Phase Plane Analysis Research Articles

Evaluation of Energy Utilization Ratio and Phase Plane Analysis of Bias Torque Function Control

Evaluation of Energy Utilization Ratio and Phase Plane Analysis of Bias Torque Function Control

Experimental study on impact coupling characteristics of high-pressure accumulator for unconstant-pressurized chamber rock drills

In order to improve the efficiency of unconstant-pressurized chamber rock drills in large-hole and hard-rock conditions, the coupling characteristics of high-pressure accumulator and the impact system were analyzed. The multi-parameter mechanism experiment was designed. Piston displacement, oil return flow, main input-oil, and internal chamber pressure were obtained. Combining the structural characteristics and working principles of rock drills, the experiment-based accumulator volume calculation methods was proposed. According to the phase plane analysis of the accumulator diaphragm and the time series analysis of impact performance parameters, which indicate that the high-pressure accumulator and impact system have typical non-contact nonlinear structures. The dynamic performances also present complex nonlinear characteristics. With the increase of gas pre-charge pressure, the bifurcation points were developed from period-one motions to period-two motions, the coupling degree of flow coupling between the accumulator and the impact piston was gradually enhanced, and the impact performance was gradually improved. With the formation of the period-one motions and the increase of bifurcation strength, the diaphragm became more susceptible to fatigue damage. The optimum choice of high-pressure accumulator gas pre-charge pressure is 80–90 bar, which provides a reference for the application and parameter setting of high-pressure accumulators in unconstant-pressurized chamber rock drills.

Novel soliton solutions and phase plane analysis in nonlinear Schrödinger equations with logarithmic nonlinearities

This paper investigates a generalized form of the nonlinear Schrödinger equation characterized by a logarithmic nonlinearity. The nonlinear Schrödinger equation, a fundamental equation in nonlinear wave theory, is applied across various physical systems including nonlinear optics, Bose–Einstein condensates, and fluid dynamics. We specifically explore a logarithmic variant of the nonlinear Schrödinger equation to model complex wave phenomena that conventional polynomial nonlinearities fail to capture. We derive four distinct forms of the nonlinear Schrödinger equation with logarithmic nonlinearity and provide exact solutions for each, encompassing bright, dark, and kink-type solitons, as well as a range of periodic solitary waves. Analytical techniques are employed to construct bounded and unbounded traveling wave solutions, and the dynamics of these solutions are analyzed through phase portraits of the associated dynamical systems. These findings extend the scope of the nonlinear Schrödinger equation to more accurately describe wave behaviors in complex media and open avenues for future research into non-standard nonlinear wave equations.

Solutions for a problem involving a ϕ-Laplacian-like operator via energy analysis, phase plane and shooting method

In this paper, we establish the existence of a countably infinite family of radially symmetric solutions that exhibit sign variations. These solutions are obtained for a Dirichlet boundary value problem incorporating a ϕ-Laplacian-like operator. Our main tools are the shooting method, phase plane and energy analysis, which demand extensive use of a Pozohaev-type identity.

Nonlinear Jeans instability analysis of gravitating astrofluids in Eddington-inspired Born-Infeld gravity framework

The nonlinear Jeans instability dynamics in viscoelastic astrofluids within the Eddington-inspired Born-Infeld (EiBI) gravitational framework excitable in star-forming interstellar cloud fluids is semi-analytically investigated. Application of multiple scaling techniques here results in a unique form of the Korteweg-de Vries (KdV) equation moderated with a peculiar set of distinctive fluid polyparametric coefficients. A numerical illustrative platform portrays the excitation of diversified compressive solitary-chain wave patterns. The evolutionary existence of such structural patterns is further confirmed geometrically with the help of illustrative phase-plane analyses. The various parametric stabilizers and destabilizers, alongside accelerating and decelerating factors on the clouds, are illustratively analysed. It is seen that the composite cloud stability is significantly influenced by the strength polarity of the gravitational EiBI-parameter (χ). The dependency of the obtained solitary wave patterns on the EiBI-polarity is further confirmed in a numerical simulation platform illustratively. Our analysis also highlights that the influence of material density on the system stability is dependent on the EiBI-polarity. The nonlinear spatiotemporal evolution of the instability reveals a unique type of soliton fission phenomenon, indicative of inherently produced additional disturbances within the astrofluid. Our obtained results could be widely helpful in application to the EiBI-centric diverse astrocosmic phenomena, such as bounded structure formation and evolution in diverse complex astroenvirons.

On differential equations with exponential nonlinearities

Two-point boundary value problems for the second order nonlinear ordinary differential equations, arising in the heat conductivity theory, are considered. Multiplicity and existence results are established. The properties of solutions are studied. Estimates of the number of solutions are obtained. A bifurcation analysis was made and the bifurcation curves were presented. The analytical technique together with the phase plane analysis is used to obtain the results.

Newton's cradle: Cell cycle regulation by two mutually inhibitory oscillators

The cell division cycle is a fundamental physiological process displaying a great degree of plasticity during the course of multicellular development. This plasticity is evident in the transition from rapid and stringently-timed divisions of the early embryo to subsequent size-controlled mitotic cycles. Later in development, cells may pause and restart proliferation in response to myriads of internal or external signals, or permanently exit the cell cycle following terminal differentiation or senescence. Beyond this, cells can undergo modified cell division variants, such as endoreplication, which increases their ploidy, or meiosis, which reduces their ploidy. This wealth of behaviours has led to numerous conceptual analogies intended as frameworks for understanding the proliferative program. Here, we aim to unify these mechanisms under one dynamical paradigm. To this end, we take a control theoretical approach to frame the cell cycle as a pair of arrestable and mutually-inhibiting, doubly amplified, negative feedback oscillators controlling chromosome replication and segregation events, respectively. Under appropriate conditions, this framework can reproduce fixed-period oscillations, checkpoint arrests of variable duration, and endocycles. Subsequently, we use phase plane and bifurcation analysis to explain the dynamical basis of these properties. Then, using a physiologically realistic, biochemical model, we show that the very same regulatory structure underpins the diverse functions of the cell cycle control network. We conclude that Newton's cradle may be a suitable mechanical analogy of how the cell cycle is regulated.

Advanced tumor growth modeling: A numerical study integrating phase plane analysis with finite volume method

This study investigates tumor density dynamics through computational modeling, focusing on key parameters: proliferation rate m, diffusion coefficient τ, and additional factors c and κ. Systematic exploration reveals nuanced interactions shaping tumor growth, regression, and dispersal. Higher m values accelerate proliferation, while increased τ facilitates dispersal. Parameters c and κ further refine our understanding of tumor behavior. These insights inform the development of computational models for oncological research, with implications for targeted therapeutic interventions. The aim can be realized by utilizing the propose Finite Volume Method to develop a representation of cancer models and improve the efficiency of their simulations. Furthermore, the numerical results will be validated by comparing them with the Tanh-Coth method formulation to assess the model's accuracy and efficiency. The findings suggest that accurate modeling of tumor dynamics can aid in predicting tumor progression and response to treatments, ultimately contributing to personalized medicine and more effective therapeutic strategies. Phase plane analysis shows the schemes are conditionally stable and accurate to the fourth order in time. The method demonstrates exceptional agreement and error-free findings compared to 3D plots, underscoring its potential utility in clinical and research settings.

Nuclear acoustic envelope soliton in a relativistically degenerate magneto-rotating stellar plasma

The study explores a model called Relativistic Degenerate Magneto-Rotating Quantum Plasma (RDMRQP), which comprises a static heavy nucleus, an inertial non-degenerate light nucleus, and warm non-relativistic or ultra-relativistic electrons. The focus is on observing the emergence of Nucleus-Acoustic Envelope Solitons (NAESs). Using the reductive perturbation method, a Nonlinear Schrödinger equation (NLSE) is derived to characterize the properties of NAESs. We have analysed the Peregrine breather soliton solution. The investigation reveals that the temperature of warm degenerate species, plasma system’s rotational speed, and the presence of heavy nucleus species can alter the fundamental features (height and width) of NAESs in the WDMRQP system. The study emphasizes the existence of only positive NA wave potential. Additionally, a phase plane analysis is conducted to gain a deeper understanding of the parametric dependencies. Through detailed mathematical and numerical analysis, the study avoids overloading with complex mathematics while demonstrating parametric dependence via phase portrait analysis. The research augments the envelop soliton model with breather mode solutions and discusses modulation instability using the Benjamin-Feir Index, highlighting the significance of solitons in star formation. Envelop solitons, stable waves within stars and proto-stars, influence energy transport and stellar evolution, playing a crucial role in the accretion process and formation of stable structures. Key findings include the effects of various parameters on NAESs’ generation and propagation in which non-relativistic and ultra-relativistic electrons support NAESs, with amplitude and width influenced by temperature, rotational frequency, inclination angle, and the presence of a static heavy nucleus. The research is applicable to hot white dwarfs and neutron stars, suggesting further exploration of quantum effects and non-planar or arbitrary amplitude NAESs.

A complete dynamical analysis of discrete electric lattice coupled with modified Zakharov–Kuznetsov equation

The behavior of nonlinear waves within a modified Zakharov–Kuznetsov equation and their interactions with discrete electric lattice structures are examined in this study. The ϕ6−model expansion method is utilized to acquire substantial knowledge into the complex dynamics of the system under consideration, particularly with regard to the discrete electric lattice and analytical electrical solitons. By incorporating higher-order effects and improving accuracy in representing specific physical conditions, the study achieves a more realistic portrayal of nonlinear wave dynamics. The investigation also sheds light on the relationship between non-linearity, discreteness, and equation dynamics by exploring the conditions that lead to the formation of solitons and other nonlinear structures. In addition, a unique set of electrical solitons is defined to explore dynamic behaviors such as chaotic, quasi-periodic, and periodic motions under various parameterized conditions, including an external damping force. Phase plane analysis is visualized by using dynamic structure 3D and 2D phase plots, is used for bifurcation and sensitivity inspections. Finally, time series graphs are offered as mathematical depictions of solitary waves, and Lyapunov exponents with real and complex eigenvalues are used to study the stability and chaotic behaviors of the system.

Existence of traveling wave solutions for density-dependent diffusion competitive systems

In this paper we are concerned with the existence of traveling wave solutions for two species competitive systems with density-dependent diffusion. Since the density-dependent diffusion is a kind of nonlinear diffusion and degenerates at the origin, the methods for proving the existence of traveling wave solutions for competitive systems with linear diffusion are invalid. To overcome the degeneracy of diffusion, we construct a nonlinear invariant region Ω near the origin. Then by using the method of phase plane analysis, we prove the existence of traveling wave solutions connecting the origin and the unique coexistence state, when the speed c is large than some positive value. In addition, when one species is density-dependent diffusive while the other is linear diffusive, via the change of variables and the central manifold theorem, we prove the existence of the minimal speed c∗ . And for c⩾c∗ , traveling wave solutions connecting the origin and the unique coexistence state still exist. In particular, when c=c∗ , we find that one component of the traveling wave solution is sharp type while the other component is smooth, which is a different phenomenon from linear diffusive systems and scalar equations.

The Control of Handling Stability for Four-Wheel Steering Distributed Drive Electric Vehicles Based on a Phase Plane Analysis

For the sake of enhancing the handling and stability of distributed drive electric vehicles (DDEVs) under four-wheel steering (4WS) conditions, this study proposes a novel hierarchical control strategy based on a phase plane analysis. This approach involves a meticulous comparison of the stable region in the phase plane to thoroughly analyze the intricate influence of the front wheel angle, rear wheel angle, road adhesion coefficient, and longitudinal speed on the complex dynamic performances of DDEVs and to accurately determine the critical stable-state parameter. Subsequently, a hierarchical control strategy is presented as an integrated solution to achieve the coordinated control of maneuverability and stability. On the upper control level, a model predictive control (MPC) motion controller is developed, wherein the real-time adjustment of the control weight matrix is ingeniously achieved by incorporating the crucial vehicle stable-state parameter. The lower control level is responsible for the optimal torque allocation among the four wheel motors to minimize the tire load rate, thereby ensuring a sufficient tire grip margin. The optimal torque distribution for the four wheel motors is achieved using a sophisticated two-level allocation algorithm, wherein the friction ellipse is employed as a judgement condition. Finally, this developed control strategy is thoroughly validated through co-simulation utilizing the CarSim 2019 and Simulink 2020b commercial software, demonstrating the validity of the developed control strategy. The comparative results indicate that the presented controller ensures a better tracking capability to the desired vehicle state while exhibiting improved handling stability under both the double lane shifting condition and the serpentine working condition.

Nonlinear spatial evolution of degenerate quartets of water waves

In this manuscript we investigate the Benjamin–Feir (or modulation) instability for the spatial evolution of water waves from the perspective of the discrete, spatial Zakharov equation, which captures cubically nonlinear and resonant wave interactions in deep water without restrictions on spectral bandwidth. Spatial evolution, with measurements at discrete locations, is pertinent for laboratory hydrodynamic experiments, such as in wave flumes, which rely on time-series measurements at fixed gauges installed along the facility. This setting is likewise appropriate for experiments in electromagnetic and plasma waves. Through a reformulation of the problem for a degenerate quartet, we bring to bear techniques of phase-plane analysis which elucidate the full dynamics without recourse to linear stability analysis. In particular we find hitherto unexplored breather solutions and discuss the optimal transfer of energy from carrier to sidebands. We show that the maximal energy transfer consistently occurs for smaller side-band separation than the fastest linear growth rate. Finally, we discuss the observability of such discrete solutions in light of numerical simulations.

Seasonal variability and stochastic branching process in malaria outbreak probability

Background: Malaria is the world’s most fatal and challenging parasitic disease, caused by the Plasmodium parasite, which is transmitted to humans by the bites of infected female mosquitoes. Bangladesh is the most vulnerable region to spread malaria because of its geographic position. In this paper, we have considered the dynamics of vector-host models and observed the stochastic behavior. This study elaborates on the seasonal variability and calculates the probability of disease outbreaks. Methods: We present a model for malaria disease transmission and develop its corresponding continuous-time Markov chain (CTMC) representation. The proposed vector-host models illustrate the malaria transmission model along with sensitivity analysis. The deterministic model with CTMC curves is depicted to show the randomness in real scenarios. Sequentially, we expand these studies to a time-varying stochastic vector-host model that incorporates seasonal variability. Phase plane analysis is conducted to explore the characteristics of the disease, examine interactions among various compartments, and evaluate the impact of key parameters. The branching process approximation is developed for the corresponding vector-host model to calculate the probability outbreak. Numerous numerical results are accomplished to observe the analytical investigation. Results: Seasonality and contact patterns affect the dynamics of disease outbreaks. The numerical illustration provides that the probability of a disease outbreak depends on the infected host or vector. Additionally, periodic transmission rates have a great influence on the probability outbreak. The basic reproduction number (R0) is derived, which is the main justification for studying the dynamical behavior of epidemic models. Conclusions: Seasonal variability significantly impacts malaria transmission, and the probability of disease outbreaks is influenced by time and the initial number of infected individuals. Moreover, the branching process approximation is applicable when the population size is large enough and the basic reproduction number is less than 1. In the future, such analysis can help decision-makers understand the impact of various parameters and their stochastic behavior in the vector-host model to prevent such types of disease outbreaks.

Exploring dust-ion acoustic shocks in a plasma in the light of phase plane analysis

This paper presents a comprehensive exploration of shock waves in a dusty plasma, incorporating the dynamics of inertial dust and ions with Maxwellian distributed electrons. By deriving the Korteweg-de Vries-Burgers (KdV-B) equation and by using its standard solution we analyze the characteristics of the shock front under varying parameters. Furthermore, we conduct a phase plane analysis to elucidate the system's behavior and dynamics by tuning in the parameters that provide various types of perturbation in the system. The investigation sheds light on the intricate behavior of shock waves in dusty plasmas and contributes valuable insights to the understanding of plasma dynamics.

On traveling wave solutions with stability and phase plane analysis for the modified Benjamin-Bona-Mahony equation.

The modified Benjamin-Bona-Mahony (mBBM) model is utilized in the optical illusion field to describe the propagation of long waves in a nonlinear dispersive medium during a visual illusion (Khater 2021). This article investigates the mBBM equation through the utilization of the rational [Formula: see text]-expansion technique to derive new analytical wave solutions. The analytical solutions we have obtained comprise hyperbolic, trigonometric, and rational functions. Some of these exact solutions closely align with previously published results in specific cases, affirming the validity of our other solutions. To provide insights into diverse wave propagation characteristics, we have conducted an in-depth analysis of these solutions using 2D, 3D, and density plots. We also investigated the effects of various parameters on the characteristics of the obtained wave solutions of the model. Moreover, we employed the techniques of linear stability to perform stability analysis of the considered model. Additionally, we have explored the stability of the associated dynamical system through the application of phase plane theory. This study also demonstrates the efficacy and capabilities of the rational [Formula: see text]-expansion approach in analyzing and extracting soliton solutions from nonlinear partial differential equations.

An early-fault diagnostic method based on phase plane for lithium-ion batteries under complex operation conditions

Timely and accurate diagnosis of lithium-ion battery faults is crucial for ensuring the safety of electric vehicles. While incremental capacity (IC)-based diagnostic methods are effective, they can only be employed under charging conditions. To address this issue, a fault diagnostic method based on phase plane analysis is proposed to extend the application to various operating conditions. In a phase plane, the battery voltage is plotted on the x-axis, while the differential in voltage is plotted on the y-axis. During charging, diagnostic vectors indicating the occurrence and type of faults are extracted from the trajectories in the phase plane. Under complex operating conditions, early faults can be detected based on the morphological and scale deviation of the trajectories. Experimental results have shown that the proposed method can effectively detect minor faults. The fault detection rate and detection accuracy rate under charging conditions are 96.25 % and 97.73 %, respectively, which are 17.5 % and 8.54 % higher than those of the IC-based diagnostic method. Furthermore, the fault detection rate and accuracy rate under UDDS (Urban Dynamometer Driving Schedule) are 91.76 % and 97.23 %, respectively. Additionally, the diagnostic speed of the proposed method is twice as fast as that of the IC-based method.

A simple modification to the classical SIR model to estimate the proportion of under-reported infections using case studies in flu and COVID-19

BackgroundUnder-reporting and, thus, uncertainty around the true incidence of health events is common in all public health reporting systems. While the problem of under-reporting is acknowledged in epidemiology, the guidance and methods available for assessing and correcting the resulting bias are obscure. ObjectiveWe aim to design a simple modification to the Susceptible – Infected – Removed (SIR) model for estimating the fraction or proportion of reported infection cases. MethodsThe suggested modification involves rescaling of the classical SIR model producing its mathematically equivalent version with explicit dependence on the reporting parameter (true proportion of cases reported). We justify the rescaling using the phase plane analysis of the SIR model system and show how this rescaling parameter can be estimated from the data along with the other model parameters. ResultsWe demonstrate how the proposed method is cross-validated using simulated data with known disease cases and then apply it to two empirical reported data sets to estimate the fraction of reported cases in Missoula County, Montana, USA, using: (1) flu data for 2016–2017 and (2) COVID-19 data for fall of 2020. ConclusionsWe establish with the simulated and COVID-19 data that when most of the disease cases are presumed reported, the value of the additional reporting parameter in the modified SIR model is close or equal to one, so that the original SIR model is appropriate for data analysis. Conversely, the flu example shows that when the reporting parameter is close to zero, the original SIR model is not accurately estimating the usual rate parameters, and the re-scaled SIR model should be used. This research demonstrates the role of under-reporting of disease data and the importance of accounting for under-reporting when modeling simulated, endemic, and pandemic disease data. Correctly reporting the “true” number of disease cases will have downstream impacts on predictions of disease dynamics. A simple parameter adjustment to the SIR modeling framework can help alleviate bias and uncertainty around crucial epidemiological metrics (e.g.: basic disease reproduction number) and public health decision making.

Adaptive Nonsingular Fast Terminal Sliding Mode-Based Direct Yaw Moment Control for DDEV under Emergency Conditions

This paper presents an innovative three-level direct yaw moment control strategy for distributed drive electric vehicles (DDEV) under emergency conditions. The phase plane analysis is used at the supervisory level to design the stability boundary function taking into account the impact of the road adhesion coefficient. To guarantee the performance of finite-time convergence and singularity-free methods, the adaptive nonsingular fast terminal sliding mode control (ANFTSMC) is developed at the decision level to determine the extra yaw moment for tracking the intended side slip angle and yaw rate. Among this, the unstable domain in the phase plane is further separated into moderately and severely unstable according to the degree of vehicle instability, which is defined by the distance between the state phase point and the stability boundary. Meanwhile, the adaptive weight between the handling and stability is obtained. At the executive level, the quadratic programming algorithm is adopted to allocate four-wheel torque with the objective of optimal tire utilization rate. Finally, the co-simulation test is executed in both closed-loop and open-loop circumstances; according to the simulation results, the presented ANFTSMC method outperforms the SMC, and it can decrease the tracking error and improve the handling and stability.

Existence and stability of solitary waves to the rotation–Camassa–Holm equation

In this paper, we investigate the existence and stability of solitary waves to the rotation–Camassa–Holm equation which can be considered as a model in the shallow water for the long‐crested waves propagating near the equator with effect of the Coriolis force due to the Earth's rotation. We prove the existence of solitary waves by performing a phase plane analysis. Moreover, utilizing the approach proposed by Grillakis–Shatah–Strauss, we prove the stability of solitary waves.