is a negative correlation between the phase-point saturation parameter and the fractal dimension, and the phase-point saturation parameter can effectively characterize the state characteristics and describe the running-in process. During the running-in process, the phase-point saturation changes from high to low and tends to be stable when it reaches the running-in state. When entering the stable wear stage, the value of the parameter is maintained near the minimum, it fluctuates little as compared with the correlation dimension, and it exhibits good stability. Therefore, the proposed phase-point saturation parameter has important theoretical significance and application value for the quantitative characterization of the running-in process and state identification.

DYNAMICS OF FORMATION OF TRANSITIONAL PRICES ON THE CHAIN OF SEQUENTIAL MARKETS: ANALYTICAL MODEL

Although the problem of formation of market prices, determination of equilibrium prices within the model “Demand – Supply” is quite known and a great number of both theoretical works and works that summarize the results of observations are devoted to its research, this problem remains relevant, especially as to the dynamics of pricing processes and the stability of equilibrium prices in relation to changes in parameters that characterize the state of the system. Most studies addressing these issues focus on either a particular local market or the global market for some products in general. The purpose of this work is to build a mathematical model that would allow us to analyze general issues related to the formation of transitional prices in the finite N-dimensional chain of sequential markets in accordance with the scheme of market equilibrium. An analytical model is proposed that makes it possible to study the dynamics of prices in adjacent markets. Within this model, which is based on the determination of processes using a system of integral equations, it was assumed that the impact on the chain of sequential markets and the response to this impact are continuous over time. The dynamic aspect of the proposed pricing model in the vertical sequence of markets is the existence of an “after-effect”, which is described in an integral form by the delay distributed over time. The issues of adequacy of the model were examined, its internal coherence was studied, the correctness of the transition from the mathematical model of dynamics as a system of integral equations to the model in the form of a system of linear algebraic equations was substantiated. The conditions for the existence of the solution for this system of equations and the area of its stability are formulated. The mathematical model proposed in this paper allows for a qualitative analysis of the system states (by phase trajectories). Examples of numerical implementation of our analytical model for two and three sequential markets are given, equilibrium prices for each link of the chain of sequential markets are determined. Applying simulation modelling, the stability of the solution in relation to changes in such parameters of the model as the elasticity of demand and supply in the market under study and cross-elasticities in adjacent markets as well as the impact of these parameters on such dynamic indicators of the market system as the rate of attainment of equilibrium was examined.

Emphasizing visual tools: using R to understand dynamic models of biology and epidemiology

ABSTRACT Epidemiological models have enhanced relevance because of the COVID-19 pandemic. In this note, we emphasize visual tools that can be part of a learning module geared to teaching the SIR epidemiological model, suitable for advanced undergraduates or beginning graduate students in disciplines where the level of prior mathematical knowledge of students may not be very strong. Visual tools – phase portrait, flow field and trajectory and line plots – available in the R software are presented in a step by step manner, moving from the exponential growth model to the logistic growth model and then to the SIR model. Code for numerical simulation of differential equations and estimation of parameters is presented for the SIR model. Suggestions for students to connect the learning from these examples with research papers on COVID-19 are provided.

Fano resonance in the essentially nonlinear classic system

The classic analogue of the Fano resonance is considered for the essentially nonlinear systems. The closed expressions for the oscillation amplitudes turn out to be universal in terms of the frequency of free oscillations for the various kinds of the nonlinearity. The conditions of the transition between different branches of the amplitude–frequency relation are formulated in the framework of the concept of the limiting phase trajectory. The problem of the instability of the oscillations that leads to the beating between excited and driven oscillators is considered. The analysis of the non-stationary dynamical equations allows us to define the regions of the instability on the amplitude–frequency plane.

Electromagnetic radiation induced non-chaotic behaviors in a Wilson neuron model

Electromagnetic radiation can impact the neuron firing activities. To disclose its dynamics effect, a memristor-based three-dimensional (3D) non-autonomous Wilson neuron model is built with the consideration of electromagnetic radiation. The number and the stability of time-varying equilibrium states change over the time, leading to the occurrence of complex neuron dynamics. Then, multiple numerical tools are employed to disclose these neuron dynamics induced by electromagnetic radiation. It is found that the proposed neuron model has different non-chaotic firing activities of periodic/quasiperiodic/chaotic spiking and periodic/quasiperiodic bursting behaviors with respect to the amplitude and frequency of the electromagnetic radiation. This reflects that the electromagnetic radiation can trigger complex non-chaotic behaviors in neurons. Afterwards, a microcontroller unit-based (MCU-based) hardware platform is implemented and hardware experiments are executed. The physically captured phase trajectories and time-domain waveforms well validate the numerically simulated ones. Studying these non-chaotic behaviors can assistant us to understand the intrinsic nature of firing activities in neurons.

METHOD FOR DETERMINING PROPERTIES OF ASPHALT CONCRETE MIXTURE COMPACTED BY VIBRATORY ROLLER

The paper investigates the process of force interaction of a vibrating roller with a layer of asphalt concrete mixture. To describe the stress-strain state of the compacted layer during the interaction of the latter with a vibration roller, the Kelvin body was used. On the basis of the law of behavior of the Kelvin body, a simulation model has been developed, the result of which is a phase trajectory in the coordinates “force–displacement” for the system “drum–asphalt concrete mixture”. An analytical study of the characteristic points of the phase trajectory was carried out, as a result of which the relationship between the properties of the asphalt concrete mixture and the force effect in the area of contact of the roller with the compacted layer was established. In addition, a condition was obtained for continuous rolling of the asphalt-concrete layer with a vibrating roller, the observance of which makes it possible to increase the resource of the roller by reducing the wear of the latter. The results obtined can be used to establish the relationship between the operating modes of the vibratory roller and the properties of the compacted layer of the asphalt concrete mixture.

Powered-Coast-Powered Guidance Reconfiguration Method of Launch Vehicle with Thrust Drop Fault

This paper describes the study of a powered-coast-powered guidance reconfiguration (PCPGR) method used to solve the autonomous rescue problem for the mission profile of a launch vehicle with a coasting phase when the thrust drop fault occurs in the first powered phase (FPP) of the final stage. We first described the constraints of the final stage and the construction of the PCP guidance problem. Then we evaluated the adaptability of the guidance reconfiguration (GR) offline with numerical optimization by adjusting the constraints of the FPP, the coast phase, and the second powered phase. To determine the fault state set where the rocket can enter the prescribed target orbit through the GR initiate, we proposed a Newton method-based rapid replanning method of the transfer orbit that transforms the complex multi-flight phase trajectory planning problem into a feasible transfer orbit search problem to produce a fast solution onboard. Combined with the adaptive adjustment of the coasting time and the iterative guidance mode, we realized the autonomous online rescue of the payload. The simulation results showed that the proposed method achieved a reliable and rapid solution and improved a launch vehicle’s adaptability to a thrust drop fault.

Piecewise-Linear Simplification for Adaptive Synaptic Neuron Model

Adaptive synaptic neuron model involves complex activation functions. These nonlinearities lead to complicated hardware implementations, which greatly hinder neuron-based applications. To effectively solve this issue, a piecewise-linear (PWL) activation function with simplified circuit implementation is presented for the adaptive synaptic neuron model in this brief. With this neuron model, the stability evolution mechanism of the equilibrium state is analyzed and the parameter- and initial condition-related neuron dynamics are numerically explored. Afterwards, an analog circuit is designed and manually made using commercially available components. The phase trajectories captured by the hardware experiments verify the feasibility of the PWL activation function. Thus, such a PWL simplification shows superiority in emulating neuron dynamics.