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This study aims to demonstrate the behaviors of a two degree-of-freedom (DOF) dynamical system consisting of attached mass to a nonlinear damped harmonic spring pendulum with a piezoelectric device. Such a system is influenced by a parametric excitation force on the direction of the spring's elongation and an operating moment at the supported point. A negative-velocity-feedback (NVF) controller is inserted into the main system to reduce the undesired vibrations that affect the system's efficiency, especially at the resonance state. The equations of motion (EOM) are derived by using Lagrangian equations. Through the use of the multiple-scales-strategy (MSS), approximate solutions (AS) are investigated up to the third order. The accuracy of the AS is verified by comparing them to the obtained numerical solutions (NS) through the fourth-order Runge-Kutta Method (RK-4). The study delves into resonance cases and solvability conditions to provide the modulation equations (ME). Graphical representations showing the time histories of the obtained solutions and frequency responses are presented utilizing Wolfram Mathematica 13.2 in addition to MATLAB software. Additionally, discusses the bifurcation diagrams, Poincaré maps, and Lyapunov exponent spectrums to show the various behavior patterns of the system. To convert vibrating motion into electrical power, a piezoelectric sensor is connected to the dynamical model, which is just one of the energy harvesting (EH) technologies with extensive applications in the commercial, industrial, aerospace, automotive, and medical industries. Moreover, the time histories of the obtained solutions with and without control are analyzed graphically. Finally, resonance curves are used to discuss stability analysis and steady-state solutions. [ABSTRACT FROM AUTHOR]

The present research explores the analytical solutions and dynamics of complex Kraenkel–Manna–Merle system, which are exploited in ceramic-like materials with magnetic characteristics in electronics. Two distinct approaches, the extended sinh-Gordon equation expansion and the modified auxiliary equation are employed to derive soliton solutions in various function forms, including hyperbolic, trigonometric, rational, and Jacobi elliptic functions. The stability and accuracy of these solutions are confirmed through modulation instability analysis. Several specific solutions are illustrated through numerical simulations after assigning values to the free parameters. By back-substituting into the original model, the newly generated solutions confirm the validity of the new findings and ensure their accuracy. The obtained multiple soliton solutions show that the two approaches are effective, efficient, reliable, and potent for studying nonlinear evolution equations. Furthermore, a transformation converted the system into a planar dynamical system, allowing phase portraits to be analyzed. Moreover, the introduction of a perturbed term uncovered chaotic behavior across a range of parameter values, illustrated through both two-dimensional and three-dimensional graphics. The study presents novel analytical solutions that offer insights into nonlinear short wave interactions, which have not been previously documented using these methodologies. [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]

This manuscript delves into the dynamic behavior of the (3 + 1) -dimensional nonlinear extended quantum Zakharov-Kuznetsov (NLEQZK) equation, exploring soliton solutions, chaotic phenomena, bifurcation, sensitivity, and stability through the lens of planar dynamical system theory. Firstly, the generalized Arnous method is used for securing some soliton solutions. Solutions obtained using the generalized Arnous method include hyperbolic, rational, and logarithmic forms. To visualize these graphically, we select appropriate parameter values and examine the graphical behavior of the obtained solutions. The visual characteristics of the solutions that were created are also evaluated in this procedure through the utilization of 3D surface graphs and line graphs representing various parameter values. The governing equation is derived using the Galilean transformation to facilitate bifurcation analysis. Chaotic behavior in the NLEQZK equation is investigated by introducing a perturbed term in the dynamical system and presenting various analyses, including Poincaré maps, time series, 2-dimensional (2D) phase portraits, and 3-dimensional (3D) phase portraits. Additionally, the Runge–Kutta method is employed for sensitivity analysis, confirming that slight adjustments to the initial conditions and check whether the system is sensitive or not. The outcomes of this study contribute valuable insights to the understanding of nonlinear dynamical systems and soliton theory. [ABSTRACT FROM AUTHOR]

ABSTRACT This study focuses on the manipulation of soliton parameters within optical waveguides characterized by a dual‐power law refractive index, drawing similarities with published papers. The precise management of solitons holds significant importance in the field of fiber‐optical communication. While previous research has primarily concentrated on modifying soliton attributes such as amplitude and width, comparatively less attention has been dedicated to controlling the wave speed of solitons. In order to bridge this gap, the researchers adopted the employment of the nonlinear Schrödinger equation with time‐dependent dispersion and two power‐law nonlinearities, mirroring similar approaches found in published literature. By employing this methodology, the researchers successfully achieved active control over the wave speed of solitons in non‐uniform optical waveguides. To effectively handle the inherent complexities of the nonlinear system, two well‐established techniques, namely, the generalized Kudryashov method and the simple equation method, were utilized to derive solutions. These techniques have been previously employed and demonstrated in published papers, further enhancing the validity and reliability of the research findings. We discuss the traveling wave solutions by finding the fixed points of the dynamical planar system. [ABSTRACT FROM AUTHOR]

This paper aims to investigate some new dynamics of a new model describing the rabies virus dynamics, taking into account the effect of proper vaccination. The model's population is divided into three main compartments, namely, susceptible S(t), infected I(t), and recovered R(t) individuals. The model is formulated and then the equilibrium points of the model are found. The local and global stabilities of equilibrium points of the proposed model are investigated where conditions of stability are attained in terms of key parameters in the model. Bifurcation analysis is performed for the possible occurrence of codimension‐one bifurcations in the model. In addition, bifurcation surfaces are plotted in the space of parameters in the model. For the numerical verification, a nonstandard finite difference method is adapted for solving the model and the accurate results of numerical simulations are depicted to reveal the dynamics of the model. The method provides realistic data for the model and these data can be used to predict the spread of the virus and to provide insight into proper procedures and control measures that can be taken. [ABSTRACT FROM AUTHOR]

This paper presents a novel viewpoint on nonlinear dust-acoustic waves (DAWs) in an unmagnetized collisionless plasma containing regularized κ distributed electrons and ions (ei) and negative dust grain. The nonlinear oscillatory system based on hybridization of the Van der Pol–Mathieu equation (VdPME) is derived by a new technique. By bifurcation analysis of the planar dynamical system (DS), the effects of parameters with the assistance of phase planes and time series of VdPME are studied. After analyzing the equation to identify the resonance region, a fourth-order Runge–Kutta method is used to solve it numerically. We explained the behavior of DA periodic, stable limit cycle, and chaotic limit cycle wave solutions with different parameters. These types of numerical solutions are illustrated in two-dimensional and three-dimensional graphics by changing the rate at which charged dust grain is produced α, as well as waste β and comparing the results with those of earlier research. A novel bifurcation analysis of VdPE and VdPME is obtained with the effects of the cut-off factor δ of regularized κ-distribution (RKD) distributed ei, the superthermality of ei particles κe,i, the ratio of ion to electron temperature σ, and the ratio of dust to electron density ρ illustrated.  It is noticed that the DAW shows promotion in width and amplitude as the frequency γ increases and it becomes rarefaction as the cut-off parameter δ increases. On the contrary, it becomes compressive under the impact of superthermality κe and κi. The obtained conclusions may help explain and comprehend a variety of applications in experimental plasma containing highly energy regularized κ dispersed ei, as well as in the interstellar medium. [ABSTRACT FROM AUTHOR]

The truncated M-fractional Kuralay (TMFK)-II equation is prevalent in the exploration of specific complex nonlinear wave phenomena. Such types of wave phenomena are more applicable in science and engineering. These equations could potentially provide insights into understanding the intricate dynamics of optical phenomena, encompassing solitons, nonlinear effects, and wave interactions. This study aims to uncover a diverse range of soliton solutions to the model, spanning trigonometric, hyperbolic, exponential, and rational expressions. These solutions are unveiled through the application of extended hyperbolic functions and improved F-expansion techniques, representing the primary objective of this research. The three-dimensional (3D) and two-dimensional (2D) combined charts are plotted for some of these solutions. The impact of the fractional parameters and time variations is also illustrated. Moreover, the models are converted into a planar dynamical system using a Galilean transformation, and the analysis of bifurcation is examined. This research underscores the versatility of the aforementioned techniques for exploring complex nonlinear phenomena across various engineering and scientific disciplines. Finally, the findings of this study hold significant implications for advancing our understanding and analysis of nonlinear wave dynamics in diverse physical systems. [ABSTRACT FROM AUTHOR]

This work investigates the Kudryashov-Sinelshchikov equation, which explains the influence of viscosity and heat transfer on pressure wave propagation. This equation is transformed into a singular dynamical system by applying the proper wave transformation. Then, by applying the proper point transformation for the independent variable, it may be transformed again into a regular dynamical system. We show that the first integral of both systems is the same. We provide a brief overview of the phase planes of both systems, taking into account the topological equivalence between their phase orbits. Furthermore, the new Kudryashov method and the sine-Gordon expansion method are considered to construct some novel exact solutions to the Kudryashov-Sinelshchikov equation. Using the wave transform, the studied equation is converted to the ODE. Then, the proposed methods are successfully applied to the equation to reveal some new solutions. The obtained solutions are stated as hyperbolic and exponential functions, which are classified as singular, bright, dark, kink, and anti-kink solutions. To better understand the physical properties of the solutions, the two-dimensional and three-dimensional graphs for the solutions are presented by selecting appropriatore values for the including physical parameters. Exploring these behaviors may play an important role in understanding the propagation of nonlinear waves in a combination of liquid with gas bubbles. [ABSTRACT FROM AUTHOR]

The spread of epidemic diseases is often considered to be bound up with human behavior. Both treatment and awareness-raising are seen as important preventive measures during an outbreak. In this paper, an SEIR model on heterogeneous network containing saturated treatment function with media information variable is constructed. Through mathematical analysis, the basic reproduction number and equilibriums of the system are derived. The global stability of disease-free equilibrium is further proved based on LaSalle's invariant principle. It is also found that with a gradual increase of the delayed treatment parameter, the system exhibits a transition from forward bifurcation to backward bifurcation with the variation of the basic reproduction number. Finally, an optimal control problem is formulated with both treatment rate and media information dissemination rate as control variables and solved using the Pontryagin's maximum principle. All the above results are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

This study delves into the exploration of the dynamics of (3 + 1) -dimensional Painlevé integrable generalized model from different perspectives, which delineates the evolution of nonlinear phenomena in three spatial dimensions and one temporal dimension, displaying the remarkable Painlevé integrability property. The ϕ 6 -expansion technique is applied to extract traveling wave solutions in the form of Jacobi elliptic functions. To give physical insights of obtained solutions, we present these solutions through graphs such as 2D, 3D and contour plots. Further, to understand the planar dynamical system, we employ the concepts of bifurcation, chaos theory and sensitivity analysis. Bifurcation analysis reveals the dependence on the solution of a planar dynamical system at critical points. Additionally, the detection of chaotic movements in the perturbed dynamical system is achieved by detecting tools. Also the sensitivity analysis of the model is investigate by three distinct initial conditions. The findings are innovative, valuable and captivating for the readers in exploring this model. [ABSTRACT FROM AUTHOR]

In this study, a bifurcation analysis of the piezoelectric neuron system (PNS) is conducted utilizing the time-domain minimum residual method (TMRM). From a biophysical standpoint, integrating piezoelectric devices into a nonlinear neuron circuit enables the conversion of external pressure or sound wave signals into electrical signals. The primary content of this research are as follows: initially, the piezoelectric neuron system is analytically solved by employing the TMRM, yielding high-precision periodic solutions. Subsequently, a detailed bifurcation analysis is performed leveraging these high-precision solutions. The findings demonstrate that the neuronal system experiences a periodic-doubling bifurcation when the amplitude of external stimuli surpasses a specific threshold. With further increases in the stimulus, this periodic-doubling response evolves into a periodic response. Moreover, changes in cell membrane resistance will lead to complex nonlinear behaviors such as stable or unstable period-1 solutions and stable or unstable and critically stable period-3 solutions in the system. These findings offer a theoretical foundation for the advanced design of intelligent functional neuron arrays and sensors. [ABSTRACT FROM AUTHOR]

In this study, the modified Sardar sub-equation method is capitalised to secure soliton solutions to the (1 + 1) -dimensional chiral nonlinear Schrödinger (NLS) equation. Chiral soliton propagation in nuclear physics is an extremely attractive field because of its wide applications in communications and ultrafast signal routing systems. Additionally, we perform bifurcation analysis to gain a deeper understanding of the dynamics of the chiral NLS equation. This highlights the complex behaviour of the system and exposes the conditions under which various types of bifurcations occur. Additionally, a sensitivity analysis is performed to assess how small changes in initial conditions and parameters influence the solutions, offering valuable perspectives on the stability and dependability of the acquired solutions. By employing the above-mentioned methodology, we derive a variety of exact solutions, including periodic, singular, dark, bright, mixed trigonometric, exponential, hyperbolic, and rational wave solutions. The study’s findings advance our theoretical knowledge of chiral NLS equations and have potential applications in optical communication and related fields. [ABSTRACT FROM AUTHOR]

One of the most important factors influencing animal growth is non-genetics, which includes factors like nutrition, management and environmental conditions. By consuming their prey, predators can directly affect their ecology and evolution, but they can also have an indirect impact by affecting their prey’s nutrition and reproduction. Preys used to change their habitats, their foraging and vigilance habits as anti-predator responses. Cooperation during hunting by the predators develops significant fear in their prey which indirectly affects their nutrition. In this work, we propose a two-species stage-structured predator–prey system where the prey are classified into juvenile and mature prey. We assume that the conversion of juvenile prey to matured prey is affected by the fear of predation risk. Non-negativity and boundedness of the solutions are demonstrated theoretically. All the biologically feasible equilibrium states are determined, and their stabilities are analyzed. The role of various important factors, e.g. hunting cooperation rate, predation rate and rate of fear, on the system dynamics is discussed. To visualize the dynamical behavior of the system, extensive numerical experiments are performed by using MATLAB and MatCont 7.3. Finally, the proposed model is extended into a harvesting model under quadratic harvesting strategy and the associated control problem has been analyzed for optimal harvesting. [ABSTRACT FROM AUTHOR]

Bifurcation analysis is an essential tool in nonlinear dynamics. Bifurcation diagrams help to discover subtle features of investigated dynamics such as chaotic and periodic regimes, hidden attractors and fixed points. However, plotting high-resolution bifurcation diagrams can be a computationally challenging task, especially in multiparametric evaluation. It should be noted that the bifurcation analysis is a task with natural parallelism and thus can be efficiently solved using hybrid central-graphics processing architectures. In this paper, we propose an advanced algorithm and special software for plotting bifurcation diagrams using calculations accelerated by graphics processing unit in combination with a highly efficient semi-implicit ordinary differential equation solver. Time series processing is based on the extraction of amplitude and phase features for the density-based spatial clustering of applications with noise to determine oscillation periodicity. We showcase the features of the application of proposed solutions on a set of test chaotic systems. The performance of the analysis algorithms is investigated in comparison with conventional solutions based on central processing unit and several approaches known from the literature. We explicitly show that the proposed algorithm outperforms known solutions in both calculation speed and precision. [ABSTRACT FROM AUTHOR]

The present paper aims to explore bifurcation analysis, modulation instability, and optical soliton solutions in nonlinear media with third-order dispersion terms. A variable coefficient third-order nonlinear Schrödinger's equation (PNLSE) with truncated M -fractional derivative is considered. We also discuss a few assets that the derivative satisfies. Initially, a novel evaluation method known as bifurcation analysis is employed to look at the complex model's dynamic behavior. Figure 1 provides an analytical and graphical study of the observed mechanism of static soliton through a saddle-node bifurcation in the nonlinear Schrödinger problem using a matching technique. Secondly, By implementing novel two analytic techniques such as unified solver and generalized unified techniques to offer insights into wave propagation and optical soliton conduct in nonlinear optics, optical communications, quantum mechanics, plasma physics, and engineering. The underlying idea of these two approaches is to transform the equation with partial derivatives into a version of the equation with ordinary derivatives, which are first required to incorporate a new wave definition. These techniques permit us to generate innovative soliton solutions that can be formulated in terms of rational, hyperbolic, and trigonometrical functions. The collected outcomes have the potential to facilitate an understanding and elucidation of the physical characteristics of waves moving within a dispersive substance. We additionally estimated conservative values related to solitons, such as energy, momentum, and power. By selecting appropriate parameter values, the graphical shapes in 3D, density, and 2D are generated using relevant parameter values to visually present the obtained results, including periodic wave soliton, interaction of kink and bell shape soliton, periodic breather wave soliton, double periodic soliton through unified solver method and interaction of kink and periodic lump wave soliton and also with the periodic soliton, double periodic wave soliton, periodic wave soliton with lump soliton, multi periodic wave with kink shape soliton, periodic wave soliton, periodic breather wave soliton, periodic wave soliton, multi periodic wave soliton, periodic lump wave soliton, breather wave soliton multi periodic wave through generalized unified technique, providing an insightful visualization of the discovered solutions. In a two-dimensional graph, we show the effect of truncated M -fractional parameters for [ g = 0.1 , 0.5 , 0.9 ]. Additionally, the modulation instability spectrum can be expressed utilizing a linear analysis technique, and the modulation instability bands are shown to be influenced by the third-order and group velocity dispersion. The findings indicate that the modulation instability disappears for negative values of the fourth order in a typical dispersion regime. Consequently, it was shown that the techniques mentioned previously could be an effective tool to generate unique, precise soliton solutions for numerous uses, which are crucial to nonlinear optics, optical communications, and engineering. [ABSTRACT FROM AUTHOR]

The current paper undertakes an in-depth exploration of the dynamics of nonlinear waves governed by a 3D-modified nonlinear wave equation, a significant model in the study of complex wave phenomena. To this end, the study employs both classical and nonclassical Lie symmetries for rigorously deriving invariant solutions of the governing equation. These symmetries enable the formal construction of exact solutions, which are crucial for understanding the complex behavior of the model. Furthermore, the research extends into the realm of bifurcation analysis through the application of planar dynamical system theory. Such an analysis reveals the conditions under which the 3D-modified nonlinear wave equation admits Jacobi elliptic function solutions. The study also delves into the impact of the nonlinear parameter on the physical characteristics of bright and kink solitary waves as well as continuous periodic waves using Maple. Overall, the comprehensive analysis presented not only enhances the understanding of complex nonlinear wave dynamics but also sets the stage for future advancements in vast areas of fluid dynamics and plasma physics. [ABSTRACT FROM AUTHOR]

In this paper, we present a dimensional reduction to obtain a one-dimensional model to analyze localized necking or bulging in a residually stressed circular cylindrical solid. The nonlinear theory of elasticity is first specialized to obtain the equations governing the homogeneous deformation. Then, to analyze the nonhomogeneous part, we include higher-order correction terms of the axisymmetric displacement components leading to a three-dimensional form of the total potential energy functional. Details of the reduction to the one-dimensional form are given. We focus on a residually stressed Gent material and use numerical methods to solve the governing equations. Two loading conditions are considered. First, the residual stress is maintained constant, while the axial stretch is used as the loading parameter. Second, we keep the pre-stretch constant and monotonically increase the residual stress until bifurcation occurs. We specify initial conditions, find the critical values for localized bifurcation, and compute the change in radius during localized necking or bulging growth. Finally, we optimize material properties and use the one-dimensional model to simulate necking or bulging until the Maxwell values of stretch are reached. [ABSTRACT FROM AUTHOR]

In this study, we present an amalgamation of solitary wave solutions for the fractional coupled Drinfeld-Sokolov-Wilson (FCDSW) equations, a versatile mathematical model with significant applications in fluid dynamics, plasma physics, and nonlinear dynamics. By leveraging the two recently developed computational approaches, namely the modified Sardar sub-equation (MSSE) method and the improved F - expansion method, we manifested the novel soliton solutions in the form of dark, dark-bright, bright-dark, singular, periodic, exponential, and rational forms. Furthermore, we also extracted W-shape, V-shape, mixed trigonometric, and hyperbolic soliton wave solutions, which are not reported in previously. Also, the modulation instability (MI) of the proposed model is also examined. To further understand the dynamics of the FCDSW equations, we conducted a detailed bifurcation analysis to investigate the bifurcation events exhibited by these equations. A sensitivity analysis was also performed to assess the model’s robustness against variations in initial conditions and parameters, offering valuable insights into the system’s susceptibility to perturbations. We proved the effectiveness of our suggested techniques in the analysis of the FCDSW equations using analytical tools and numerical simulations. Our results provide new insights into the behavior and solutions of the FCDSW equations and enhance the mathematical tools for investigating nonlinear partial differential equations (NLPDEs). These findings have potential applications in fields like fluid flow modeling, wave propagation in plasmas, and the study of nonlinear optical phenomena in physics and applied mathematics. [ABSTRACT FROM AUTHOR]

The stability of the transportation system refers to the structural stability of the system. When the system structure is unstable, local or global bifurcation phenomena will occur, which is one of the main reasons for nonlinear traffic phenomena such as congestion. To truly understand the internal mechanism of the formation of these phenomena, it is necessary to analyze the bifurcation of traffic flow. In this paper, the Hopf bifurcation control of a modified viscous macroscopic traffic flow model is studied by using the linear state feedback method, which changes the characteristics of the bifurcation phenomenon of the dynamic system and obtains the required dynamic behavior of the system. First, we can convert the original traffic model into the nonlinear ordinary differential form suitable for bifurcation analysis, solve the equilibrium point of the system, and carry out phase plane analysis. Then, the linear state feedback term is added and the corresponding controlled system is generated, the existence and type of Hopf bifurcation and the existence of saddle node bifurcation are proved. Numerical simulation results show that the analysis and control of Hopf bifurcation in the traffic model are well realized in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we delve into a predator–prey model incorporating a fear effect in prey reproduction, influenced by both delay and harvesting. The model accounts for delayed fear dynamics to capture more realistic dynamics. Initially, our focus lays on the nondelayed model, examining each biologically plausible equilibrium points and assessing their stability concerning the parameters of the model. Next, detailed mathematical results are provided, encompassing the asymptotic stability of all equilibria, Hopf bifurcation, and the direction and stability of bifurcated periodic solutions. Also, the stability analysis of the Hopf-bifurcating periodic solution is confirmed through the computation of first Lyapunov coefficient. Furthermore, we observed that the nondelayed system experiences Bogdanov–Takens bifurcation in a two-parameter space. Subsequently, we analyzed the corresponding delayed system, establishing the existence of a stable limit cycle through Hopf bifurcation concerning the delay parameter. Additionally, the inclusion of delay can prompt critical dynamics within the system, resulting in period-doubling routes toward chaotic oscillations. To validate our analytical findings, we conducted comprehensive and meticulous numerical simulations. The findings of the numerical simulations suggest that the impact of fear can be used as a measure of chaos control. [ABSTRACT FROM AUTHOR]

This study focuses on revealing closed-form solutions for traveling waves in the nonlinear dynamical model of dust-acoustic waves in multispecies of dusty plasma. Plasma consists of cold dust grains, along with isothermal electrons and positive ions for balance. Fluid equations simplify through reductive perturbation, yielding the modified Korteweg–de Vries equation for analyzing weak dust-acoustic solitons. Applying this method to a specific scenario involving negatively charged dust grains and protons with all electrons bound to the dust grains shows support for rarefactive supersonic solitons, contrasting with typical compressive ion-acoustic solitons. Introducing streaming reveals a subtle soliton amplitude decrease. Using the Generalized exp(-S(φ)) expansion method effectively derives various soliton solutions, including kink, anti-kink, singular, hyperbolic functions, and diverse solitary waves. Furthermore, the equation’s dynamic behaviors are explored through equilibrium point bifurcations. Employing a Galilean transformation, the model is transformed into a planar dynamical system, and qualitative inspection is performed. Phase portraits depict bifurcation outcomes. Our methodology is significant for tackling the novel problem and applying previously untested approaches in this context, resulting in the development of multiple exact and novel optical soliton solutions. This highlights the efficiency and broad application of our new strategy to addressing nonlinear challenges in astrophysics, laboratories, space, and technology.

This study delves into the exploration of the dynamics of (3+1)-dimensional Painlevé integrable generalized model from different perspectives, which delineates the evolution of nonlinear phenomena in three spatial dimensions and one temporal dimension, displaying the remarkable Painlevé integrability property. The ϕ6-expansion technique is applied to extract traveling wave solutions in the form of Jacobi elliptic functions. To give physical insights of obtained solutions, we present these solutions through graphs such as 2D, 3D and contour plots. Further, to understand the planar dynamical system, we employ the concepts of bifurcation, chaos theory and sensitivity analysis. Bifurcation analysis reveals the dependence on the solution of a planar dynamical system at critical points. Additionally, the detection of chaotic movements in the perturbed dynamical system is achieved by detecting tools. Also the sensitivity analysis of the model is investigate by three distinct initial conditions. The findings are innovative, valuable and captivating for the readers in exploring this model.

The truncated M-fractional Kuralay (TMFK)-II equation is prevalent in the exploration of specific complex nonlinear wave phenomena. Such types of wave phenomena are more applicable in science and engineering. These equations could potentially provide insights into understanding the intricate dynamics of optical phenomena, encompassing solitons, nonlinear effects, and wave interactions. This study aims to uncover a diverse range of soliton solutions to the model, spanning trigonometric, hyperbolic, exponential, and rational expressions. These solutions are unveiled through the application of extended hyperbolic functions and improved F-expansion techniques, representing the primary objective of this research. The three-dimensional (3D) and two-dimensional (2D) combined charts are plotted for some of these solutions. The impact of the fractional parameters and time variations is also illustrated. Moreover, the models are converted into a planar dynamical system using a Galilean transformation, and the analysis of bifurcation is examined. This research underscores the versatility of the aforementioned techniques for exploring complex nonlinear phenomena across various engineering and scientific disciplines. Finally, the findings of this study hold significant implications for advancing our understanding and analysis of nonlinear wave dynamics in diverse physical systems.

Abstract In this study, the modified Sardar sub-equation method is capitalised to secure soliton solutions to the $$(1+1)$$ ( 1 + 1 ) -dimensional chiral nonlinear Schrödinger (NLS) equation. Chiral soliton propagation in nuclear physics is an extremely attractive field because of its wide applications in communications and ultrafast signal routing systems. Additionally, we perform bifurcation analysis to gain a deeper understanding of the dynamics of the chiral NLS equation. This highlights the complex behaviour of the system and exposes the conditions under which various types of bifurcations occur. Additionally, a sensitivity analysis is performed to assess how small changes in initial conditions and parameters influence the solutions, offering valuable perspectives on the stability and dependability of the acquired solutions. By employing the above-mentioned methodology, we derive a variety of exact solutions, including periodic, singular, dark, bright, mixed trigonometric, exponential, hyperbolic, and rational wave solutions. The study’s findings advance our theoretical knowledge of chiral NLS equations and have potential applications in optical communication and related fields.

Abstract The current paper undertakes an in-depth exploration of the dynamics of nonlinear waves governed by a 3D-modified nonlinear wave equation, a significant model in the study of complex wave phenomena. To this end, the study employs both classical and nonclassical Lie symmetries for rigorously deriving invariant solutions of the governing equation. These symmetries enable the formal construction of exact solutions, which are crucial for understanding the complex behavior of the model. Furthermore, the research extends into the realm of bifurcation analysis through the application of planar dynamical system theory. Such an analysis reveals the conditions under which the 3D-modified nonlinear wave equation admits Jacobi elliptic function solutions. The study also delves into the impact of the nonlinear parameter on the physical characteristics of bright and kink solitary waves as well as continuous periodic waves using Maple. Overall, the comprehensive analysis presented not only enhances the understanding of complex nonlinear wave dynamics but also sets the stage for future advancements in vast areas of fluid dynamics and plasma physics.

Low aspect ratio flying wings have attracted significant attention due to high maneuverability and potential for stealth applications, and their flight dynamics are complex and dominated by nonlinearities. Wing rock is a common quasi-periodic oscillation in low aspect ratio aircraft, which can exhibit chaotic motion under certain conditions. However, the specific characteristics of wing rock in low aspect ratio flying wings and its potential chaotic behavior remain unclear. In this paper, the wing rock motion of a low aspect ratio flying wing is investigated through bifurcation analysis and time-domain responses analysis based on a full-order nonlinear aircraft model. It is found that elevator deflection induces a successive period-doubling bifurcation path, which transitions the wing rock from single-period to multi-period, ultimately evolving toward chaos. Multi-period and chaotic wing rock poses a higher level of danger compared to single-period wing rock, as their trajectories exhibit rapid helical descent similar to spin. These results significantly enhance our comprehension of wing rock in low aspect ratio flying wings and contribute valuable insights to the design of effective recovery strategies for wing rock. [ABSTRACT FROM AUTHOR]

Abstract The COVID-19 pandemic came with many setbacks, be it to a country’s economy or the global missions of organizations like WHO, UNICEF or GTFCC. One of the setbacks is the rise in cholera cases in developing countries due to the lack of cholera vaccination. This model suggested a solution by introducing another public intervention, such as adding Chlorine to water bodies and vaccination. A novel delay differential model of fractional order was recommended, with two different delays, one representing the latent period of the disease and the other being the delay in adding a disinfectant to the aquatic environment. This model also takes into account the population that will receive a vaccination. This study utilized sensitivity analysis of reproduction number to analytically prove the effectiveness of control measures in preventing the spread of the disease. This analysis provided the mathematical evidence for adding disinfectants in water bodies and inoculating susceptible individuals. The stability of the equilibrium points has been discussed. The existence of stability switching curves is determined. Numerical simulation showed the effect of delay, resulting in fluctuations in some compartments. It also depicted the impact of the order of derivative on the oscillations.

Abstract This research examines pseudoparabolic nonlinear Oskolkov-Benjamin-Bona-Mahony-Burgers (OBBMB) equation, widely applicable in fields like optical fiber, soil consolidation, thermodynamics, nonlinear networks, wave propagation, and fluid flow in rock discontinuities. Wave transformation and the generalized Kudryashov method is utilized to derive ordinary differential equations (ODE) and obtain analytical solutions, including bright, anti-kink, dark, and kink solitons. The system of ODE, has been then examined by means of bifurcation analysis at the equilibrium points taking parameter variation into account. Furthermore, in order to get insight into the influence of some external force perturbation theory has been employed. For this purpose, a variety of chaos detecting techniques, for instance poincaré diagram, time series profile, 3D phase portraits, multistability investigation, lyapounov exponents and bifurcation diagram are implemented to identify the quasi periodic and chaotic motions of the perturbed dynamical model. These techniques enabled to analyze how perturbed dynamical system behaves chaotically and departs from regular patterns. Moreover, it is observed that the underlying model is quite sensitivity, as it changing dramatically even with slight changes to the initial condition. The findings are intriguing, novel and theoretically useful in mathematical and physical models. These provide a valuable mechanism to scientists and researchers to investigate how these perturbations influence the system’s behavior and the extent to which it deviates from the unperturbed case.

In this research work, we investigate a new three-dimensional jerk system with three parameters in which one of the nonlinear terms is a sinusoidal nonlinearity. We show that the new jerk system has two unstable equilibrium points on the ��-axis. Numerical integrations show the existence of periodic and chaotic states, as well as unbounded solutions. Consideration of the Poincaré sphere at infinity found no periodic states. We show that the new jerk system exhibits multistability with coexisting attractors. We also present results for the offset boosting of the proposed chaotic jerk system. Using MultiSim version 14.1, we design an electronic circuit for the new jerk system with a sinusoidal nonlinearity. As a control application, we design complete synchronization for the master-slave jerk systems using backstepping control technique. Simulations are presented to illustrate the main results of this research work.

The aim of this paper is to show that Tilman's graphical method for the study of competition between two species for two resources can be advantageously used for the study of commensalism or syntrophy models, where a first species produces the substrate necessary for the growth of the second species. The growth functions of the species considered are general and include both inhibition by the other substrate and inhibition by the species' limiting substrate, when it is at a high concentration. Because of their importance in microbial ecology, models of commensalism and syntrophy, with or without self-inhibition, have been the subject of numerous studies in the literature. We obtain a unified presentation of a large number of these results from the literature. The mathematical model considered is a differential system in four dimensions. We give a new result of local stability of the positive equilibrium, which has only been obtained in the literature in the case where the removal rates of the species are identical to the dilution rate and the study of stability can be reduced to that of a system in two dimensions. We describe the operating diagram of the system: this is the bifurcation diagram which gives the asymptotic behavior of the system when the operating parameters are varied, i.e., the dilution rate and the substrate inlet concentrations.

This paper focuses on obtaining exact solutions of nonlinear Akbota equation through the application of the modified Khater method and Sardar sub-equation method. Renowned as one of the latest and precise analytical schemes for nonlinear evolution equations, this method has proven its efficacy by generating diverse solutions for the model under consideration. The equation is crucial in the study of optical solitons, which are stable pulses of light that maintain their shape over long distances. The Akbota equation helps in understanding the behavior and stability of these solitons. The governing equation undergoes transformation into an ordinary differential equation through a well-suited wave transformation. This analytical simplification paves the way for the derivation of trigonometric, hyperbolic, and rational solutions through the proposed methods. To illuminate the physical behavior of the model, the study presents graphical plots of the selected solutions of Khater and Sardar sub-equation method. This visual representation, achieved by selecting appropriate values for arbitrary parameters, enhances the understanding of the system’s dynamics. All calculations in this study are meticulously conducted using the Mathematica and Maple software, ensuring accuracy and reliability in the analysis of the obtained solution. Furthermore we investigate the sensitivity analysis of the dynamical system. [ABSTRACT FROM AUTHOR]

This paper proposes a novel nonuniform continuous traffic flow model, which takes into account the differences in drivers' expected headway and further improves the model by incorporating the influences of relative distance and relative velocity of vehicles through linear weighting. Additionally, the model also considers the dynamic response time of drivers for further refinement. With this model, bifurcation theory can also be applied in the stability analysis of traffic systems to study the stability mutation behavior of traffic systems at bifurcation points. Through linear and nonlinear analysis methods, the stability conditions of the model and the KdV–Burgers equation can be derived, the type of equilibrium point of the model can be judged, the conditions for the existence of Hopf bifurcations and the type of bifurcations can be proved, and the traffic flow problem can be transformed into the stability analysis problem of the system. Through numerical simulations in both high-density and low-density scenarios, it is shown that the model can well describe the actual traffic phenomena and can describe the stability abrupt behavior of the traffic system at Hopf bifurcation points and saddle-node bifurcation points through density-time and phase plane diagrams. [ABSTRACT FROM AUTHOR]

In this paper we study a two infection SIR-SIR compartmental model, considering biological features described in dengue fever epidemiology. Due to a progressive loss of protective antibodies there is waning immunity in the first infection stage and disease enhancement or protection effects by the second infection stage. Bifurcation analysis reveals two codim-2 bifurcations as organizing centers. The unfolding of a cusp bifurcation describes the transition of the disease-free equilibrium into an endemic equilibrium by varying a parameter. These equilibria allow an analytical solution with explicit expressions which allow for a full geometrical interpretation of the occurring bifurcations related to stationary dynamics. A Bogdanov-Takens point is the starting point in the parameter space where oscillatory endemic dynamics occurs including a homoclinic connection. These findings bring additional insights on biological mechanisms able to generate rich and complicated dynamical behavior in simple epidemic models that are, so far, largely unexplored. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a discrete-time prey-predator model. The model is formulated by using the piecewise constant argument method for differential equations and taking into account Holling type III. The existence and local behavior of equilibria are studied. We established that the system experienced both Neimark-Sacker and perioddoubling bifurcations analytically by using bifurcation theory and the center manifold theorem. In order to control chaos and bifurcations, the state feedback method is implemented. Numerical simulations are also provided for the theoretical discussion. [ABSTRACT FROM AUTHOR]

The COVID-19 pandemic came with many setbacks, be it to a country's economy or the global missions of organizations like WHO, UNICEF or GTFCC. One of the setbacks is the rise in cholera cases in developing countries due to the lack of cholera vaccination. This model suggested a solution by introducing another public intervention, such as adding Chlorine to water bodies and vaccination. A novel delay differential model of fractional order was recommended, with two different delays, one representing the latent period of the disease and the other being the delay in adding a disinfectant to the aquatic environment. This model also takes into account the population that will receive a vaccination. This study utilized sensitivity analysis of reproduction number to analytically prove the effectiveness of control measures in preventing the spread of the disease. This analysis provided the mathematical evidence for adding disinfectants in water bodies and inoculating susceptible individuals. The stability of the equilibrium points has been discussed. The existence of stability switching curves is determined. Numerical simulation showed the effect of delay, resulting in fluctuations in some compartments. It also depicted the impact of the order of derivative on the oscillations. [ABSTRACT FROM AUTHOR]

In this paper, we have analyzed the bifurcation of stagnation points in the non-Newtonian flow of the FENE-P fluid through a channel induced by peristalsis. The stream function for the flow under consideration is developed under negligible inertia and streamline curvature effects. A nonlinear autonomous system is developed for scrutiny of the critical points. A dynamical system approach is used to examine the stability of bifurcation points. Streamline patterns, and local and global bifurcation diagrams are plotted for analysis of the backward flow, trapping, and augmented flow regions. The impacts of important embedded parameters, i.e., extensibility parameter and Deborah number on local and global bifurcation diagrams, are also examined. The whole analysis reveals that two critical conditions appear in the flow field that are actually responsible for the conversion of backward flow to trapping and trapping to augmented flow. It is further seen that bifurcations occur at lower flow rates for increasing Deborah's number while an opposite trend prevails for increasing the extensibility parameter. A comparison of bifurcations between planar and axisymmetric flows is also performed. It is observed that a greater area of the trapping region is encountered for axisymmetric flow than that for planar flow. [ABSTRACT FROM AUTHOR]

This research examines pseudoparabolic nonlinear Oskolkov-Benjamin-Bona-Mahony-Burgers (OBBMB) equation, widely applicable in fields like optical fiber, soil consolidation, thermodynamics, nonlinear networks, wave propagation, and fluid flow in rock discontinuities. Wave transformation and the generalized Kudryashov method is utilized to derive ordinary differential equations (ODE) and obtain analytical solutions, including bright, anti-kink, dark, and kink solitons. The system of ODE, has been then examined by means of bifurcation analysis at the equilibrium points taking parameter variation into account. Furthermore, in order to get insight into the influence of some external force perturbation theory has been employed. For this purpose, a variety of chaos detecting techniques, for instance poincaré diagram, time series profile, 3D phase portraits, multistability investigation, lyapounov exponents and bifurcation diagram are implemented to identify the quasi periodic and chaotic motions of the perturbed dynamical model. These techniques enabled to analyze how perturbed dynamical system behaves chaotically and departs from regular patterns. Moreover, it is observed that the underlying model is quite sensitivity, as it changing dramatically even with slight changes to the initial condition. The findings are intriguing, novel and theoretically useful in mathematical and physical models. These provide a valuable mechanism to scientists and researchers to investigate how these perturbations influence the system's behavior and the extent to which it deviates from the unperturbed case. [ABSTRACT FROM AUTHOR]

In this research, we discussed the different chaotic phenomena, sensitivity analysis, and bifurcation analysis of the planer dynamical system by considering the Galilean transformation to the Lonngren wave equation (LWE) and the (2 + 1)-dimensional stochastic Nizhnik–Novikov–Veselov System (SNNVS). These two important equations have huge applications in the fields of modern physics, especially in the electric signal in data communication for LWE and the mechanical signal in a tunnel diode for SNNVS. A different chaotic nature with an additional perturbed term was also highlighted. Concerning the theory of the planer dynamical system, the bifurcation analysis incorporating phase portraits of the dynamical systems of the declared equations was performed. Additionally, a sensitivity analysis was used to monitor the sensitivity of the mentioned equations. Also, we extracted new, abundant solitary wave structures with the graphical phenomena of the mentioned nonlinear mathematical models. By conducting an expansion method on the abovementioned equations, we generated three types of soliton structures, which are rational function, trigonometric function, and hyperbolic function. By simulating the 3D, contour, and 2D graphs of these obtained solitons, we scrutinized the behavior of the waves affecting the nonlinear terms. The figures show that the solitary waves obtained from LWE are efficient in analyzing electromagnetic wave signals in the cable lines, and the solitary waves from SNNVS are essential in any stochastic system like a sound wave. Moreover, by taking some values of the parameters, we found some interesting soliton shapes, such as compaction soliton, singular periodic solution, bell-shaped soliton, anti-kink-shaped soliton, one-sided kink-shaped soliton, and some flat kink-shaped solitons, etc. This article will have a great impact on nonlinear science due to the new solitary wave structures with different complex phenomena, sensitivity analysis, and bifurcation analysis. [ABSTRACT FROM AUTHOR]

Ca 2 + is a ubiquitous signaling mechanism across different cell types. In T-cells, it is associated with cytokine production and immune function. Benson et al. have shown the coexistence of competing Ca 2 + oscillations during antigen stimulation of T-cell receptors, depending on the presence of extracellular Ca 2 + influx through the Ca 2 + release-activated Ca 2 + channel (Benson in J Biol Chem 29:105310, 2023). In this paper, we construct a mathematical model consisting of five ordinary differential equations and analyze the relationship between the competing oscillatory mechanisms.. We perform bifurcation analysis on two versions of our model, corresponding to the two oscillatory types, to find the defining characteristics of these two families. [ABSTRACT FROM AUTHOR]

Recent developments of eco-evolutionary models have shown that evolving feedbacks between behavioral strategies and the environment of game interactions, leading to changes in the underlying payoff matrix, can impact the underlying population dynamics in various manners. We propose and analyze an eco-evolutionary game dynamics model on a network with two communities such that players interact with other players in the same community and those in the opposite community at different rates. In our model, we consider two-person matrix games with pairwise interactions occurring on individual edges and assume that the environmental state depends on edges rather than on nodes or being globally shared in the population. We analytically determine the equilibria and their stability under a symmetric population structure assumption, and we also numerically study the replicator dynamics of the general model. The model shows rich dynamical behavior, such as multiple transcritical bifurcations, multistability, and anti-synchronous oscillations. Our work offers insights into understanding how the presence of community structure impacts the eco-evolutionary dynamics within and between niches. [ABSTRACT FROM AUTHOR]

Inertial focusing in curved microfluidic ducts exploits the interaction of the drag force from the Dean flow with the inertial lift force to separate particles or cells laterally across the cross-section width according to their size. Experimental work has identified that using a trapezoidal cross section, as opposed to a rectangular one, can enhance the sized based separation of particles/cells over a wide range of flow rates. Using our model, derived by carefully examining the way the Dean drag and inertial lift forces interact at low flow rates, we calculate the leading order approximation of these forces for a range of trapezoidal ducts, both vertically symmetric and nonsymmetric, with an increasing amount of skew towards the outside wall. We then conduct a systematic study to examine the bifurcations in the particle equilbira that occur with respect to a shape parameter characterizing the trapezoidal cross section. We reveal how the dynamics associated with particle migration are modified by the degree of skew in the cross-section shape, and show the existence of cusp bifurcations (with the bend radius as a second parameter). Additionally, our investigation suggests an optimal amount of skew for the trapezoidal cross section for the purposes of maximizing particle separation over a wide range of bend radii. [ABSTRACT FROM AUTHOR]

This investigation discusses the modified M-truncated form of the perturbed Chen–Lee–Liu (pCLL) dynamical equation. The pCLL equation is a generalization of the original CLL equation, which describes the propagation of optical solitons in optical fibers. The pCLL equation includes additional terms that account for various influences such as chromatic dispersion, nonlinear dispersion, inter-modal dispersion, and self-steepening. A new version of the generalized exponential rational function method is utilized to obtain multifarious types of soliton solutions. Moreover, the planar dynamical system of the concerned equation is created using a Hamiltonian transformation, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. Numerical simulations of the solutions are added to the analytical results to better understand the dynamic behavior of these solutions. The study's findings could be extremely useful in solving additional nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

A system of partial differential equations is developed to study the spreading of tau pathology in the brain for Alzheimer’s and other neurodegenerative diseases. Two cases are considered with one assuming intracellular diffusion through synaptic activities or the nanotubes that connect the adjacent cells. The other, in addition to intracellular spreading, takes into account of the secretion of the tau species which are able to diffuse, move with the interstitial fluid flow and subsequently taken up by the surrounding cells providing an alternative pathway for disease spreading. Cross membrane transport of the tau species are considered enabling us to examine the role of extracellular clearance of tau protein on the disease status. Bifurcation analysis is carried out for the steady states of the spatially homogeneous system yielding the results that fast cross-membrane transport combined with effective extracellular clearance is key to maintain the brain’s healthy status. Numerical simulations of the first case exhibit solutions of travelling wave form describing the gradual outward spreading of the pathology; whereas the second case shows faster spreading with the buildup of neurofibrillary tangles quickly elevated throughout. Our investigation thus indicates that the gradual progression of the intracellular spreading case is more consistent with the clinical observations of the development of Alzheimer’s disease. [ABSTRACT FROM AUTHOR]

Contracting cancer typically induces a state of terror among the individuals who are affected. Exploring how chemotherapy and anxiety work together to affect the speed at which cancer cells multiply and the immune system's response model is necessary to come up with ways to stop the spread of cancer. This paper proposes a mathematical model to investigate the impact of psychological scare and chemotherapy on the interaction of cancer and immunity. The proposed model is accurately described. The focus of the model's dynamic analysis is to identify the potential equilibrium locations. According to the analysis, it is possible to establish three equilibrium positions. The stability analysis reveals that all equilibrium points consistently exhibit stability under the defined conditions. The bifurcations occurring at the equilibrium sites are derived. Specifically, we obtained transcritical, pitchfork, and saddle-node bifurcation. Numerical simulations are employed to validate the theoretical study and ascertain the minimum therapy dosage necessary for eradicating cancer in the presence of psychological distress, thereby mitigating harm to patients. Fear could be a significant contributor to the spread of tumors and weakness of immune functionality. [ABSTRACT FROM AUTHOR]

The increasing complexity and integration of systems present challenges in understanding and managing these systems, as highlighted by the INCOSE Systems Engineering Vision for 2035. This complexity risks system failures due to insufficient understanding. Resilience Engineering is a field that focuses on system behavior in the face of disruptions. However, existing methods to measure resilience, such as probability‐based measures and linear recovery models, are limited by the non‐linear and dynamic nature of modern systems. Bifurcation Analysis, a mathematical system dynamics technique, offers a different perspective by examining how systems behave under changing conditions. This approach, developed in ecology and traditionally applied in fields including power systems and neural networks, can provide insights into the resilient characteristics of non‐linear systems. Despite its promise, Bifurcation Analysis is not usually associated with Resilience Engineering. This paper proposes using Bifurcation Analysis to understand system resilience better and support Resilience Engineering practitioners. It aims to bridge the gap between the two fields, offering a framework for integrating both approaches. This framework is then applied to a model system to demonstrate its potential to enhance the understanding of system resilience. Contributions are an overview of current Resilience Engineering and Bifurcation Analysis, a general‐use framework for practitioners, and an application of this framework. Future work will focus on applying the new framework to more cases, allowing for improved Resilience Engineering, and improving the framework to make it more accessible, versatile, and reliable. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show that Tilman’s graphical method for the study of competition between two species for two resources can be advantageously used for the study of commensalism or syntrophy models, where a first species produces the substrate necessary for the growth of the second species. The growth functions of the species considered are general and include both inhibition by the other substrate and inhibition by the species’ limiting substrate, when it is at a high concentration. Because of their importance in microbial ecology, models of commensalism and syntrophy, with or without self-inhibition, have been the subject of numerous studies in the literature. We obtain a unified presentation of a large number of these results from the literature. The mathematical model considered is a differential system in four dimensions. We give a new result of local stability of the positive equilibrium, which has only been obtained in the literature in the case where the removal rates of the species are identical to the dilution rate and the study of stability can be reduced to that of a system in two dimensions. We describe the operating diagram of the system: this is the bifurcation diagram which gives the asymptotic behavior of the system when the operating parameters are varied, i.e., the dilution rate and the substrate inlet concentrations. [ABSTRACT FROM AUTHOR]

This work studies the behavior of electrical signals in resonant tunneling diodes through the application of the Lonngren wave equation. Utilizing the method of Lie symmetries, we have identified optimal systems and found symmetry reductions; we have also found soliton wave solutions by applying the tanh technique. The bifurcation and Galilean transformation are found to determine the model implications and convert the system into a planar dynamical system. In this experiment, the equilibrium state, sensitivity, and chaos are investigated and numerical simulations are conducted to show how the frequency and amplitude of alterations affect the system. Furthermore, local conservation rules are demonstrated in more detail to unveil the whole system of movements. [ABSTRACT FROM AUTHOR]

This study aims to investigate a diffusive predator–prey system incorporating additional food for predators, prey refuge, fear effect, and its carry-over effects. For the temporal model, the well-posedness and persistence of the system have been discussed. We investigated the existence and the stability behavior of the various equilibria. Furthermore, we explored the bifurcations of codimension-1 including transcritical, saddle-node, and Hopf, concerning the crucial parameters. The system also presents codimension-2 bifurcations such as Bogdanov–Takens and cusp bifurcation along with the global homoclinic bifurcation. We observed the bubbling phenomena, which illustrate the fluctuations in the amplitudes of the periodic oscillations. For the spatiotemporal system, we established the non-negativity and boundedness of the solutions. We derived the conditions for the diffusion-driven instabilities in a confined region with Neumann boundary conditions. Extensive numerical simulations have been conducted to depict the various stationary patterns in Turing space. It is observed that incorporating cross-diffusion divides the bi-parametric plane into various sub-regions and dynamic patterns are analyzed in these different regions. The intricate spatiotemporal dynamics exhibited by prey–predator interactions are crucial for unraveling the intricacies within ecological systems. [ABSTRACT FROM AUTHOR]

This work delves into the behavior of cold bosonic atoms within a zigzag optical lattice which offers a rich platform for studying quantum many‐body physics and has potential applications in various fields including quantum simulation, quantum computing, and precision measurement. The study aims to derive exact solutions and explore qualitative properties through dynamical analysis. Exact solutions of the nonlinear differential equation model can provide insights into the behavior of bosonic atoms in complex optical lattice configurations, allowing researchers to simulate and study phenomena such as quantum phase transitions, many‐body localization, and exotic states of matter. The invariant analysis has been performed for the first time on the specified equation, yielding a reduced system of equations with forms that are easier to handle. Novel techniques are applied to extract solutions from the reduced system of equations obtained via invariant analysis. The results reveal a rich set of solutions, including various traveling wave solutions and doubly periodic functions in the form of Jacobian elliptic functions. Bifurcation analysis, conducted through phase portraits, provides insights into the system's long‐term behavior under different parameter values. This work contributes to a deeper understanding of the dynamics of cold bosonic atoms in optical lattices via 2D and 3D plots of obtained solutions depicting the change in the behavior of soliton solutions with fractional derivative parameter. [ABSTRACT FROM AUTHOR]