Phase Plane Analysis Research Articles

Steady-state solutions for a reaction–diffusion equation with Robin boundary conditions: Application to the control of dengue vectors

AbstractIn this paper, we investigate an initial-boundary value problem of a reaction–diffusion equation in a bounded domain with a Robin boundary condition and introduce some particular parameters to consider the non-zero flux on the boundary. This problem arises in the study of mosquito populations under the intervention of the population replacement method, where the boundary condition takes into account the inflow and outflow of individuals through the boundary. Using phase plane analysis, the present paper studies the existence and properties of non-constant steady-state solutions depending on several parameters. Then, we prove some sufficient conditions for their stability. We show that the long-time efficiency of this control method depends strongly on the size of the treated zone and the migration rate. To illustrate these theoretical results, we provide some numerical simulations in the framework of mosquito population control.

Dynamical behavior of chaos, bifurcation analysis and soliton solutions to a Konno-Onno model.

The fractional coupled Konno-Onno model, which is frequently used in numerous fields of scientific and engineering disciplines, is being investigated in the current study in order to gain an understanding of complex phenomena and systems. The two main goals of this study are to be accomplished. Firstly, the research aims to identify novel solitons for the fractional coupled Konno-Onno model using the unified technique, which is currently absent from the literature. Secondly, a novel strategy that hasn't been previously investigated is phase portrait analysis for both perturbed and non-perturbed dynamical systems. The current study uses appropriate parametric values in phase plane analysis, 2D, 3D, and density plots to ensure the results are physically compatible. The results validate the claim that the technique used in this research to produce complete and uniform responses is not only simple to use and effective, but also substantially faster in computing. The technique is useful for resolving more complex phenomena that arise in engineering and mathematical physics.

Dynamical profile and multi-stability of ion-acoustic waves with soliton solution arising in plasma physics

The present exploration studies the dynamical profile, multistability and soliton solution of ion-acoustic waves (IAWs) in plasma physics. The new auxiliary equation approach is implemented to compute a wide range of solitary wave structures of different types. The dynamical profile is studied by using the bifurcation theory. This system is converted into a planar dynamical system via the Galilean transformation. Taking different cases of its coefficients, the phase plane analysis of this resultant dynamical system is performed. The phase diagrams show the different characteristics of the equilibrium points of the planar dynamical system. The quasi-periodic behaviors, multistability of this planar dynamical system are done by adding the perturbation term. The sensitivity assessment of this perturbed dynamical system is performed by taking different initial constraints.

Particle dynamics around the Einstein–Maxwell–Yang–Mills black hole

The presence of matter in general relativity can cause spacetime to curve. This paper investigates the spacetime properties of the Einstein–Maxwell–Yang–Mills (EMYM) black hole by discussing the geodesic motion of neutral and charged particles. By studying the Lagrangian of these particles, equations of motion and effective potential are obtained in the asymptotic flat and (A)dS spacetimes. Moreover, for the motion of a neutral particle, it is found that there exist stable elliptic/hyperbolic orbits and unstable orbits with different energies by discussing the effective potential and the phase-plane analysis. However, for the motion of a charged particle, there exists an equilibrium point that separates these orbits. In addition, the effects of the system parameters on the potential are analyzed, including the masses of the black hole and charged particles, charges, the cosmological constant, and the strength of the external magnetic field.

Painlevé Test, Phase Plane Analysis and Analytical Solutions of the Chavy–Waddy–Kolokolnikov Model for the Description of Bacterial Colonies

The Chavy–Waddy–Kolokolnikov model for the description of bacterial colonies is considered. In order to establish if the mathematical model is integrable, the Painlevé test is conducted for the nonlinear ordinary differential equation which corresponds to the fourth-order partial differential equation. The restrictions on the mathematical model parameters for ordinary differential equations to pass the Painlevé test are obtained. It is determined that the method of the inverse scattering transform does not solve the Cauchy problem for the original mathematical model, since the corresponding nonlinear ordinary differential equation passes the Painlevé test only when its solution is stationary. In the case of the stationary solution, the first integral of the equation is obtained, which makes it possible to represent the general solution in the quadrature form. The stability of the stationary points of the investigated mathematical model is carried out and their classification is proposed. Periodic and solitary stationary solutions of the Chavy–Waddy–Kolokolnikov model are constructed for various parameter values. To build analytical solutions, the method of the simplest equations is also used. The solutions, obtained in the form of a truncated expansion in powers of the logistic function, are represented as a closed formula using the formula for the Newton binomial.

An extension of the Poincaré–Birkhoff Theorem coupling twist with lower and upper solutions

In 1983, Conley and Zehnder proved a remarkable theorem on the periodic problem associated with a general Hamiltonian system, giving a partial answer to a conjecture by Arnold. Their pioneering paper has been extended in different directions by several authors. In 2017, Fonda and Ureña established a deeper relation between the results by Conley and Zehnder and the Poincaré–Birkhoff Theorem. In 2020, Fonda and Gidoni pursued along this path in order to treat systems whose Hamiltonian function includes a nonresonant quadratic term. It is the aim of this paper to further extend this fertile theory to Hamiltonian systems which, besides the periodicity-twist conditions always required in the Poincaré–Birkhoff Theorem, also include a term involving a pair of well-ordered lower and upper solutions. Phase-plane analysis techniques are used in order to recover a saddle-type dynamics permitting us to apply the above mentioned results.

Solitary waves in a quantum droplet-bearing system

We unravel the existence and stability properties of dark soliton solutions as they extend from the regime of trapped quantum droplets towards the Thomas-Fermi limit in homonuclear symmetric Bose mixtures. Leveraging a phase-plane analysis, we identify the regimes of existence of different types of quantum droplets and subsequently examine the possibility of black and gray solitons and kink-type structures in this system. Moreover, we employ the Landau dynamics approach to extract an analytical estimate of the oscillation frequency of a single dark soliton in the relevant extended Gross--Pitaevskii model. Within this framework, we also find that the single soliton immersed in a droplet is stable, while multisoliton configurations exhibit parametric windows of oscillatory instabilities. Our results pave the way for studying dynamical features of nonlinear multisoliton excitations in a droplet environment in contemporary experimental settings.

Spatial spread of epidemic with Allee effect.

The spatial spread of an epidemic is investigated in the case of a bistable dynamics, where the effective transmission rate depends on the fraction of infected and the state of no epidemic is linearly stable. The front propagation phenomenon is investigated both numerically and theoretically, by an analysis in a four-dimensional phase plane. A good agreement between numerical and theoretical results has been found both for the front profiles and for the speed of invasion. We discovered a novel phenomenon of front stoppage: In some regime of parameters, the front solution ceases to exist, and the propagating pulse of infection decays despite the initial outbreak.

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.

Slow negative feedback enhances robustness of square-wave bursting

Square-wave bursting is an activity pattern common to a variety of neuronal and endocrine cell models that has been linked to central pattern generation for respiration and other physiological functions. Many of the reduced mathematical models that exhibit square-wave bursting yield transitions to an alternative pseudo-plateau bursting pattern with small parameter changes. This susceptibility to activity change could represent a problematic feature in settings where the release events triggered by spike production are necessary for function. In this work, we analyze how model bursting and other activity patterns vary with changes in a timescale associated with the conductance of a fast inward current. Specifically, using numerical simulations and dynamical systems methods, such as fast-slow decomposition and bifurcation and phase-plane analysis, we demonstrate and explain how the presence of a slow negative feedback associated with a gradual reduction of a fast inward current in these models helps to maintain the presence of spikes within the active phases of bursts. Therefore, although such a negative feedback is not necessary for burst production, we find that its presence generates a robustness that may be important for function.

On a modified Korteweg–de Vries equation for electrostatic structures in relativistic degenerate electron–positron plasma

In this paper, the propagation of acoustic solitons in an electron–positron (e–p) degenerate plasma with the relativistically degenerate electrons and positrons is studied. Moreover, we considered stationary positive ions for neutralizing the plasma background. Using the reductive perturbation method, we obtained a modified Korteweg–de Vries equation. By applying a Jacobi elliptic function expansion method, we have proposed the exact analytical solitary, kink, and periodic solutions for the nonlinear wave structures. Also, using the phase plane analysis three equilibrium points have been perceived two of them should be centers and the last is a saddle for quantum acoustic waves. The numerical and computational results show that firstly some of them should be discarded by physical values and then these solutions have been influenced by the plasma parameters such as the electron degenerate density parameter (ne0) and the ion-to-electron density ratio (δ) which peaked out in white dwarf stars. It is found that both the amplitude and the width of the nonlinear structures are affected by the mentioned parameters. This investigation is useful for the relativistically degenerate plasma media in ultra-dense astrophysical environments (such as neutron stars, white dwarfs, etc.), inertia fusion science, experiments involving the interaction of laser and solid density plasma, and inertial confinement fusion in which the relativistic effects play a major role.

Regular solutions to elliptic equations

A review of results and techniques on the existence of regular radial solutions to second order elliptic boundary value problems driven by linear and quasilinear operators is presented. Of particular interest are results where the solvability of a given elliptic problem can be analyzed by the relationship between the spectrum of the operator and the behavior of the nonlinearity near infinity and at zero. Energy arguments and Pohozaev type identities are used extensively in that analysis. An appendix with a proof of the contraction mapping principle best suited for using continuous dependence to ordinary differential equations on initial conditions is presented. Another appendix on the phase plane analysis as needed to take advantage of initial conditions is also included. For studies on singular solutions the reader is referred to Ardila et al., Milan J. Math (2014) and references therein. See also https://ejde.math.txstate.edu/special/02/c2/abstr.html

Utilization of entire new dynamical system approach for peristalsis of Power law fluid with temperature-dependent viscosity

Peristalsis of non-Newtonian fluid with variable viscosity and viscous dissipation is considered in this paper. Modified Brinkman number highlighting both the variable viscosity and viscous dissipation is introduced. By utilizing the long wavelength and small Reynold number approximations, an analytical solution is obtained for stream function. The qualitative analysis is presented using the theory of dynamical system. The phase portraits representing topological structures like trapping, augmented flow and the backward flow are drawn. The change in qualitative flow patterns is discussed from the bifurcation diagrams. The exact size of the bolus is numerically determined. HIGHLIGHTS Analytical solution for Power law fluid with temperature-dependent viscosity. The theory of dynamical system is utilized for stability and bifurcation analysis. The topological structure of the flow is presented using bifurcation theory and phase plane analysis. The size of the bolus is calculated numerically.

Quantitative and qualitative analyses of the mKdV equation and modeling nonlinear waves in plasma

In this paper, nonlinear electrostatic structures on the ion time scale in plasma consisting of two populations of electrons (cold and hot), positrons, and warm adiabatic ions are investigated. The multiple scale method is used to derive the modified Korteweg–de Vries equation (mKdVE). The Jacobi elliptic function expansion method (JEFEM) is employed to find some exact analytical solutions such as periodic, solitonic, and shock solutions. It is shown that the variation in the plasma parameters of interest, for our model, allows the existence of solitary and periodic structures and no shocks. It is also shown that the most important plasma parameters for the plasma model under consideration are positron concentration, α, and the percentage of cold and hot electrons, represented by the parameters μ and ν, respectively. Additionally, the qualitative behavior of the mKdVE is studied using dynamical system theory. The topological structure of the solution is discussed in the phase plane. In this work, the phase plane analysis, which is restricted to the discrete values of the parameter, is extended to the continuous range of the parameter using a bifurcation diagram. Bifurcation diagrams are drawn to forecast the behavior of the solution for specifically chosen essential plasma parameters. The analytical solution and the qualitative behavior of the solution presented in this paper are shown to be compatible with each other. The results presented here are general and can be gainfully employed to study a variety of nonlinear waves in space, laboratory plasmas, and astrophysical plasmas.

Mechanism Analysis and Control of Lateral Instability of 4WID Vehicle Based on Phase Plane Analysis Considering Front Wheel Angle

The current vehicle stability judgment methods in the stability control of distributed drive electric vehicles focuses on using the phase plane method, but the existing phase plane stability boundary delineation methods ignore the influence of the front wheel rotation angle. Therefore, a new phase plane partitioning method (NPPPM) is proposed in this paper, and a new phase plane stability domain library is established. Based on this, the direct yaw moment control of distributed drive electric vehicle through phase plane analysis is established. Firstly, the influence of the front wheel turning angle on the phase plane is considered in the design of the phase plane stability boundary, and a new stability boundary function is fitted by the least squares method. This approach avoids the lateral instability caused by too large a front-wheel turning angle and improves the accuracy of vehicle stability judgment. Next, a hierarchical direct transverse moment controller is constructed using Matlab/Simulink. Among them, the upper layer controller adopts sliding mode control with the sideslip angle error of the center of mass as the tracking target. The lower controller adopts the optimal configuration algorithm with the optimal road adhesion utilization as the objective function. Finally, the effectiveness and superiority of the proposed phase plane partitioning method (NPPPM) for distributed drive electric vehicles with sinusoidal and angular order inputs for the front wheel rotation angle are verified by simulation.

Bifurcation of backward and forward solitary waves in helicoidal Peyrard–Bishop–Dauxois model of DNA

In this work, we investigate the bifurcation of backward and forward solitary waves in helicoidal Peyrard–Bishop–Dauxois model of deoxyribonucleic acid (DNA). We start by showing that this model can adopt right-handed or left-handed behavior depending on the wave frequency that it propagates. Then, using the semi-discrete approximation, we show that the dynamics of modulated waves in the network is governed by a quintic nonlinear Schrödinger (QNLS). Applying the phase imprint technique, this QNLS equation is first reduced into a derivative QNLS, and then to a cubic–quintic Duffing oscillator equation. Based on the phase plane analysis, we present all phase portraits of the dynamical system. The obtained results show a number of new phase portraits which cannot exist without the phase imprint approach. The exact representations of the nonlinear localized waves corresponding to the homoclinic and heteroclinic orbits in the phase portrait of the dynamical system are given. These waves include bright soliton, kink and anti-kink solitons, dark soliton and grey solitons. In addition, the effects of phase imprint parameter on the wave-shape profile of these solutions are investigated. We find out that the phase imprint parameter considerably affects the amplitude and the width of each of the above enumerated solitary waves, while the phase imprint method leads to new type solutions.

Global-in-time existence of unbounded separatrices and blow-up time of solutions for Liénard systems whose restoring force is an odd power function

The existence, uniqueness, and number of limit cycles in Liénard dynamics have been intensely studied since the 1940s because they relate to Hilbert's 16th problem. Limit cycles sharply distinguish the nature of adjacent orbits. Moreover, they are essential for understanding dynamical systems. In contrast, the importance of unbounded separatrices is less well recognized even though they play an identical role. Therefore, this study focuses on the Liénard dynamical system and investigates whether all orbits, including unbounded separatrices, exist globally in time. The blow-up times of solutions that cannot be extended up to negative or positive infinity are also discussed. The result allows classification of all solutions by the time for which they exist. For example, for the van der Pol oscillator known to have the sole limit cycle, all solutions that pass through any point in the region bounded by that limit cycle and the two solutions that correspond to the unbounded separatrices continue to exist for all time, whereas the other solutions have positive or negative blow-up times. This fact cannot be judged solely from the shapes of the orbits obtained by simulation. Phase plane analysis is applied to prove the main theorem. Furthermore, several examples and their plane figures are provided to facilitate the understanding of the proof.

Higher-order shock wave structures and phase plane analysis in multicomponent dusty plasma

In this investigation, dust acoustic (DA) shocks and dressed shocks formed due to the contribution of higher-order effects in a non-Maxwellian plasma have been examined. The reductive perturbation method is employed to derive the nonlinear Burgers and Burgers-type inhomogeneous equations with the contribution of higher-order effects of nonlinearity and dissipation. The properties of dust acoustic shocks and higher order (dressed shocks) are examined from the solutions of these nonlinear equations under the effect of various plasma parameters. Furthermore, the travelling wave analysis is used to obtain the dynamical system for the Burgers equation using bifurcation theory. The characteristic properties of DA periodic waves have also been analysed under the influence of various plasma parameters.

Wave propagation and stabilization in the Boussinesq–Burgers system

This paper considers the existence and stability of traveling wave solutions of the Boussinesq–Burgers system describing the propagation of bores. Assuming the fluid is weakly dispersive, we establish the existence of three different wave profiles by the geometric singular perturbation theory alongside phase plane analysis. We further employ the method of weighted energy estimates to prove the nonlinear asymptotic stability of the traveling wave solutions against small perturbations. The technique of taking antiderivative is utilized to integrate perturbation functions because of the conservative structure of the Boussinesq–Burgers system. Using a change of variable to deal with the dispersion term, we perform numerical simulations for the Boussinesq–Burgers system to showcase the generation and propagation of various wave profiles in both weak and strong dispersions. The numerical simulations not only confirm our analytical results, but also illustrate that the Boussinesq–Burgers system can generate numerous propagating wave profiles depending on the profiles of initial data and the intensity of fluid dispersion, where in particular the propagation of bores can be generated from the system in the case of strong dispersion.

Ultradiscrete hard-spring equation and its phase plane analysis

Ultradiscretization enables us to construct a piecewise linear equation which approximates a given subtraction-free difference equation. Recently proposed “ultradiscretization with parity variables” (pUD) can treat an equation with subtraction. However, its solution may have an infinite number of branches under some specific conditions. In this paper, solutions of pUD for the hard-spring equation is investigated. The pUD equation is reinterpreted as the mapping which maps a set on the phase plane to another one, and the behaviour of the solutions are analyzed through approximative transition diagrams.As a result, the infinite branches are translated into a few finite branches and allowed easier understanding.