Phase Portrait Analysis Research Articles

Using Harvesting Models to Teach Modeling with Differential Equations

: Harvesting models based on ordinary differential equations are commonly used in the fishery industry and wildlife management to model the evolution of a population depleted by harvest mortality. We present a project consisting of a series of scenarios based on fishery harvesting models to teach the application of theoretical concepts learned in a differential equations course to scenarios encountered in real fisheries. These projects require a thorough understanding of simplifying assumptions inherent in various models, as well as a qualitative analysis of phase portraits, bifurcations, and stability of steady states. Parameters are estimated and equations are sometimes solved both analytically and numerically. Students learn to respond to a professional request from a fishery in the form of a scientific report, which requires organizing and communicating assumptions, models, solution methods, results, and a final recommendation with clarity and professionalism.

Primary and Secondary Resonance Analyses of Viscoelastic Nanoplates Based on Strain Gradient Theory

In this study, the nonstationary oscillation, secondary resonance and nonlinear dynamic behavior of viscoelastic nanoplates with linear damping are investigated based on the modified strain gradient theory extended for viscoelastic materials. The viscous component of the nonclassical and classical stress tensors are evaluated on the basis of the Leaderman viscoelastic model. Then, incorporating the size-dependent potential energy, kinetic energy and an external excitation force work, the governing equations of the oscillations are obtained based on the Hamilton’s principle. The governing formula is obtained as a nonlinear second-order integro-differential partial equation. This size-dependent viscoelastic formula is solved using analytical Harmonic balance method (HBM) and the fourth-order Runge–Kutta technique after applying the expansion theory. Additionally, the stability of the steady-state response is examined by means of HBM. Then, the secondary resonance conditions due to the super-harmonic motion are determined by performing frequency response, force response, Poincare map and phase portrait analyses. In addition, the nonstationary transient vibration of viscoelastic nanosystem is analyzed by performing Hilbert–Huang transform.

Appropriate Boundary Condition for Dupuit‐Boussinesq Theory on the Steady Groundwater Flow in an Unconfined Sloping Aquifer With Uniform Recharge

AbstractThe application of Dupuit‐Boussinesq theory on the two‐dimensional steady groundwater flow in an unconfined sloping aquifer with uniform recharge is theoretically investigated. The examination of conditions at the downstream and upstream is performed. The method of phase plane is used to geometrically interpret the characteristics of the nonlinear theory by various initial conditions. For phase portrait analysis, three configurations for aquifers are considered. In each type of aquifer three conditions of recharges as well as downstream groundwater tables are also considered for comprehensive analysis. Based on the phase portrait analysis, one explicit formula in terms of known parameters and downstream boundary condition is derived to easily and straightforwardly determine an appropriate upstream boundary condition. The derived explicit formula can facilitate correct modeling using Boussinesq theory.

Vehicle control synthesis using phase portraits of planar dynamics

ABSTRACTPhase portraits provide control system designers strong graphical insight into nonlinear system dynamics. These plots readily display vehicle stability properties and map equilibrium point locations and movement to changing parameters and system inputs. This paper extends the usage of phase portraits in vehicle dynamics to control synthesis by illustrating the relationship between the boundaries of stable vehicle operation and the state derivative isoclines in the yaw rate–sideslip phase plane. Closed-loop phase portraits demonstrate the potential for augmenting a vehicle's open-loop dynamics through steering and braking. The paper concludes by applying phase portrait analysis to an envelope control algorithm for yaw stability and a sliding surface controller for stabilising a saddle point equilibrium in drifting.

Geometric analysis of a model for cross‐feeding in the chemostat

A model for competition between bacterial strains in a chemostat is studied using a model reduction argument and phase portrait analysis for the specific case of trophic chain. The first strain feeds on glucose but secretes a metabolic intermediate (acetate), which the second strain consumes. However, the metabolic intermediate imposes a cost to growth to both strains. The geometric and asymptotic analysis of the reduced model allows a rigourous treatment of the role of this assumption in the emergence of coexistence (cross‐feeding) equilibria. Conditions for non‐existence of cross‐feeding equilibria are given for certain parameter ranges. In the case of trophic chains, existence of cross‐feeding equilibria does not rely only on the presence of the acetate specialist scavenging for the intermediate metabolite inside the resource‐limited chemostat environment. The glucose specialist must, in addition, possess a certain degree of tolerance to the metabolic intermediate. Relaxing the assumption of the cost of growth helps establish a cross‐feeding equilibrium.

Phase portrait analysis of super solitary waves and flat top solutions

The phase portrait analysis of super solitary waves has revealed a new kind of intermediate solution which defines the boundary between the two types of super solitary waves, viz., Type I and Type II. A Type I super solitary wave is known to be associated with an intermediate double layer while a Type II solution has no such association. The intermediate solution at the boundary has a flat top structure and is called a flat top solitary wave. Its characteristics resemble an amalgamation of a solitary wave and a double layer. It was found that, mathematically, such kinds of structures may emerge due to the presence of an extra nonlinearity. Although they are relatively unfamiliar in the realm of plasma physics, they have much wider applications in other physical systems.

Oscillation phenomenon and its mechanism of an energy-saving and emission-reduction system

PurposeThis paper aims to study the oscillation phenomenon before chaos as well as its mechanism of occurrence in the energy-saving and emission-reduction system.Design/methodology/approachThe system dynamics analysis, phase portrait analyses, equilibrium point analysis and bifurcation curve were applied to this paper.FindingsFirst, the authors find an oscillation phenomenon previous to chaos. Second, on the one hand, the existence of two unstable saddles is the reason for the occurrence of oscillation phenomenon. On the other hand, the increasing of carbon emissions can arouse oscillation phenomenon.Originality/valueThis paper finds an oscillation phenomenon previous to chaos in the energy-saving and emission-reduction system. The mechanism of occurrence of oscillation phenomenon is studied. The existence of two unstable saddles is the reason for the occurrence of such oscillation phenomenon. The oscillation is related with fold bifurcation. The study also provides a theoretical basis for the further study of chaos control.

Dependence of chaotic actuation dynamics of Casimir oscillators on optical properties and electrostatic effects

With Casimir and electrostatic forces playing a crucial role for the performance and stability of microelectromechanical systems (MEMS), the presence of chaotic behavior, which is often unavoidable, leads to device malfunction due to stiction. Therefore, we investigate here how the optical properties of different materials influence the chaotic behavior of electrostatic torsional MEMS due to changes in magnitude of the Casimir forces and torques. We consider the materials Au, which is a good conductor, AIST, which is a phase change material being close to metal in the crystalline state, and finally doped SiC as a very poor conductor. For the conservative systems, there is no chaotic behavior and the analysis of phase portraits and bifurcation diagrams reveal the strong sensitivity of stable actuation dynamics on the material optical properties, while applied electrostatic potentials lead faster to instability and stiction for higher conductivity materials. For the driven systems, the Melnikov method is used to study the chaotic behavior. The results from this method are supported by the study of the contours of the transient time to stiction in the phase plane, which reveal a substantially increased chaotic behavior for higher conductivity materials, associated with stronger Casimir torques and applied electrostatic potentials.

SYNTHESIS OF THE LAWS GOVERNING THE NON-HOLONOMIC MODEL OF A TWO-LINK ROAD TRAIN WITH REVERSE MOTION (OFF-AXLE HITCHING MODEL)

The complexity of the control of the road train is due to the pronounced nonlinearities, as well as the instability of the control object during the movement in the backward motion (jackknifing). For the road trains, the location of the towing device behind the tractor's rear axle is quite typical. In this study, a synthesis of control laws for road trains with offset of coupling devices relative to the rear axle of the tractor (off-axle hitching) is proposed. The controllers have been implemented both to ensure a stable circular motion and for rectilinear motion with a given orientation angle, and the behavioral features of this model have been studied on the basis of them. Based on the analysis of the approaches to the synthesis of the laws governing the road train with the coupling out, it was decided to synthesize the required control laws using the Lyapunov function method. Synthesized controllers can be directly used to program the robotic systems of the respective models. It is also possible to use them for the development of the Dubins machine for the investigated model. They can be used to build automatic control systems that would help the driver to drive a car with a trailer while driving backward. In this research, a study was made of the state of the solution of the problem associated with the reverse movement of a road train consisting of a tractor and a semitrailer with a coupling, synthesized laws made it possible to study the features of such model, determined by its linear dimensions. For comparison of the synthesized laws, the analysis of phase portraits of trajectories, angles of folding and control, orientation angles was carried out, and also the analysis of the quality of transient processes with the change in the speed of the road train was performed.

Variable Structure Control of a Mass Spring Damper Subjected to a Unilateral Constraint: Application to Radio-Frequency MEMS Switches

A feedback control strategy is presented for improving the transient response of the ubiquitous mass-spring-damper (MSD) system; the closed-loop system has a small settling time with no overshoot for a step input. This type of response is ideal for MSD systems subjected to a unilateral constraint such as radio-frequency micro-electro-mechanical-system (RF MEMS) switches, which are required to close in a short period of time without bouncing. The control strategy switches the stiffness of the MSD between its nominal value and a negative value, resulting in a hybrid dynamical system. A phase portrait analysis of the hybrid system is carried out to establish the asymptotic stability property of the equilibrium and quantify the transient response. Simulation results are presented using parameter values of a real RF MEMS switch from the literature. As compared to open-loop strategies that are currently used, the proposed feedback control strategy promises to provide comparable switch-closing times with robust performance and eliminate bouncing.

Solitary wave solution and dynamic transition of dust ion acoustic waves in a collisional nonextensive dusty plasma with ionization effect

The solitary wave solution and dynamic transition of dust ion acoustic waves (DIAWs) are studied in an unmagnetized collisional dusty plasma, which consists of negatively charged dust fluid, positively charged ions, q-nonextensive electrons, and background neutral particles, considering the ionization effect, ion loss, ion–neutral, ion–dust, and dust–neutral collisions. Using reductive perturbation technique (RPT) the damped Korteweg–de Vries (DKdV) equation and the damped modified Korteweg–de Vries (DMKdV) equation are derived. Using momentum conservation law, dust ion acoustic solitary wave solutions (DIASWSs) are obtained for the DKdV and MDKdV equations. Effects of nonextensive parameter (q) and ion–dust collisional frequency (νid) are presented on the DIASWSs. Effects of ion–dust collisional frequency (νid), ion–neutral collisional frequency (νin) and ion loss rate (νl) on DIAWs are investigated considering an external periodic perturbation through three-dimensional phase portrait and time-series analysis. It is found that νid, νin and νl significantly affect on the dynamics of the perturbed DIAWs and play a pivotal role in the transition from quasiperiodic and chaotic motions to periodic motion.

Spontaneous imbibition in fractal tortuous micro-nano pores considering dynamic contact angle and slip effect: phase portrait analysis and analytical solutions

Shales have abundant micro-nano pores. Meanwhile, a considerable amount of fracturing liquid is imbibed spontaneously in the hydraulic fracturing process. The spontaneous imbibition in tortuous micro-nano pores is special to shale, and dynamic contact angle and slippage are two important characteristics. In this work, we mainly investigate spontaneous imbibition considering dynamic contact angle and slip effect in fractal tortuous capillaries. We introduce phase portrait analysis to analyse the dynamic state and stability of imbibition. Moreover, analytical solutions to the imbibition equation are derived under special situations, and the solutions are verified by published data. Finally, we discuss the influences of slip length, dynamic contact angle and gravity on spontaneous imbibition. The analysis shows that phase portrait is an ideal tool for analysing spontaneous imbibition because it can evaluate the process without solving the complex governing ordinary differential equations. Moreover, dynamic contact angle and slip effect play an important role in fluid imbibition in fractal tortuous capillaries. Neglecting slip effect in micro-nano pores apparently underestimates imbibition capability, and ignoring variations in contact angle causes inaccuracy in predicting imbibition speed at the initial stage of the process. Finally, gravity is one of the factors that control the stabilisation of the imbibition process.

Design and Development of Surgeon Augmented Endovascular Robotic System.

Inadequate visual and force feedback while navigating surgical tools elevate the risk of endovascular procedures. It also poses occupational hazard due to repeated exposure to X-rays. A teleoperated robotic system that augments surgeon's actions is a solution. We have designed and developed an endovascular robotic system that augments surgeon's actions using conventional surgical tools, as well as generates feedback in order to ensure safety during the procedure. The reaction force from vasculature is estimated from motor current that drives the surgical tool. Calibration required for force estimation is based on bilevel optimization. Input shaping is used in conjunction with a cascaded controller to avoid large responses due to faster inputs and to track tool position. The design, realization, and testing of our system are presented. The responses of the system in comparison with the dynamics model is similar vis-à-vis the same input commands. Any error in the position tracking is reduced by the cascaded controller. Phase-portrait analysis of the system showed that the system is stable. The reaction force estimation is validated against load cell measurements. The safety mechanism in the events of excessive reaction forces while interacting with vasculature is demonstrated. Our system is a step toward intelligent robots that can assist surgeons during endovascular procedures by monitoring and alerting the surgeons regarding detrimental parameters. It arrests any unintended excursions of the surgical tools or surgeon's actions. This will also eliminate the need for surgeons to be in radiation environment.

Fast magnetohydrodynamic cnoidal waves and solitons in electron-positron plasma

Linear and nonlinear propagation of fast magnetohydrodynamic waves (or magnetoacoustic waves) are studied in homogeneous, magnetized and warm collisionless electron-positron (e-p) plasma by using two fluid magnetohydrodynamic model. In the linear limit, the wave dispersion relation is obtained and wave dispersion effect which appears through inertial length in e-p plasma system is also discussed. Using reductive perturbation method, the Korteweg-de Vries (KdV) equation for small but finite wave amplitude of magnetoacoustic waves is derived with appropriate boundary conditions. The cnoidal wave and soliton solutions are obtained using well known Sagdeev potential approach for magnetoacoustic waves in e-p plasmas propagating in the direction perpendicular to the external magnetic field. The phase portrait analysis and numerical illustration of magnetoacoustic cnoidal waves and solitons is also presented by using the parameters such as magnetic field intensity, plasma density and temperature of electron and positron fluids for astrophysical plasma situations exist in the literature.

On the Stability Analysis of a Coupled Rigid Body

In this research article, we investigate the stability of a complex dynamical system involving coupled rigid bodies consisting of three equal masses joined by three rigid rods of equal lengths, hinged at each of their bases. The system is free to oscillate in the vertical plane. We obtained the equation of motion using the generalized coordinates and the Euler-Lagrange equations. We then proceeded to study the stability of the dynamical systems using the Jacobian linearization method and subsequently confirmed our result by phase portrait analysis. Finally, we performed MathCAD simulation of the resulting ordinary differential equations, describing the dynamics of the system and obtained the graphical profiles for each generalized coordinates representing the angles measured with respect to the vertical axis. It is discovered that the coupled rigid pendulum gives rise to irregular oscillations with ever increasing amplitude. Furthermore, the resulting phase portrait analysis depicted spiral sources for each of the oscillating masses showing that the system under investigation is unstable.

Dust acoustic cnoidal waves in a polytropic complex plasma

The nonlinear characteristics of dust acoustic (DA) waves in an unmagnetized collisionless complex plasma containing adiabatic electrons and ions and negatively charged dust grains (including the effects of modified polarization force) are investigated. Employing the reductive perturbation technique, a Korteweg–de Vries–Burgers (KdVB) equation is derived. The analytical solution for the KdVB equation is discussed. Also, the bifurcation and phase portrait analyses are presented to recognize different types of possible solutions. The dependence of the properties of nonlinear DA waves on the system parameters is investigated. It has been shown that an increase in the value of the modified polarization parameter leads to a fast decay and diminishes the oscillation amplitude of the DA damped cnoidal wave. The relevance of our findings and their possible applications to laboratory and space plasma situations is discussed.

Convection Dynamics of Fe3O4 Nanoparticles in Blood Fluid Flow

In this paper, the convection dynamics of super paramagnetic iron oxide nanoparticles (MNPs) in blood fluid is studied. The system under consideration consists of a fluid which contains the layer of blood with MNPs having iron oxide core (Fe3O4) nanoparticle which is moving through the cavity of the blood vessel which has depositions of cholesterol on the inner lining. In its passing through the blood vessel it is subjected to external magnetic field and heat. The nonlinear three dimensional governing equations for the system under study are derived from the partial differential equations of conservation of momentum and energy. The effect of magnetic field for Hartman number on the chaotic convection of the MNPs in blood fluid is studied using phase portrait, time series and stability analysis. It is assumed that a minimum magnetic field is necessary to stabilise the convection of the MNPs to regulate its flow in specific directions so that the drug can be delivered to specific locations by magnetic dragging. Rectangular cavity structure is considered for the study based on non-uniform cholesterol deposition. It is observed that with magnetisation by external applied magnetic field the convection of MNPs are stabilised in a particular direction. This indicates towards the growing utilization as contrast agents in non-invasive imaging technique of magnetic resonance imaging (MRI) and magnetic drug delivery (MDD).

ФАЗОВ ПОРТРЕТ НА ДИНАМИЧНИ СИСТЕМИ ОТ ВТОРИ РЕД

The phase portrait of the second and higher order differential equations presents in graphical form the behavior of the solution set without solving the equation. In this way, the stability of a dynamic system and its long-time behavior can be studied. The article explores the capabilities of Mathcad for analysis of systems by the phase plane method. A sequence of actions using Mathcad's operators to build phase portrait and phase trace analysis is proposed. An example is given by a differential equation of the second order. The approach is also applicable to systems of higher order.

PHASE PORTRAIT OF SECOND-ORDER DYNAMIC SYSTEMS

The phase portrait of the second and higher order differential equations presents in graphical form the behavior of the solution set without solving the equation. In this way, the stability of a dynamic system and its long-time behavior can be studied. The article explores the capabilities of Mathcad for analysis of systems by the phase plane method. A sequence of actions using Mathcad's operators to build phase portrait and phase trace analysis is proposed. The approach is illustrated by a model of plasma renin activity after treatment of experimental animals with nicardipine. The identified process is a differential equation of the second order. The algorithm is also applicable to systems of higher order.

Generic phase portrait analysis of finite-time singularities and generalized Teleparallel gravity**Supported by the Egyptian Ministry of Scientific Research (24-2-12)

We analyze the four common types of finite-time singularity using a generic framework of the phase portrait geometric approach. This technique requires the Friedmann system to be written as a one-dimensional autonomous system. We employ a scale factor that has been used widely in the literature to realize the four finite-time singularity types, then we give a detailed discussion for each case showing possible novel models. Moreover, we show how different singularity types can play essential roles in different cosmological scenarios. Among several modified gravity theories, we show that the f(T) cosmology is compatible with the phase portrait analysis, since the field equations include Hubble derivatives only up to first order. Therefore, we reconstruct the f(T) theory which generates these phase portraits. We also perform a complementary analysis using the effective equation of state. Furthermore, we investigate the role of the torsion fluid in realizing the cosmic singularities.