Phase Plane Portraits Research Articles

Modeling and Stability Analysis of the Dynamics of Prey-Predator Population

The aim of this work is to investigate the numerical study of prey-predator model and give a thorough overview of the existing research. Here we see instances that involve stability analysis, phase plane, and phase portrait, among others. Furthermore, a numerical simulation is shown to illustrate the results of the prey-predator model. This is to provide light on the procedures followed in order to arrive at the conclusions. The one-prey, one-predator principle is discussed in this article from multiple theoretical perspectives. Finally, for various parameter values and time intervals, we have demonstrated the phase portraits and the dynamic behavior of the pre-predator model. Jagannath University Journal of Science, Volume 11, Number 1, June 2024, pp. 81−93

Nonlinear analysis of hydrodynamics of a shallow-draft floating wind turbine

AbstractThis study investigates numerically the dynamic responses of the T-Omega Wind novel concept of Floating Offshore Wind Turbine. The turbine is light-weight, has a shallow-draft and a relatively high centre of gravity that allows it to glide over harsh marine environments. The turbine responses are studied under regular wave excitation, considering most probable ranges of discrete sea wave heights and periods representative of real ocean conditions. A multibody virtual model is developed, simplified to a rigid 6 DOF system and experimentally validated in the state-of-art Marine Simulator to define the types of dynamical responses for both “Low” and “High” Sea States. The dynamics of coupled heave and pitch DOFs are evaluated with time histories, phase-plane portraits, Poincaré sections and FFT analyses to conclude that period-1 stable solutions exist for all studied cases of “Low Sea States”, whereas period-2, period-3 and period-4 periodic responses are identified for short wave periods of excitation under “High Sea States” conditions. Simulation results show that regions where period-1 responses exist are highly sensitive to wave height and can widen as the wave amplitude reduces. Finally, the turbines’ nonlinearities generated by the floats’ geometry are observed in this dynamical system, which are identified to be related to variation in float waterplane area and particularly observable for “High Sea States”.

Chaotic dynamics of flexible graphene electronic membranes with variable density in motion

With advancements in artificial intelligence and wearable technology, flexible electronic devices characterized by their flexibility and extensibility have found widespread applications in fields such as information technology, healthcare, and military. Printing technology can accurately print a circuit diagram onto a flexible membrane substrate by the pressure transfer of a conductive ink, which makes the large-scale printing of flexible graphene electronic membranes possible. However, during the roll-to-roll printing process used to prepare flexible graphene electron membranes, the density of electron membranes is variable due to the uneven distribution of inkjet-printed circuits, which limits the printing speed of flexible graphene electron membranes. Hence, investigating the dynamic properties of flexible graphene electron materials with different densities is of paramount importance to improve the production efficiency and quality of flexible graphene electron membranes. This paper takes roll-to-roll intelligent graphene electronic membranes as the research object. According to Hamilton’s principle, nonlinear vibration partial differential equations for the motion of flexible graphene electron membranes with varying densities were established and subsequently discretized using the assumed displacement function and the Bubnov–Galerkin method. Through numerical calculations, the simulation results obtained based on the fourth-order Runge–Kutta method and the multiscale algorithm were compared, and the multiscale algorithm was verified to be more correct and effective. The primary resonance amplitude–parameter characteristic curve, along with phase-plane portraits, Poincaré maps, power spectrum, time history plots, and bifurcation diagrams, for the nonlinear behavior of the membrane was obtained. The impacts of the density coefficient, velocity, damping ratio, excitation force, and detuning parameters on the nonlinear primary resonance and chaotic behavior of the moving graphene electron membrane were determined, and the stable operational region was identified, laying a theoretical foundation for the development of flexible graphene electronic membranes.

Parametric controllable planar multi-scroll chaotic attractors in a 3-D memristive tabu learning single neuron model

In this paper, a three-dimensional (3D) autonomous tabu learning single neuron model is proposed, which is achieved by using a sinusoidal activation function and introducing a memristor synapse. This model exhibits the remarkable capability to produce a series of planar multi-scroll chaotic attractors, and its unique feature lies in the ability to control the number of scrolls. The investigation of the planar multi-scroll chaotic attractors and its dynamical behaviors is conducted through the analysis of phase plane portraits, bifurcation diagrams, and spectral entropies. The numerical simulations unveil a compelling relationship between the number of chaotic scrolls and specific control parameters governing the model. To further validate the findings, a 3D autonomous tabu learning single neuron model is implemented on a digital hardware platform. In an effort to extend the practical significance of this research, the multi-scroll chaotic phenomenon generated by the proposed model is deployed for image encryption. The fusion of mathematical modeling, digital hardware implementation, and practical application underscores the universality and significance of the proposed single neuron model in the fields of chaotic systems and engineering applications.

A simple model of nutrient recycling and dormancy in a chemostat: Mathematical analysis and a second-order nonstandard finite difference method

A chemostat is an apparatus that sustains a homogeneous environment through continuous inflow and outflow. Presented is a chemostat model that characterizes the dynamics of dormancy-capable microorganisms. This model of coupled systems of nonlinear ordinary differential equations (ODEs) can apply to various types of organisms, such as different species of bacteria, archaea, algae, fungi, viruses, and protozoa. However, these species reside in different environments and rely on different sets of nutrients. Thus, the model adapts to each species’ limiting nutrient through nutrient recycling. This paper includes a complete stability analysis, supporting phase plane portraits, and accompanying bifurcation diagrams. The paper also proposes an advanced second-order, positivity-preserving, and elementary stable nonstandard finite difference method for solving the mathematical model. Series of numerical simulations are presented that support the theoretical results and explore different biological scenarios. The stability analysis reveals that (1) if the overall dilution, death, and conversion are less than the overall growth, both the dormant and active populations persist when introduced to a chemostat; and (2) if the overall dilution, death, and conversion are more than the overall growth, the microorganism population in its entirety will die off when introduced to a chemostat. Furthermore, the model study suggests that neither dormancy nor nutrient recycling provides substantial survival advantages in a basic chemostat when no threat to the active microorganism is present.

Effect of Subsensory Vibration on Postural Control of Healthy Elderly and Subjects with Simulated Reduced Plantar Feet Sensation: A Randomized Crossover Controlled Trial

Abstract Introduction Postural instability is considered a normal result of aging and is caused by some diseases such as neuropathy. Aim The purpose of the present study was to analyze and compare the effect of subthreshold vibration that was applied to the plantar surface of feet on balance control of healthy elderly and young subjects with simulated decreased plantar feet sensation. Methods Twenty-four subjects (12 elderly persons and 12 young persons with decreased plantar feet sensation) were included in this randomized, double-blinded crossover study. Participants were divided into two intervention groups: 1) orthopedic shoe with vibration mechanism on; and 2) orthopedic shoe with vibration off. Primary outcomes were center of pressure (COP) parameters including mean velocity, phase plane portrait, standard deviation (SD) of amplitude, and SD of velocity. Secondary outcomes were Berg Balance Scale (BBS) score and timed up and go test (TUG). The standing balance was tested with different conditions: eye opened/closed and foam/rigid surface. General linear mixed models tests were used for statistical analysis. The level of statistical significance was set at 0.05 for all analyses. Discussion This study showed very large effect sizes (partial η2 > 0.2) with application of vibration as compared with vibration off for COP parameters with greater effects in condition with more baseline fluctuations. There was significant change in BBS test (Cohen d, 0.6; confidence interval [CI], −0.215 to −1.421). The time on TUG test was decreased and approached significance (P = 0.06). Conclusions The immediate effect of the study showed that subthreshold vibration can improve static and dynamic balance in healthy elderly people and people with decreased plantar feet sensation. However, long-term studies are needed to establish the clinical value of the subthreshold vibration on balance control. Clinical Relevance Subthreshold vibration applied to the feet soles via custom-made shoes can reduce the postural sway in subjects with decreased feet sensation. By conducting more studies with conclusive results in the field of vibration effectiveness in the elderly, we can hope to reduce the rate of falls in the elderly.

Comparing the effectiveness of flexi-bar and stability exercises on postural control in chronic nonspecific low back pain: A randomized controlled study.

This study aimed to compare the effect of flexi-bar and stabilization exercises on static and dynamic postural control in patients with chronic nonspecific low back pain. In this randomized controlled study conducted between November 2019 and March 2020, 38 patients (19 males, 19 females; mean age: 33.8±6.2 years; range, 20 to 45 years) were randomly assigned into flexi-bar (n=19) and stabilization (n=19) groups. Both groups received general physiotherapy for three sessions per week, a total of 10 sessions. Besides, the flexi-bar group received flexi-bar exercises, and the stabilization group received stabilization exercises. Postural sway was assessed with a force platform in three difficult conditions, including open eye, close eye, and one-leg standing and dynamic posture with the modified Star Excursion Balance Test. After the intervention, both groups showed a significant improvement in static and dynamic postural control (p<0.05). However, no significant differences were found between groups after treatment, while only the phase-plane portrait of opened eyes condition was significantly improved (p=0.03), in the flexi-bar group compared to the stabilization group. Both flexi-bar and stabilization exercises effectively improved static and dynamic postural control, but none of the exercises was superior to the other. Flexi-bar is recommended as an effective tool in low back pain rehabilitation.

Synchronization of discrete time fractional order neuromuscular models in the presence and absence of stimulus

This article proposes a fractional order discrete-time neuromuscular model incorporating calcium kinetics to study the essential role of the neuromuscular system in facilitating rapid information transmission between nerve cells and muscles for bodily functions. The study conducts a dynamical analysis to investigate the stability and periodic oscillations of the system in the presence and absence of tetanus stimuli from neurons. Bifurcation diagrams are generated for different fractional orders and rate constants, highlighting the occurrence of chaotic dynamics. The region of chaos is determined using the largest Lyapunov exponents and the Jacobian matrix method. The study also achieves synchronization of the systems by employing nonlinear control functions with feedback gains. Emphasizing the significance of constructing the biological model using fractional order operators, the article provides a chemical interpretation of the analytical findings. Moreover, the article employs dynamic plots to visually depict the changing system variables over time, allowing for an examination of the concurrent presence of multiple attractors through bifurcation diagrams and phase plane portraits. By studying how chemical kinetics and muscle activation are interrelated, the article uncovers captivating revelations that could guide the formulation of strategies to combat muscle disorders. The article also underscores the potential utility of synchronization in managing neuromuscular conditions, thus emphasizing its practical implications in disease control.

Quasi-periodic vibration of an axially moving beam under conveying harmonic varying mass

In this work, quasi-periodic vibration of an axially moving beam under conveying harmonic varying mass are studied. Coupled modelling equations of nonlinear vibration of an axially moving beam are established with the aid of the generalized Hamilton principle. A nonlinear partial-integro-differential equation as the transverse vibration modelling equation is deduce by decoupled method. After obtaining the equilibrium buckled state of the axially moving beam, a transverse vibration governing equation around the buckled state is presented. The Galerkin method is applied to truncate the governing equation into a set of nonlinear ordinary differential equations. The harmonic balance method with three time variables is applied to analyze quasi-periodic vibration of the nonlinear dynamic system with combined parametric and forced external excitation. Time traces and phase-plane portraits of quasi-periodic vibration are obtained, which are in excellent agreement with those of the direct time integration method. Frequency amplitude responses of the vibration are determined by the harmonic balance method with three time variables. Effects of the velocity of the axially moving beam, amplitude of the varying mass, and the external excitation frequency on the vibration amplitude of axially moving beam and the ratio of the two fundamental frequencies are investigated.

Nonlinear analysis of the flow-induced vibration of a circular cylinder with a splitter-plate attachment

The nonlinear characteristics of the flow-induced vibration of an elastically-supported cylinder-plate assembly are investigated using numerical simulations of the flow past the assembly at a low-Reynolds number Re=100. The beating phenomenon in the time series of the transverse (lift) force coefficient for both vortex-induced vibration and galloping is studied. The dynamical characteristics of the beating are elucidated using local attributes (local frequency), global attributes (periodicity and symmetry), and the properties of the energy transfer (per unit time) between the cylinder-plate assembly and the surrounding fluid. From this analysis, five distinct types of beating for a cylinder-plate assembly are identified. The nonlinear dynamical characteristics is investigated using a number of methodologies such as the power spectral density, the phase-plane portrait and the Poincaré section. The analysis implies the existence of a close relationship between the multiple characteristic frequencies in the power spectrum and the beating observed in the time variations. Three types of nonlinear oscillations in a cylinder-plate assembly have been identified and characterized in terms of the nature of their limit cycles in the phase plane and the point set distribution in the Poincaré section. The wake mode of a cylinder-plate assembly during beating is found to be invariably asymmetric with highly irregular vortex shapes, e.g., a right-angled vortex designated as SA and a crescent-shaped vortex designated as SB—multiplets (groups) of vortices involving either SA or SB with two elliptically-shaped vortices are closely associated with the observed beating.

Imaging phase plane models

In application-oriented mathematics, particularly in the context of nonlinear system analysis, phase plane analysis through SageMath offers a visual display of the qualitative behaviour of solutions to differential equations without inundating students with cumbersome calculations of the plane-phase. A variety of examples is usually given to illustrate phase-plane behaviour. We approach these problems by considering a problem containing a single real parameter that exemplifies the various situations clearly and simply. We developed two computer programs in SageMath: one program calculates aspects of the Jacobian matrix and displays the phase plane portraits, the other determines the centre manifold. Computations and images generated with computer codes are useful in understanding dynamic models in biology, physics and engineering that involve planar non-linear autonomous differential equations. This paper covers analytical and computational skills that are helpful for students and teachers.

Viscoelastic Effects on the Nonlinear Oscillations of Hard-Magnetic Soft Actuators

AbstractThe hard-magnetic soft materials (HMSMs) belong to the magnetoactive category of smart polymers that undergo large actuation strain under an externally applied magnetic field and can sustain a high residual magnetic flux density. Because of these remarkable characteristics, HMSMs are promising candidates for the remotely controlled actuators. The magnetic actuation behavior of the hard-magnetic soft actuators (HMSAs) is considerably affected by the viscoelastic material behavior of HMSMs. In this article, we aim at developing an analytical dynamic model of a typical planar model of HMSAs concerning the viscoelasticity of HMSMs. A Zener rheological model in conjunction with an incompressible neo-Hookean model of hyperelasticity and Rayleigh dissipation function is employed for defining the constitutive behavior of the viscoelastic HMSA. The governing equations of dynamic motion are deduced by implementing the nonconservative form of the Euler–Lagrange equation. The established dynamic model is utilized for providing preliminary insights pertaining to the effect of the viscoelasticity on the nonlinear oscillations of the actuator. The phase–plane portraits, Poincaré maps, and the time–history response are plotted to investigate the stability, resonant behavior, and periodicity of the actuator. The results and inferences reported here should provide the initial step toward the design and the development of modern actuators for diverse futuristic applications in the medical and engineering fields.

Two-dimensional non-autonomous neuron model with parameter-controlled multi-scroll chaotic attractors

This work presents a two-dimensional (2-D) non-autonomous tabu learning single neuron (TLSN) model based on sinusoidal activation function (SAF), which can generate a class of multi-scroll chaotic attractors with parameters controlling the number of scrolls. The SAF-based TLSN (SAF-TLSN) model has a periodically alterable equilibrium point closely related to the model parameters. Its equilibrium trajectories with different stability types can be distributed in the phase plane, resulting in the generation of multi-scroll chaotic attractors. The multi-scroll chaotic attractors and their dynamical behaviors are investigated by phase plane portrait, maximal Lyapunov exponent, bifurcation diagram, and cross section. The numerical results demonstrate that each scroll of multi-scroll chaotic attractors is associated with the equilibrium trajectories composed of infinitely many unstable equilibrium points, and the scroll distributions of multi-scroll chaotic attractors are confined to the distribution ranges of equilibrium trajectories that are controlled by the model parameters. Besides, using a digital hardware platform, the SAF-TLSN model is implemented and the multi-scroll chaotic attractors with different scrolls are obtained experimentally to verify the numerical ones.

Dynamic modeling and nonlinear oscillations of a rotating pendulum with a spinning tip mass

In this work, a dynamic model of a rotating pendulum with a spinning tip mass is developed. The pendulum is rotating with a prescribed angular velocity around the vertical axis, while the tip mass spins around an arbitrary axis with constant angular velocity. The dynamic stability and bifurcation of the system are examined first for the pendulum when rotating with a constant angular velocity. The effects of the tip mass spin on the dynamic equilibrium and stability are thoroughly examined by constructing the bifurcation diagrams and phase plane portrait plots. It is found the tip mass spin considerably affects the qualitative nonlinear behavior and stability of the system. The frequency response of the pendulum due to sinusoidal angular rotation of the pendulum around the vertical axis is obtained using numerical integration combined with an arc-length continuation scheme. The effects of the magnitudes and directions of the tip mass spin on the nonlinear dynamic behavior of the system are subsequently investigated.

Distinction of non-specific low back pain patients with proprioceptive disorders from healthy individuals by linear discriminant analysis.

The central nervous system (CNS) dynamically employs a sophisticated weighting strategy of sensory input, including vision, vestibular and proprioception signals, towards attaining optimal postural control during different conditions. Non-specific low back pain (NSLBP) patients frequently demonstrate postural control deficiencies which are generally attributed to challenges in proprioceptive reweighting, where they often rely on an ankle strategy regardless of postural conditions. Such impairment could lead to potential loss of balance, increased risk of falling, and Low back pain recurrence. In this study, linear and non-linear indicators were extracted from center-of-pressure (COP) and trunk sagittal angle data based on 4 conditions of vibration positioning (vibration on the back, ankle, none or both), 2 surface conditions (foam or rigid), and 2 different groups (healthy and non-specific low back pain patients). Linear discriminant analysis (LDA) was performed on linear and non-linear indicators to identify the best sensory condition towards accurate distinction of non-specific low back pain patients from healthy controls. Two indicators: Phase Plane Portrait ML and Entropy ML with foam surface condition and both ankle and back vibration on, were able to completely differentiate the non-specific low back pain groups. The proposed methodology can help clinicians quantitatively assess the sensory status of non-specific low back pain patients at the initial phase of diagnosis and throughout treatment. Although the results demonstrated the potential effectiveness of our approach in Low back pain patient distinction, a larger and more diverse population is required for comprehensive validation.

줄넘기 시 템포가 협응에 미치는 영향

The purpose of this study was to compare and analyze the effect of tempo on coordination during jump rope. Thirty healthy adult males participated. The task was to jump rope single under with both feet synchronize with metronome tempo(80, 100, 120, 140, 160 bpm). The joint angles and angular velocities of wrist, elbow, shoulder, hip, knee, and ankle were calculated using data stably performed 5 times consecutively among the data performed 10 times in a row synchronize with metronome tempo. And angle-angle diagram, phase plane portraits, and correlation analysis were analyzed using each joint angle and angular velocity. The results of this study were as follows: First, The angle-angle diagrams show that the joint angles decrease as getting faster the tempo. In the wrist-shoulder and hip-ankle diagrams, coupled coordination was observed as getting faster the tempo. Second, In the phase plane portraits, the angular-angular velocity of all joints showed a convex shape. As getting faster the tempo, the angular velocities of the shoulders, hips, and knees decreased. Third, Elbow-shoulder, hip-knee, knee-ankle, and hip-ankle correlation coefficients were high. It was found that the correlation coefficient of hip-knee and hip-ankle decreased as getting faster the tempo. In conclusion, Wrist, elbow, and ankle joints played a more important role in jumping rope than the shoulder, knee, and hip joints. And, When the tempo of jumping rope was 120 bpm or higher, a stable coordination pattern appeared, but when the jumping rope was slowed at a slow tempo, an unstable coordination pattern was observed.

A Novel Multiscroll Memristive Hopfield Neural Network

Memristors are usually introduced into neuron models as neural synapses to investigate firing activities. In this paper, a novel generic memristor with smooth cosine memductance is proposed, and its dynamic characteristic concerning multistability, which is completely different from any known memristors, is analyzed and validated by numerical and PSIM simulations. The PSIM simulations intuitively reflect the memory properties of the constructed memristor emulator. To investigate the application of the proposed memristor, a multiscroll memristive Hopfield neural network is modeled by introducing the memristor into a tri-neuron Hopfield neural network as a synapse weight. Homogeneous single-scroll and double-scroll multistability phenomena are revealed by utilizing some analytical tools, such as bifurcation diagrams, local attraction basins, phase plane portraits, and so on. It is found that there is a multi-double-scroll attractor with growing scrolls for appropriate parameters. Furthermore, the average Hamiltonian energy, dependent on the homogeneous dynamics, is analyzed based on Helmholtz’s theorem. It is discovered that the homogeneous dynamics is closely related to the energy transition, which may provide a new explanation for the occurrence of multistability in the human brain. Finally, several circuit experiments are carried out to confirm the dynamical behaviors, and it is found that the circuit can show similar dynamics as the numerical simulations.

Dynamics simulation of rolling bearing system with pedestal looseness in speed-up train

The paper mainly studies the dynamics characteristics of the roller bearing system with pedestal looseness in speed-up train. Considering eccentricity, damping coefficient, stiffness of shaft, and support forces of roller bearing, a centralized mass model of roller bearing system of speed-up train is presented. Moreover, numerical analysis method is adopted to solve the dynamics responses. Based on the trajectory cures, phase plane portraits, Poincaré maps, and amplitude spectrum cures, the dynamics characteristics are analysed when the motor rotating speed and gap of the looseness are changed. The simulation results reveal some complex dynamic behaviour such as periodic motion, quasi-periodic motion, multi-periodic motion and chaotic motion. These results will help us understand and harness these dynamics behaviour of the proposed bearing system of speed-up train.

Supercritical nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading

Dynamics of a hyperelastic beam in a buckled state subjected to a harmonic axial load is investigated in this work. For the static case, the buckled configuration of the hyperelastic beam is first determined through an asymptotic method when the axial load is in excess of its critical load. Then, the governing equation of vibration of the buckled beam is derived, which is a complex nonlinear partial differential equation with varying coefficients. The first natural frequency of the buckled beam is obtained by linear analysis and effects of material and geometrical parameters on it are numerically investigated. By applying the Galerkin method, the governing equation for nonlinear transverse vibration of the beam in the buckled state is transformed into a series of strongly nonlinear ordinary differential equations. Dynamic characteristics of the hyperelastic beam are investigated by the Runge–Kutta method. Bifurcation diagrams, time traces, phase-plane portraits, and Poincare sections are obtained for different values of the external excitation frequency and amplitude of variation of the axial load. Results show that different dynamic behaviors of the hyperelastic buckled beam such as periodic, multi-periodic, quasiperiodic, and chaotic motions can be found when the amplitude of the axial load is varied. Amplitude–frequency responses of the hyperelastic buckled beam are obtained by the Runge–Kutta method and harmonic balance method. Results from the Runge–Kutta method are in good agreement with those from the harmonic balance method. Lastly, effects of the external mean axial load, amplitude of variation of the axial load, geometrical and material parameters on amplitude–frequency responses of the buckled beam are numerically investigated.

Nonlinear dynamics phenomenon detection in a polydyne cam with an offset flat-faced follower mechanism using multi shocks absorbers systems

The influences of the internal distance of the guide from inside (I.D.), the constant speeds of the cam (N), and the followers' offsets (O) on the chaotic phenomena are studied in this research. The chaotic phenomena is investigated using Phase-plane portrait with Poincare’ Maps, Power Density Function of Fast Fourier Transform (FFT), and Largest Lyapunov Exponent. The chaos analysis of the offset flat-faced follower due to the contact with the polydyne cam profile is controlled using multi linear and nonlinear shocks absorbers systems. The suspensions of the shocks absorbers are described using (spring-damper-mass) systems. The nonlinear response of the flat-faced follower due to the offset shift is calculated at different internal distance of the follower guide (I.D), different speeds of the cam (N), and different followers’ offsets (O). The analytic solution of the nonlinear follower motion is done using the second-order ordinary differential equation. The numerical results of the nonlinear response of the follower is determined using SolidWorks program, while OPTOTRAK 30/20 equipment with high speed camera is used in the experiment inspection. Largest Lyapunov exponent has a maximum values when the follower offsets to the right, while the largest Lyapunov exponent has a minimum values when the follower offsets to the left. The transition to chaos when the follower offsets to the left is faster than when the follower offsets to the right at (O = 50 mm). The chaos phenomenon in cam follower system in the presence of follower offset is increased with the increasing of (I.D.), (N), and (O).