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A microcrystalline material refers to a crystallized substance or rock comprised of tiny crystals that can only be observed under a microscope. The strain wave equation is a fourth-order nonlinear partial differential equation encountered in the examination of non-dissipative strain wave propagation within microstructured solids. In this paper, the transmission of waves in microcrystalline materials is dictated by the non-dissipative case of strain wave equation’s structure, accounting for multiple dimensions within microcrystalline structures. The simplest equation method is employed to extract multi-soliton solutions, while the modified Sardar subequation method is applied to identify additional soliton solutions, including bright, combined dark–bright, combined dark-singular, periodic singular, and singular solitons. Furthermore, the dynamical system bifurcation theory approach is utilized to investigate the phase diagrams of the governing equation. Further elaboration on the physical dynamical representation of the presented solutions is provided through profile illustrations. A comparison with the existing literature is also provided, highlighting the efficacy of our work. The significance of the acquired outcomes lies in their capacity to portray a wide array of intricate and diverse phenomena observed in both mathematical and physical systems. [ABSTRACT FROM AUTHOR]

In this paper, we present a method for constructing vector fields whose phase portraits have finite sets of prescribed special trajectories (limit cycles, simple and complex singular points, separatrices) and prescribed topological structures in limited domains of the phase plane. The problem of constructing such vector fields is a generalization of a number of well-known inverse problems of the qualitative theory of ordinary differential equations. The proposed method for solving it expands the possibilities of mathematical modeling of dynamic systems with prescribed properties in various fields of science and technology. [ABSTRACT FROM AUTHOR]

One of the most important tasks in the qualitative theory of differential equations in the plane is the study of global asymptotic stability: an equilibrium point that is globally attractive. It is known that if an equilibrium point of a planar vector field is globally asymptotically stable, then the vector field is bounded. A planar vector field is said to be bounded if the forward orbit of every point enters and remains in a compact set. In this paper, we study cubic Kukles systems that are bounded which is the first step toward the characterization of cubic Kukles systems that are globally asymptotically stable. We emphasize that these systems form a seven-parameter family of polynomial differential systems. We obtain a total of 25 cubic Kukles sub-systems that are bounded and we provide 11 phase portraits of such sub-systems in a neighborhood of infinity. [ABSTRACT FROM AUTHOR]

We deal with a Leslie-Gower predator–prey model with a generalist or alternating food for predator and linear functional response. Using a topological equivalent polynomial system we prove that the system is not Liouvillian (hence also not Darboux) integrable. In order to study the global dynamics of this model, we use the Poincaré compactification of R 2 to characterize all phase portraits in the Poincaré disc, obtaining two different topological phase portraits. [ABSTRACT FROM AUTHOR]

In this paper, the Shynaray-IIA equation, which characterizes phenomena like tidal waves and tsunamis, is considered. In the first stage, the relevant equation is transformed into a planer dynamic system by using the Galilean transformation. Thanks to the well-known bifurcation theory, phase portraits for nonlinear traveling wave solutions have been investigated. As far as is known, before this study, there is no research in which this review is conducted. In addition, an external force is added to the resulting dynamic system and its effect is examined. The model’s dynamic behaviour is investigated through bifurcation, periodic quasi-periodic and chaotic behaviour, and sensitivity. These include methods like phase portrait rendering, time series scrutiny, Lyapunov exponents calculation, and the assessment of multi-stability. To examine the solitonic wave solutions to the underlying equation, the improved tan (ϕ / 2) -expansion method has been utilized. The solutions retrieved by this method are expressed in the form of hyperbolic, trigonometric, rational and exponential functions. In addition, some of the various kinds of solitons such as dark and bright wave obtained have been visualized with 3D graphics for different values of the parameters in order to better understand their dynamic behavior. Furthermore, the modulation instability of the governing equation is investigated through the application of linear stability analysis. [ABSTRACT FROM AUTHOR]

The article analyzes existing schemes of chaos generators. Numerical and simulation modeling is carried out aimed at identifying chaotic dynamics. Based on wellknown concepts, a chaos generator is developed and a simulation model is built. A mathematical description of the generator is given and phase portraits are obtained. Diagrams of chaotic oscillations of the Lorentz model and the Colpitts model are presented. The operation of the Van der Pol generator is considered and the chaotic processes that arise during external harmonic oscillations are shown. Chua's model is presented, its mathematical description is given, and an analysis of oscillations in the deterministic chaos regime is presented.

Abstract The main purpose of this article is to analyze the bifurcation, chaotic behaviors, and solitary wave solutions of the fractional Twin-Core couplers with Kerr law non-linearity by using the planar dynamical system method. This equation has profound physical significance and application value in the areas of optics and optical communication. Firstly, the traveling wave transformation is applied to convert the beta-derivative Twin-Core couplers with Kerr law non-linearity into the ordinary differential equations. Secondly, phase portraits and Poincaré sections of two-dimensional dynamical system and its perturbation system are plotted by using mathematical software. For different initial values, the planar phase diagram and three-dimensional phase diagram in red and blue are plotted, respectively. Finally, the solitary wave solutions of the fractional Twin-Core couplers with Kerr law non-linearity are obtained by using theory of planar dynamical system. In addition, three-dimensional graphs, two-dimensional graphs, and the contour graphs of the solitary wave solutions are drawn.

In this article, the phase portraits, chaotic patterns, and traveling wave solutions of the conformable extended Korteweg–de-Vries (KdV) model are investigated. First, the conformal fractional order extended KdV model is transformed into ordinary differential equation through traveling wave transformation. Second, two-dimensional (2D) planar dynamical system is presented and its chaotic behavior is studied by using the planar dynamical system method. Moreover, some three-dimensional (3D), 2D phase portraits and the Lyapunov exponent diagram are drawn. Finally, many meaningful solutions are constructed by using the complete discriminant system method, which include rational, trigonometric, hyperbolic, and Jacobi elliptic function solutions. In order to facilitate readers to see the impact of fractional order changes more intuitively, Maple software is used to draw 2D graphics, 3D graphics, density plots, contour plots, and comparison charts of some obtained solutions.

The global dynamics of an asymmetrical rigid body with an axisymmetric rotor are discussed. Compared with a free rigid body, a gyrostat has an extra degree of freedom, and for its integrability, an additional first integral is required. Two cases are considered: either the relative angular velocity of the rotor is constant, or the projection of angular momentum of the rotor onto its axis is constant. In both cases, phase portraits are formed by curves lying at the intersection of an ellipsoid with a family of concentric spheres offset relative to the center of the ellipsoid. The structurally stable phase portraits with differing numbers and types of singular points are classified. Rearrangements of these portraits for the first type of gyrostat occur when the rotor rotation speed changes. For the second type, a discontinuous bifurcation is possible when the rotor is released; in this case, the inertia tensor changes abruptly. In previous studies, the stability of stationary gyrostat motions and their local bifurcations were studied in detail. Global rearrangements were apparently not considered. The obtained results can be used to control the orientations of satellites and mobile robots. [ABSTRACT FROM AUTHOR]

Our research on fractional-order nonlinear Schrödinger equations (FONSEs) reveals several new findings, which may contribute to our comprehension of wave dynamics and hold practical importance for the field of ocean engineering. We have employed an innovative approach to derive double periodic Weierstrass elliptic function solutions for FONSEs, thereby offering exact solutions for these equations. Additionally, we have observed that fractional derivatives significantly impact the dynamics of solitary waves, potentially holding significance for the design of ocean structures. Our research reveals the previously unknown phenomenon of oblique wave variations, which can impact the reliability and lifespan of offshore structures. Our findings highlight the significance of taking into account various fractional derivatives in future studies. Using the planar dynamical system technique, we gain a deeper understanding of the behavior of FONSEs, revealing critical thresholds and regions of chaotic behavior. Linear stability analysis provides a strong framework for studying the modulation instability of dynamical systems, shedding light on the conditions and mechanisms of modulated behavior. Applying this analysis to the FONSEs offers insights into the critical parameters, growth rates, and formation of modulated patterns, with potential implications for innovative research in ocean engineering. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to analyze the bifurcation, chaotic behaviors, and solitary wave solutions of the fractional Twin-Core couplers with Kerr law non-linearity by using the planar dynamical system method. This equation has profound physical significance and application value in the areas of optics and optical communication. Firstly, the traveling wave transformation is applied to convert the beta-derivative Twin-Core couplers with Kerr law non-linearity into the ordinary differential equations. Secondly, phase portraits and Poincaré sections of two-dimensional dynamical system and its perturbation system are plotted by using mathematical software. For different initial values, the planar phase diagram and three-dimensional phase diagram in red and blue are plotted, respectively. Finally, the solitary wave solutions of the fractional Twin-Core couplers with Kerr law non-linearity are obtained by using theory of planar dynamical system. In addition, three-dimensional graphs, two-dimensional graphs, and the contour graphs of the solitary wave solutions are drawn. [ABSTRACT FROM AUTHOR]

This paper presents a global study of the class Q E S ̂ of all real quadratic polynomial differential systems possessing exactly one elemental infinite singular point and one triple infinite singular point, which is either an infinite nilpotent elliptic-saddle or a nilpotent saddle. This class can be divided into three different families, namely, Q E S ̂ (A) of phase portraits possessing three real finite singular points, Q E S ̂ (B) of phase portraits possessing one real and two complex finite singular points, and Q E S ̂ (C) of phase portraits possessing one real triple finite singular point. Here, we provide a comprehensive study of the geometry of these three families. Modulo the action of the affine group and time homotheties, families Q E S ̂ (A) and Q E S ̂ (B) are three-dimensional and family Q E S ̂ (C) is two-dimensional. We study the respective bifurcation diagrams of their closures with respect to specific normal forms, in sub-sets of real Euclidean spaces. The bifurcation diagram of family Q E S ̂ (A) (resp., Q E S ̂ (B) and Q E S ̂ (C)) yields 1274 (resp., 89 and 14) sub-sets with 91 (resp., 27 and 12) topologically distinct phase portraits for systems in the closure Q E S ̂ (A) ¯ (resp., Q E S ̂ (B) ¯ and Q E S ̂ (C) ¯) within the representatives of Q E S ̂ (A) (resp., Q E S ̂ (B) and Q E S ̂ (C)) given by a specific normal form. [ABSTRACT FROM AUTHOR]

The light pressure of an evanescent electromagnetic wave formed by total internal reflection near the flat interface of a dielectric and a liquid on a dielectric spherical nanoparticle located in a liquid medium is considered. Phase portraits of a two-dimensional system of equations that is equivalent to the equation of nanoparticle transportation under the influence of the force gradient of the light pressure of the evanescent field taking into account the medium resistance force are plotted. Various phase portraits can be realized both without equilibrium points and with one equilibrium point (stable focus, stable node or saddle) on the phase plane, depending on the parameters of the laser radiation and the material of the nanoparticle suspended in the water. [ABSTRACT FROM AUTHOR]

In this paper we study the topological phase portraits of polynomial differential systems with a uniform isochronous center, whose nonlinear parts are homogeneous polynomials of degree 6. We obtain all the distinct topological phase portraits in the Poincaré disc. For each phase portrait, we give a precise system to realize it. [ABSTRACT FROM AUTHOR]

In this article, the phase portraits, chaotic patterns, and traveling wave solutions of the stochastic Sasa–Satsuma equation are investigated. Firstly, the stochastic Sasa–Satsuma equation is transformed into an ordinary differential equation through traveling wave transformation. Secondly, two-dimensional planar dynamical system is presented by using the theory of planar dynamical systems. Then, the three-dimensional and two-dimensional phase portraits of the dynamical system are drawn by using Maple software. Finally, the complete discriminant system method is used to solve the stochastic Sasa-Satsuma equation, resulting in many solutions that other methods cannot obtain, including rational, trigonometric, hyperbolic, and Jacobi elliptic function solutions. Moreover, three-dimensional-surface plots and two-dimensional-shape plots for the module length of some solutions under different parameters are drawn by using Maple software. The innovation of this article lies in introducing stochastic parameters into the Sasa–Satsuma equation, obtaining more diverse and comprehensive conclusions. [ABSTRACT FROM AUTHOR]

Due to the difference in output characteristics between the line-commutated converter-based high-voltage direct current (LCC-HVDC) and voltage-source converter-based high-voltage direct current (VSC-HVDC), the hybrid multi-infeed high-voltage direct current (HMIDC) presents complex coupling characteristics. As the AC side is disturbed, the commutation failure (CF) occurring on the LCC side is the main factor threatening the safe operation of the system. In this paper, the simplified equivalent network model of HMIDC is established by analyzing the output characteristics of VSC and LCC. Hereafter, based on the derived model and the control system of LCC-HVDC, the dynamic equations of the extinction angle are deduced. Consequently, by applying the phase portrait method, the causes of CF occurring in the HMIDC system as well as the impacts of control parameters on the transient stability are revealed. Furthermore, the stabilization boundaries for the reference value of the DC voltage are obtained via the above analysis. Finally, the theoretical analysis is verified by the simulations in the PSCAD/EMTDC. [ABSTRACT FROM AUTHOR]

[Display omitted] • Proposes a surrogate sample entropy-based method to assess air quality index. • Air quality of most polluted state capital in the world, Delhi, is investigated. • Temporal data of the pollutants are subjected to time series analysis. • Unveils the dependence of PM2.5 concentration on humidity and rain. • Phase portrait's complexity and sample entropy increase with pollutants level. Pollution and its impacts on human health have become a crisis in regions with poor air quality index (AQI), which is an indicator of concentrations of pollutants, prompting the United Nations (UN) to set sustainable development goals (SDG). The present study proposes a surrogate sample entropy-based method in tune with UN's SDG, to assess AQI from the time series of any of the pollutants. New Delhi is one of the world's most polluted state capital, with a higher level of particulate matter (PM). The temporal data of the pollutants in New Delhi, recorded in the one-hour interval during the years 2016 and 2017, are subjected to time series analysis. The data collected from the Central Pollution Control Board of India are analyzed with special reference to PM and compared with the World Air Quality Report 2021 and University of Washington data. The dependence of PM2.5 concentration on humidity and rain is also studied. The study reveals the increase in complexity with the concentration of pollutants through the phase portrait. The sample entropy analysis of the nonlinear time series of the pollutants exhibits a linear relation with AQI suggesting the possibility of using sample entropy as a surrogate measure of AQI. [ABSTRACT FROM AUTHOR]

In this paper, we study one class of Volterra quadratic stochastic operators with continuous time. Fixed points are investigated, numerical and analytical solutions are found, and, as a special case, the solution of the main problem is given. Numerical and analytical solutions of the problem at various initial values are analyzed using the MathCAD mathematical system. [ABSTRACT FROM AUTHOR]

This paper explores novel ways of predicting and classifying the dynamics of piecewise smooth maps using various deep learning models. Moreover, we have used machine learning models such as Decision Tree Classifier, Logistic Regression, K-Nearest Neighbor, Random Forest, and Support Vector Machine for predicting the border collision bifurcation in the 1D normal form map and the 1D tent map. The decision tree classifier best predicts the border collision bifurcation for the 1D normal form map, the random forest, and the 1D tent map. This study introduces a novel application of deep learning models to cobweb diagrams and phase portraits, which provides a new perspective for classifying regular and chaotic behaviour. Further, we classified the regular and chaotic behaviour of the 1D tent map and the 2D Lozi map using deep learning models like Convolutional Neural Network (CNN), ResNet50, and ConvLSTM via cobweb diagram and phase portraits, where CNN exhibits better performance than other models. We also classified the chaotic and hyperchaotic behaviour of the 3D piecewise smooth map using deep learning models such as the Feed Forward Neural Network (FNN), Long Short-Term Memory (LSTM), and Recurrent Neural Network (RNN). We have shown that LSTM performs best for classifying chaotic and hyperchaotic behaviour. Additionally, LSTM outperforms other models in accuracy and computational efficiency, making it highly effective for real-time analysis. Finally, deep learning models such as Long Short Term Memory (LSTM) and Recurrent Neural Network (RNN) are used for reconstructing the two-parameter bifurcation charts of 2D normal form map, in which LSTM is more precise than RNN in reconstructing the two-parametric charts.

Abstract The main purpose of this article is to investigate the qualitative behavior and traveling wave solutions of the fractional stochastic Kraenkel–Manna–Merle equations, which is commonly used to simulate the zero conductivity nonlinear propagation behavior of short waves in saturated ferromagnetic materials. Firstly, fractional stochastic Kraenkel–Manna–Merle equations are transformed into ordinary differential equations by using the traveling wave transformation. Secondly, the phase portraits, sensitivity analysis, and Poincaré sections of the two-dimensional dynamic system and its perturbation system of ordinary differential equations are drawn. Finally, the traveling wave solutions of fractional stochastic Kraenkel–Manna–Merle equations are obtained based on the analysis theory of planar dynamical system. Moreover, the obtained three-dimensional graphs of random solutions, two-dimensional graphs of random solutions, and three-dimensional graphs of deterministic solutions are drawn.

The problem of constructing n-linear (n ≥ 2) plane vector fields with an isolated critical point and given separatrices of prescribed types is considered. Such constructions are based on the use of vector algebra, the qualitative theory of second-order dynamic systems and classical methods for investigating their critical points. This problem is essentially an inverse problem of the qualitative theory of ordinary differential equations, and its solution can be used to synthesize mathematical models of controlled dynamical systems of various physical nature. [ABSTRACT FROM AUTHOR]

This study investigates the shift in dynamical states of the thermo-acoustic oscillations for the turbulent syngas combustor having a bluff body for the flame anchoring. In this paper, an analysis was conducted to look into the effects of three different syngas compositions on the variation of Reynolds number (Re) in the range of 2289 to 8009. The analysis involved simultaneous, unsteady pressure measurement and OH* chemiluminescence. The investigation reveals that as the controlled parameters vary, the system exhibits a sequence of dynamic states characterized by distinct nonlinear oscillations. This study aims to explore the infrequently observed transitions from low-frequency instability (LFI) to high-frequency instability (HFI) by examining various time-series data and post-processing techniques. Additionally, it aims to understand how these transitions ultimately lead to the emergence of combustion noise as a result of a change in Reynolds number. To ascertain the characteristics of thermo-acoustic oscillations under investigation, a comprehensive analysis is conducted utilizing nonlinear time-series analysis techniques like phase portrait and recurrence plots. The investigation of flame behavior in response to changes in Reynolds number has been conducted using time-resolved OH* chemiluminescence. The results obtained from this study reveal distinct flame behavior patterns. The combustion instability of syngas at HFI is driven by flame modulated by small-scale structures and its anchoring in the shear layer of the bluff-body whereas the LFI is due to larger flame modulations near the wall of the combustion chamber. In addition, the recurrence analysis method is employed to monitor the progression of the dynamical states to understand the nature of the dynamical states. Such analysis will ultimately contribute to the establishment of a stable or nearly stable combustion system. [ABSTRACT FROM AUTHOR]

In this paper, we characterize and study dynamical properties of cubic vector fields on the sphere S 2 = { (x , y , z) ∈ R 3 | x 2 + y 2 + z 2 = 1 } . We start by classifying all degree three polynomial vector fields on S 2 and determine which of them form Kolmogorov systems. Then, we show that there exist completely integrable cubic vector fields on S 2 and also study the maximum number of various types of invariant great circles for homogeneous cubic vector fields on S 2 . We find a tight bound in each case. Further, we also discuss phase portraits of certain cubic Kolmogorov vector fields on S 2 . [ABSTRACT FROM AUTHOR]

The main purpose of this article is to investigate the dynamic behavior and optical soliton for the M-truncated fractional paraxial wave equation arising in a liquid crystal model, which is usually used to design camera lenses for high-quality photography. The traveling wave transformation is applied to the M-truncated fractional paraxial wave equation. Moreover, a two-dimensional dynamical system and its disturbance system are obtained. The phase portraits of the two-dimensional dynamic system and Poincaré sections and a bifurcation portrait of its perturbation system are drawn. The obtained three-dimensional graphs of soliton solutions, two-dimensional graphs of soliton solutions, and contour graphs of the M-truncated fractional paraxial wave equation arising in a liquid crystal model are drawn. [ABSTRACT FROM AUTHOR]

The main object of this paper is to study the traveling wave solutions of the fractional coupled Konopelchenko–Dubrovsky model by using the complete discriminant system method of polynomials. Firstly, the fractional coupled Konopelchenko–Dubrovsky model is simplified into nonlinear ordinary differential equations by using the traveling wave transformation. Secondly, the trigonometric function solutions, rational function solutions, solitary wave solutions and the elliptic function solutions of the fractional coupled Konopelchenko–Dubrovsky model are derived by means of the polynomial complete discriminant system method. Moreover, a two-dimensional phase portrait is drawn. Finally, a 3D-diagram and a 2D-diagram of the fractional coupled Konopelchenko–Dubrovsky model are plotted in Maple 2022 software. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a wide family of dynamical systems whose characteristic feature is a polynomial right-hand side containing coprime forms of the phase variables of the system. One of the equations of the system contains a third-degree polynomial (cubic form), the other equation contains a quadratic form. We consider the problem of constructing all possible phase portraits in the Poincaré disk for systems from the family considered and establish criteria for the implementation of each portrait that are close to coefficient criteria. This problem is solved by using the central and orthogonal Poincaré methods of sequential mappings and a number of other methods developed by the authors for the purposes of this study. We obtained rigorous qualitative and quantitative results. More than 250 topologically distinct phase portraits of various systems were constructed. The absence of limit cycles of systems of this family is proved. Methods developed can be useful for the further study of systems with polynomial right-hand sides of other forms. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the exact solutions of a modified generalized multidimensional fractional Kadomtsev–Petviashvili (KP) equation by the bifurcation method. First, the equation is converted into a planar dynamical system through fractional complex wave transformation. The phase portraits of the equation and qualitative analysis are presented under different bifurcation conditions. Then, the bounded and unbounded traveling wave solutions, including periodic, kink, anti-kink, dark-solitary, bright-solitary and breaking wave solutions, are acquired by integrating along different orbits. Finally, numerical simulations of the dynamic behaviors of the solutions obtained are graphically illustrated by choosing appropriate parameters. [ABSTRACT FROM AUTHOR]

A reliable algorithm based on an adaptation of the standard differential transform method (DTM) is presented, which is the multi-step differential transform method (Ms-DTM) since it may be difficult to directly apply differential transform method (DTM) to obtain the series solutions for the present two- dimensional coupled asymmetric van der Pol oscillator. The solutions of a two- dimensional coupled asymmetric van der Pol oscillator were obtained by MsDTM. Figurative comparisons between the Ms-DTM and the classical fourth order Runge-Kutta method (RK4) are given. The obtained results reveal that the proposed technique is a promising tool to solve the considered van der Pol oscillator and yield same information on the phase portrait confirming the stability of the system, effectively. It can be said that the considered approach can be easily extended to other nonlinear van der Pol oscillator systems and therefore is widely applicable in engineering and other sciences. [ABSTRACT FROM AUTHOR]

This paper deals with the global dynamics of planar piecewise smooth differential systems constituted by two different vector fields separated by one straight line that passes through the origin. From a quasi-canonical family of piecewise quadratic differential systems with a pseudo-focus point at the origin, which has six parameters, we investigate the subfamilies where the origin is indeed a pseudo-center. For such subfamilies, we classify its global phase portraits in the Poincaré disk and the associated bifurcation sets. [ABSTRACT FROM AUTHOR]

A systematic analytical study of the mathematical properties of the previously constructed nonlinear model of the shear flow of thixotropic viscoelastic-plastic media, which takes into account the mutual influence of the deformation process and structure evolution, is carried out. A set of two nonlinear differential equations describing shear at a constant rate and stress relaxation was obtained. Assuming six material parameters and an (increasing) material function that control the model are arbitrary, the basic properties of the families of stress-strain curves at constant strain rates, stress relaxation curves (Part 2) and creep curves (Part 3) generated by the model, and the features of the evolution of the structuredness under these types of loading were analytically studied. The dependences of these curves on time, shear rate, stress level, initial strain and initial structuredness of material (for example, degree of physical crosslinking), as well as on material parameters and function governing the model, were studied. Several indicators of the model applicability are found, which are convenient to check with experimental data. It was examined what effects typical for viscoelastic-plastic media can be described by the model and what unusual effects (properties) are generated by structuredness changes in comparison to typical stress-strain, relaxation and creep curves of structurally stable materials. The analysis proved the ability of the model to describe behavior of not only liquid-like viscoelastoplastic media, but also solid-like (thickening, hardening, hardened) media: the effects of creep, relaxation, recovery, a number of typical properties of experimental relaxation curves, creep and stress-strain curves at a constant rate, strain rate and strain hardening, flow under constant stress, etc. The first part of the article is devoted to formulation of the model and preparation of basis for the second part: the proof of the uniqueness and stability of the equilibrium point of the nonlinear equations set, analytical study of the equilibrium point dependence on all material parameters, possible types of phase portraits and the properties of integral and phase curves of the model. [ABSTRACT FROM AUTHOR]

This study computationally investigates small-scale flickering buoyant diffusion flames in externally swirling flows and focuses on identifying and characterizing various distinct dynamical behaviors of the flames. To explore the impact of finite rate chemistry on flame flicker, especially in sufficiently strong swirling flows, a one-step reaction mechanism is utilized for investigation. By adjusting the external swirling flow conditions (the intensity R and the inlet angle α ), six flame modes in distinct dynamical behaviors were computationally identified in both physical and phase spaces. These modes, including the flickering flame, oscillating flame, steady flame, lifted flame, spiral flame, and flame with a vortex bubble, were analyzed from the perspective of vortex dynamics. The numerical investigation provides relatively comprehensive information on these flames. Under the weakly swirling condition, the flames retain flickering (the periodic pinch-off of the flame) and are axisymmetric, while the frequency nonlinearly increases with the swirling intensity. A relatively high swirling intensity can cause the disappearance of the flame pinch-off, as the toroidal vortex sheds around either the tip or the downstream of the flame. The flicker vanishes, but the flame retains axisymmetric in a small amplitude oscillation or a steady stay. A sufficiently high swirling intensity causes a small Damköhler number, leading to the lift-off of the flame (the local extinction occurs at the flame base). Under the same swirling intensity but large swirling angles, the asymmetric modes of the spiral and vortex bubble flames were likely to occur. With R and α increasing, these flames exhibit axisymmetric and asymmetric patterns, and their dynamical behaviors become more complex. To feature the vortical flows in flames, the phase portraits are established based on the velocity information of six positions along the axis of the flame, and the dynamical behaviors of various flames are presented and compared in the phase space. Observing the phase portraits and their differences in distinct modes could help identify the dynamical behaviors of flames and understand complex phenomena. [ABSTRACT FROM AUTHOR]

Bifurcation analysis is a powerful method for investigating the steady-state nonlinear dynamics of systems. There are software programmes that allow for the numerical continuation of steady-state solutions while the system's parameters are changed. The aim of this manuscript is to deal a mixed derivative nonlinear Schrödinger's equation (MD-NLSE), α -helix NLSE and Zoomeron model utilising the bifurcation theory technique of dynamical systems. A mathematical method called bifurcation analysis is used to examine how a system behaves when a parameter is changed. It is used to examine how the behaviour of the system changes as one or more parameters are altered and to find critical values of the parameters that cause appreciable changes in the behaviour of the system. Bifurcations are places in parameter space when the system's qualitative behaviour abruptly shifts. A system could, for instance, go from having a stable equilibrium to oscillating between two states when a parameter is increased. To investigate these changes and comprehend how the system's behaviour would alter if the parameter is further altered, bifurcation analysis is performed. An example of a phase portrait is a plot of a dynamic system's phase space, which graphically depicts the behaviour of the system. A two-dimensional depiction of the system's variables on the x and y axes is a common way to depict the phase space, which is the space of all conceivable states of the system. The paths of the system are shown as curves or lines in the phase space in a phase picture. These paths show how the system has changed over time and may be used to examine the stability and behaviour of the system. Phase portraits are often employed in the study of the behaviour of complex systems in physics, engineering, and mathematics. Numerous systems, including mechanical systems, electrical circuits, chemical processes, and biological systems, may be studied using them. Researchers can learn more about a system's dynamics and forecast how it will behave in various situations by looking at its phase picture. [ABSTRACT FROM AUTHOR]

The following differential quadratic polynomial differential system   d x d t = y − x ,   d y d t = 2 y − y γ − 1 2 − γ y − 5 γ − 4 γ − 1 x , when the parameter γ ∈ (1 , 2 ] models the structure equations of an isotropic star having a linear barotropic equation of state, being x = m (r) / r where m (r) ≥ 0 is the mass inside the sphere of radius r of the star, y = 4 π r 2 ρ where ρ is the density of the star, and t = ln (r / R) where R is the radius of the star. First, we classify all the topologically non-equivalent phase portraits in the Poincaré disc of these quadratic polynomial differential systems for all values of the parameter γ ∈ R ∖ { 1 } . Second, using the information of the different phase portraits obtained we classify the possible limit values of m (r) / r and 4 π r 2 ρ of an isotropic star when r decreases. [ABSTRACT FROM AUTHOR]

Ensuring noise immunity is one of the main tasks of radio engineering and telecommunication. The main task of signal receiving comes down to the best recovery of useful information from a signal that is destructed during propagation and received together with interference. Currently, the interference and noise control comes to the fore. Modern elements and methods of processing, related to intelligent systems, strengthen the role of the verification and recognition of targets. This makes noise control particularly relevant. The most-important quantitative indicator that characterizes the quality of the useful signal is the signal-to-noise ratio. Therefore, determining the noise parameters is very important. In the present paper, a signal model is used in the form of an additive mixture of useful signals and Gaussian noise. It is an ordinary model of a received signal in radio engineering and communications. It is shown that the phase portrait of this signal has the shape of an ellipse at the low noise level. For the first time, an expression of the width of the ellipse line is obtained, which is determined by the noise dispersion. Currently, in electroencephalography, diagnosis is based on the Fourier transform. But, many brain diseases are not detected by this method. Therefore, the search and use of other methods of signal processing is relevant. [ABSTRACT FROM AUTHOR]

In nature, the vast majority of species live in ecosystems that are not isolated, and the same is true for predator-prey ecological systems. With this work, we extend a predator-prey model by considering the inclusion of an immigration term in both species. From a biological point of view, that allows us to achieve a more realistic model. We consider a system with a Holling type Ⅰ functional response and study its global dynamics, which allows to not only determine the behavior in a region of the plane R 2 , but also to control the orbits that either go or come to infinity. First, we study the local dynamics of the system, by analyzing the singular points and their stability, as well as the possible behavior of the limit cycles when they exist. By using the Poincaré compactification, we determine the global dynamics by studying the global phase portraits in the positive quadrant of the Poincaré disk, which is the region where the system is of interest from a biological point of view. [ABSTRACT FROM AUTHOR]

This paper is part of a series of works whose ultimate goal is the complete classification of phase portraits of quadratic differential systems in the plane modulo limit cycles. It is estimated that the total number may be around 2000, so the work to find them all must be split in different papers in a systematic way so to assure the completeness of the study and also the non intersection among them. In this paper we classify the family of phase portraits possessing one finite saddle-node and a separatrix connection and determine that there are a minimum of 77 topologically different phase portraits plus at most 16 other phase portraits which we conjecture to be impossible. Along this paper we also deploy a mistake in the book (Artés et al. in Structurally unstable quadratic vector fields of codimension one, Birkhäuser/Springer, Cham, 2018) linked to a mistake in Reyn and Huang (Separatrix configuration of quadratic systems with finite multiplicity three and a M 1 , 1 0 type of critical point at infinity. Report Technische Universiteit Delft, pp 95–115, 1995). [ABSTRACT FROM AUTHOR]

Background. Applying parametric systems of ordinary differential equations to the dynamic modelling of competitive interaction is of current interest. There are lots of publications dedicated to two-dimensional systems in this connection. Investigation of higher-dimensional system often demands the methods of numerical analysis; actual scientific literature represents some difficulties in applying classic methods of qualitative theory to higher-dimensional systems. The purpose of this article is to study a 3-dimentional system being applied for modelling of three competing groups interaction, just by the qualitative theory method. Materials and methods. A review of publications about differential equations in competition dynamics modelling is done. We consider a 6-parametric 3- dimentional system on the invariant triangle of frequencies. Definitions of approaching and retiring areas of singular points of the system are formulated. Through consideration of level surfaces of specially constructed functions, we demonstrate how to find out whether an arbitrary point of the frequency triangle belongs to approaching or retiring areas relatively to all singular points located on the triangle sides (but not at its apexes). Results. The equations of the bounds of these areas have been derived. Some theorems describing mutual disposition of those curves (approaching-retiring areas bounds) and singular points of system are proved. A numeric example is given to illustrate the theoretical results. This example is based on linguistic problem data. Conclusions. The developed and theoretically grounded method allows to precise phase portrait of the system under consideration without solving analytically nor numerically.

The main purpose of this article is to study the qualitative analysis and explicit solutions of perturbed Chen–Lee–Liu equation with refractive index. By utilizing wave transformation, two-dimensional dynamics in a plane are obtained. Using the qualitative theory of planar dynamical systems, a series of planar phase portraits are presented. By analyzing the bifurcations of these phase portraits and the characteristics of equilibrium points, some explicit solutions such as Jacobian function solutions and hyperbolic function solutions are constructed. These can further understand the dynamic behavior of perturbed Chen–Lee–Liu equation and their wave propagation.

The following differential quadratic polynomial differential system dxdt=y−x, dydt=2y−yγ−12−γy−5γ−4γ−1x, when the parameter γ∈(1,2] models the structure equations of an isotropic star having a linear barotropic equation of state, being x=m(r)/r where m(r)≥0 is the mass inside the sphere of radius r of the star, y=4πr2ρ where ρ is the density of the star, and t=ln(r/R) where R is the radius of the star. First, we classify all the topologically non-equivalent phase portraits in the Poincaré disc of these quadratic polynomial differential systems for all values of the parameter γ∈R∖{1}. Second, using the information of the different phase portraits obtained we classify the possible limit values of m(r)/r and 4πr2ρ of an isotropic star when r decreases.

In nature, the vast majority of species live in ecosystems that are not isolated, and the same is true for predator-prey ecological systems. With this work, we extend a predator-prey model by considering the inclusion of an immigration term in both species. From a biological point of view, that allows us to achieve a more realistic model. We consider a system with a Holling type Ⅰ functional response and study its global dynamics, which allows to not only determine the behavior in a region of the plane $ \mathbb{R}^2 $, but also to control the orbits that either go or come to infinity. First, we study the local dynamics of the system, by analyzing the singular points and their stability, as well as the possible behavior of the limit cycles when they exist. By using the Poincaré compactification, we determine the global dynamics by studying the global phase portraits in the positive quadrant of the Poincaré disk, which is the region where the system is of interest from a biological point of view.

In this article, the fractional perturbed Gerdjikov–Ivanov equation is investigated. Firstly, the fractional perturbed Gerdjikov–Ivanov equation is transformed into an ordinary differential equation through traveling wave transformation. Secondly, using the trial method of rank homogeneous equation polynomials and the principle of homogeneous equilibrium, a two-dimensional planar dynamic system is presented and its bifurcation behavior is studied. Then, its two-dimensional phase portraits are drawn by using Maple software. Finally, disturbance factors are introduced into the planar dynamical system to study its chaotic behavior, and some two-dimensional phase portraits, three-dimensional phase portraits, Poincaré sections, and sensitivity analysis graphs of the perturbed system are plotted by using Maple software. The novelty lies in studying the dynamic behavior of the objective equation without the need for solving.

Asymmetric faults are a common voltage fault in the power grid, with the rapid development of wind and photovoltaic power generation, grid-connected power generation systems based on voltage source converters (VSCs) will also be affected by asymmetric faults. Asymmetric faults can cause transient power imbalances and generate negative sequence components that have a negative impact on power quality. Therefore, it is necessary to limit overcurrent under asymmetric faults. At present, there are many studies on current limitations during faults, and it has a good inhibitory effect on overcurrent phenomena during faults. As is well known, during power grid faults, the voltage at the PCC will undergo sudden changes, and the frequency of the PCC will also change. Therefore, not only overcurrent needs to be suppressed, but frequency fluctuations also need to be suppressed. This article proposes a control strategy of positive and negative sequence variable virtual impedance control (PN-VVIC), generate positive and negative sequence impedances based on the magnitude of overcurrent and frequency fluctuations to enhance the ability to suppress transient overcurrent and frequency fluctuations, and proposes a power reference recalculation during faults to adjust the output power of VSC according to the severity of the fault, ensure no overcurrent during the fault, and inject reactive power into the power grid to assist in fault recovery. Subsequently, discuss the influence of the PN-VVIC’s parameters on VSC’s stability through small signal modelling and analyze VSC’s stability with the proposed method under asymmetric faults using phase portraits. Finally, the effectiveness of the control strategy proposed in this article is verified through experiments, and the experimental results are analyzed.

This study focuses on the variant Boussinesq equation, which is used to model waves in shallow water and electrical signals in telegraph lines based on tunnel diodes. The aim of this study is to find closed-form wave solutions using the extended direct algebraic method. By employing this method, a range of wave solutions with distinct shapes, including shock, mixed-complex solitary-shock, singular,mixed-singular, mixed trigonometric, periodic, mixed-shock singular, mixed-periodic, and mixed-hyperbolic solutions, are attained. To illustrate the propagation of selected exact solutions, graphical representations in 2D, contour, and 3D are provided with various parametric values. The equation is transformed into a planar dynamical structure through the Galilean transformation. By utilizing bifurcation theory, the potential phase portraits of nonlinear and super-nonlinear traveling wave solutions are investigated. The Hamiltonian function of the dynamical system of differential equations is established, revealing the system’s conservative nature over time. The graphical representation of energy levels offers valuable insights and demonstrates that the model has closed-form solutions.

The present work strives to explore some qualitative analysis for the governing equation describing the dynamic response of a piezoelectro-magnetic circular rod. As a result of the integrability study of the governed equation, which furnishes valuable insights into its structure, solutions, and applications in various fields, we apply the well-known Ablowitz–Ramani–Segur (ARS) algorithm to prove the non-integrability of the governed equation in a Painlevé sense. The qualitative theory for planar integrable systems is applied to study the bifurcation of the solutions based on the values of rod material properties. Some new solutions for the governing equation are presented and they are categorized into solitary and double periodic functions. We display a 3D representation of the solutions in addition to investigating the influence of wave velocity on the obtained solution for the particular material of the rod. [ABSTRACT FROM AUTHOR]

The article represents methods and results of the original investigation of some hierarchical family of cubic dynamic differential systems in the Poincaré disk. For a family of systems, whose right-hand sides are reciprocal polynomial forms of the phase variables (a cubic form in the first equation and a square form in another one) all possible topologically different phase portraits in the Poincaré disk were investigated and constructed using approaches of the qualitative theory of ODEs and dynamic systems. Close to coefficient criteria of all phase portraits appearance for each of the existing subfamilies of systems under consideration were obtained. Research methods invented especially for the goals of this study are described in the paper. [ABSTRACT FROM AUTHOR]