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We carry out qualitative analysis of a fourth-order tumor growth control model using ordinary differential equations. We show that the system has one positive equilibrium point, and its stability is independent of the feedback gain. Using a Lyapunov function method, we prove that there exist realistic parameter values for which the systems admit limit cycle oscillations due to a supercritical Hopf bifurcation. The time evolution of the state variables is also represented. [ABSTRACT FROM AUTHOR]

Here, we investigate the impact of space-time reverse (STR) problems, a result of parity-time symmetry in optics and quantum mechanics, on soliton propagation in optical fibers. The STR problems are characterized by the existence of a field and its reverse. The research introduces a new classification of two scenarios: non-interactive and interactive fields and reverse fields. The solutions for the generalized Fokas–Lenells equation (gFLE) with STR and third-order dispersion are derived. To tackle this, adaptive transformations for the field and its reverse are introduced, employing a unified method. In the non-interactive scenario, both exact and approximate solutions are found. However, in the interactive case, only exact solutions are discovered. This work reveals that the presence of the field and its reverse unveils new soliton structures, including bright–dark envelope solitons and right and left envelope-solitons. In the non-interactive case, the field displays a right envelope-soliton, while the reverse field exhibits a left envelope-soliton (or vice versa). The study hypothesizes that the presence of a reverse field might impede soliton propagation in optical fibers.The research also includes an analysis of modulation instability (MI), determining that MI is initiated when the coefficient of Raman scattering exceeds a critical value. Furthermore, the study examines the modulated wave gain and explores global bifurcation through phase portrait by constructing the Hamiltonian function. [ABSTRACT FROM AUTHOR]

A periodic boundary-value problem for an equation with a deviating spatial argument is considered. Using the Poincaré–Dulac method of normal forms, the method of integral manifolds, and asymptotic formulas, we examine a number of bifurcation problems of codimension 1 and 2. For homogeneous equilibrium states, we analyze possibilities of implementing critical cases of various types. The problem on the stability of homogeneous equilibrium states is studied and asymptotic formulas for spatially inhomogeneous solutions and conditions for their stability are obtained. [ABSTRACT FROM AUTHOR]

The biped robot has two identical legs that undergo alternating stance and swing phases during walking. The robot's motion is facilitated by actuators positioned at the ankle and hip joints. The paper investigates various fault scenarios in dynamic walker models, specifically situations where one or both actuators experience failure while the robot is in motion. The first model assumes faults in both the hip and ankle joints, while the second model assumes a fault only in the ankle joint. The dynamic walker typically undergoes two steps in each cycle. This study specifically examines the different types of faults that may occur during the second step. The goal is to identify the potential walking gaits that can result from these faults. It is observed that walking with faults at both the hip and ankle joints exhibited greater stability and a wider range of stable states compared to walking with a fault only at the ankle joint. This suggests that incorporating hip joint faults or underactuation into robotic or prosthetic gait patterns could improve their stability and overall performance. The local and global stability of the dynamic walker were analyzed using Poincare's method and the basin of attraction plot, respectively. Additionally, a control algorithm based on kinetic energy shaping is proposed for the second model, which resulted in an increased basin of attraction. Notably, the second model exhibited the ability to walk with variable step length and speed, akin to human locomotion. [ABSTRACT FROM AUTHOR]

The torsional vibration phenomenon of the crankshaft system can trigger faults such as burned tile and flywheel deformation. The coupling is an important structure for regulating the torsional vibration of a shaft system. Most of the literature has only studied the effects of coupling misalignment, damping nonlinearity, and excitation nonlinearity on the torsional response of the system, while ignoring the vibration bifurcation caused by the coupling structure and fit parameters. To address the above limitations, this paper establishes a torsional vibration mechanics model of crankshaft, coupling, and rotor based on Lagrange dynamics. Mathematical models of coupling inclination angle, excess, and torsional stiffness are constructed using the finite element method. The effects of coupling inclination angle, excess, and angular velocity on the vibration bifurcation of the shaft system are discussed separately by the 4th-order Runge–Kutta methods. The results show that with an increase in the inclination angle, the vibration state of the shaft system switches between chaos and divergence, which is not conducive to the torsional vibration control of the shaft system. At the same time, the increase in the excess will lead to an increase in the angular velocity of vibration, which in turn will lead to increased wear of the shaft system and bearings. In addition, after considering the nonlinear characteristics of the coupling stiffness, the vibration amplitude of the shaft system rises with the increase in angular velocity, and the vibration state transforms from periodic to divergent. Finally, a coupling field vibration test study was carried out to verify the accuracy of the numerical model. The research results of this paper have a theoretical reference value for determining the coupling structure-fit parameters and the angular velocity of the shaft system. [ABSTRACT FROM AUTHOR]

A harmonically excited oscillator with symmetric constraints is considered and the constraints are assumed to be rigid. To calculate the coexisting periodic motions of the impact oscillator and carry out their stability and bifurcation analysis, the smooth flow maps and impact maps are constructed and the calculation methods of their Jacobi matrices are presented. The Jacobi matrix of global Poincaré map for various types of periodic motions can be obtained according to the chain rule of compound map. The two-parameter transition characteristics between 1–p–pS and 1–(p + 1)–(p + 1)S orbits and the global dynamics in beat motion and hysteresis regions are investigated by combining shooting, continuation and cell mapping approaches as well as numerical simulation. The periodic saddles are computed and traced to help explain the evolutions of periodic and chaotic motions. The grazing bifurcation of 1–p–p orbit not only creates U1–(p + 1)–(p + 1)S saddle but also can give birth to Un–(np + 1)–(np + 1)S saddles as well as a pair of anti-symmetric n–(np + 1)–np and n–np–(np + 1) saddles, and thereby forming the beat motion regions in the transition from 1–p–pS orbit to 1–(p + 1)–(p + 1)S orbit. The collision between the stable attractor and periodic saddle can give rise to rich dynamical behaviors, such as grazing and saddle-node bifurcations as well as interior, boundary and attractor merging crises. The unique characteristics of chaotic crises in impact oscillator with symmetric rigid constraints are revealed. A comparative analysis of two-parameter dynamical characteristics in the impact oscillators with symmetric rigid and elastic constraints is carried out. The pitchfork bifurcation can exhibit subcritical characteristics since it is followed very closely by a saddle-node bifurcation. This type of pitchfork bifurcation is defined as SN-type pitchfork bifurcation, which will further enrich the dynamics of non-smooth systems. [ABSTRACT FROM AUTHOR]

Predator foraging facilitation or hunting cooperation and the antipredator behavior of prey are essential mechanisms in evolutionary biology and ecology and may strongly influence the predator–prey dynamics. In a real-world scenario, this behavioral tendency is well documented, but less is known about how it could affect the dynamics between predator and prey. Here, we investigate the impact of the fear of predator on prey and the hunting cooperation in predator on the predator–prey dynamics, where the predator is assumed to be of generalist type. We observe that without fear, even with the high level of hunting cooperation, both populations may coexist, though the increasing level of hunting cooperation reduces the prey density at coexistence equilibrium. Moreover, increasing level of fear also destabilizes the system with and without hunting cooperation. Further, in the presence of hunting cooperation and fear effect, the model shows three different types of bistability phenomena: bistability between two coexisting equilibria, bistability between coexisting equilibria and prey-free equilibrium, and bistability between stable limit cycle and coexisting equilibria. In addition, saddle-node, Hopf, transcritical bifurcation of codimension one, Bautin (generalized Hopf), Bogdanov–Takens, and cusp bifurcation of codimension two are observed. [ABSTRACT FROM AUTHOR]

This paper investigates a two-species amensalism model that includes the fear effect on the first species and the Beddington–DeAngelis functional response. The existence and stability of possible equilibria are investigated. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, global dynamics analysis of the model is performed. It is observed that under certain parameter conditions, when the intensity of the fear effect is below a certain threshold value, as the fear effect increases it will only reduce the density of the first species population and will have no influence the extinction or existence of the first species population. However, when the fear effect exceeds this threshold, the increase of the fear effect will accelerate the extinction of the first species population. Finally, numerical simulations are performed to validate theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a Holling-III predator–prey model considering the perturbation of slow-varying, carrying capacity parameters. The study aims to address how the slow changes in carrying capacity influence the dynamics of the model. Based on the bifurcation theory and the slow–fast analysis method, the existence and the equilibrium of the autonomous system are explored, and then, the critical condition of Hopf bifurcation and transcritical bifurcation is established for the autonomous system. The slow–fast coupled nonautonomous system has quasiperiodic oscillations, single Hopf bursting oscillations, and transcritical–Hopf bursting oscillations within a certain range of perturbation amplitude variation if the carrying capacity perturbation amplitude crosses some critical values, such that the predator–prey management is challenging for the extinction of predator populations under the critical value. The motion pattern of the nonautonomous system is closely related to the transcritical bifurcation, Hopf bifurcation and attractor type of the autonomous system. Finally, the effects of changes in parameters related to predator aggressiveness on system behavior are investigated. These results show how crucial the predator–prey control is for varying carrying capacities. [ABSTRACT FROM AUTHOR]

Background: Coronary bifurcation lesions are commonly encountered during coronary angiography. The management of bifurcation lesions remains challenging, with various bifurcation techniques being available and outcomes varying depending on the Medina classification and operator experience. Methods: We present a short case series and the outcomes of a new bifurcation technique for the management of simple Medina '0,0,1' and '0,0,1' bifurcation lesions using the kissing balloon-stent technique (kissing BS). Results: We retrospectively identified 8 patients who underwent bifurcation stenting using the kissing Balloon-Stent technique, along with their clinical and angiographic follow-up outcomes. We also describe the benefits and limitations of the technique, delineate the potential mechanisms of target lesion failure, and outline appropriate patient selection. Conclusions: Kissing Balloon-Stent technique is a simple single stent technique that is safe and feasible in select patients with Medina '0,0,1' and '0,0,1' bifurcation lesions. [ABSTRACT FROM AUTHOR]

In this article, a partial differential equation (PDE) model for atherosclerosis disease is presented that analyzes the anti-oxidative activity of high-density lipoprotein (HDL) during the reverse cholesterol transport (RCT) process. The model thoroughly investigates the complex interplay between oxidized low-density lipoprotein (ox-LDL) and high-density lipoprotein in the context of atherosclerosis, emphasizing their combined impact on plaque formation, disease progression, and regression. In addition to this, we considered that monocytes are also attracted by the presence of ox-LDL within the intima. Detailed discussions on stability analyses of the reaction dynamical system at non-inflammatory and chronic equilibrium are provided, followed by a bifurcation analysis for the proposed system. Furthermore, stability analysis for the PDE model in the presence of diffusion is conducted. Our study reveals that the oxidation rate of LDL by monocytes (δ) and the influx rate of HDL (ϕ) due to drugs/diet are primarily responsible for the existence of bi-stability of equilibrium points. In the numerical results, we observe that non-inflammatory or chronic equilibrium points exist for either a short or a long time, and these findings are validated with existing results. The biological elucidation shows the novelty in terms of enhancing our ability to assess intervention efficacy to generate therapeutic strategies resulting in the reduction of the atherosclerotic burden and associated cardiovascular risks. [ABSTRACT FROM AUTHOR]

Within the confines of a two-fluid quantum magnetohydrodynamic model, the investigation of magnetoacoustic shock and solitary waves is conducted in an electron-ion magnetoplasma that considers electrons of spin 1/2. When the plasma system is nonlinearly investigated using the reductive perturbation approach, the Korteweg de Vries-Burgers (KdVB) equation is produced. Sagdeev’s potential is created, revealing the presence of solitary solutions. However, when dissipative terms are included, intriguing physical solutions can be obtained. The KdVB equation is further investigated using the phase plane theory of a planar dynamical system to demonstrate the existence of periodic and solitary wave solutions. Predicting several classes of traveling wave solutions is advantageous due to various phase orbits, which manifest as soliton-shock waves, and oscillatory shock waves. The presence of a magnetic field, the density of electrons and ions, and the kinematic viscosity significantly alter the properties of magnetoacoustic solitary and shock waves. Additionally, electric fields have been identified. The outcomes obtained here can be applied to studying the nature of magnetoacoustic waves that are observed in compact astrophysical environments, where the influence of quantum spin phenomena remains significant, and also in controlled laboratory plasma experiments. [ABSTRACT FROM AUTHOR]

Aiming at the nonlinear vibration problem caused by the dynamic misalignment of rolling bearings in the rotating machinery, a rotor dynamic model with bearing dynamic misalignment is established, and the effects of the dynamic misalignment of single and two bearings and the phase relationship of two dynamic misalignment bearings on the rotor vibration characteristics are researched. The effects of bearing clearance, raceway curvature radius, the ball number and radial load on the nonlinear vibration response of the dynamic misalignment bearing-unbalance rotor system are further analyzed. The results show that the dynamic misalignment of the bearing can excite the vibration at twice the rotor frequency, raise the resonance speed and enhance the hardening-type nonlinearity characteristics of the rotor. The influence of dynamic misalignment under two bearings with out-of-phase on rotor resonance is greater than that of the in-phase. The stability of the system can be improved by decreasing the bearing clearance and increasing the curvature radius coefficient. The vibration caused by bearing dynamic misalignment can be reduced by selecting the bearing with more balls and appropriately increasing the bearing load. [ABSTRACT FROM AUTHOR]

Design load and vibration for parked conditions are gaining in importance for large-scale modern wind turbines with increasing flexibility, especially edgewise vibration when the blade is at a high angle of attack. In this work, flow-induced vibration of the wind turbine airfoil at 90 degrees of attack angle is studied with the fluid-structure interaction (FSI) simulation. The unsteady aerodynamic force due to flow separation and vortex shedding at the high angle of attack causes the chordwise vibration of the airfoil. When the vortex shedding frequency f v gets close to the chordwise natural frequency f n of the airfoil, vortex-induced vibration (VIV) of high amplitude occurs accompanied with the frequency lock-in phenomenon. In the post lock-in regime, it is found that period-3 and torus bifurcation occur successively and the vibration response becomes aperiodic. Dynamic mode decomposition(DMD) technique is used to investigate the mechanism of bifurcation from the perspective of energy balance, through analyzing the vorticity field in the wake and pressure distribution on the airfoil surface. For the certain incoming velocity in the post lock-in regime, since the frequency of the DMD mode f = 2 f v / 3 is close to the natural frequency f n , both the vibration of frequency 2 f v / 3 and f v get excited, leading to the onset of bifurcation. The Lissajou curves are obtained through reconstructing the transient pressure of each DMD mode, which indicates that energy transfer mainly exists in modes f = f v . In addition, the reconstructed Lissajou curves based on the leading DMD modes agree well with the original time-domain Lissajou curves. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviors of a discrete-time fractional-order population model are considered. The stability analysis and the topological classification of the model at the fixed point have been investigated. It is shown that the model undergoes flip and Neimark–Sacker bifurcations around the co-existence fixed point by using the bifurcation and the normal form theory. These bifurcations lead to chaos when the parameter changes at critical point. In order to control chaotic behavior in the model result from Neimark–Sacker bifurcation, the OGY feedback method has been used. Furthermore, some numerical simulations, including bifurcation diagrams, phase portraits and maximum Lyapunov exponents of the presented model are plotted to support the correctness of the analytical results. The positive Lyapunov exponents demonstrate that chaotic behavior exists in the considered model. [ABSTRACT FROM AUTHOR]

The paper discusses the nutrient-plankton system with effect of time delay in nutrient recycling and toxin-determined function response (TDFR). The designed model system explores the delay-induced system dynamics. We present the local stability analysis of interior equilibrium points in absence as well as in presence of time delay. Further, the direction of Hopf bifurcation is obtained. We perform the numerical computation and observe that time delay in nutrient recycling can generate the periodic solution in a stable nutrient-plankton system. Some other essential parameters, such as input concentration of nutrients and natural removal rate of nutrients, also regulate the dynamical system. The system shows Hopf and double-Hopf bifurcation in the presence of time delay. Our study shows that the delay in the nutrient recycling causes instability transition phenomenon. The delay-induced nutrient recycling and different input concentrations of nutrients can regulate the estuarine system. Finally, the stability switching is observed for delayed system. [ABSTRACT FROM AUTHOR]

We have combined cooperative hunting, inspired by recent experimental studies on birds and vertebrates, to develop a predator-prey model in which the fear effect simultaneously influences the birth and mortality rates of the prey. This differs significantly from the fear effect described by most scholars. We have made a comprehensive analysis of the dynamics of the model and obtained some new conclusions. The results indicate that both fear and cooperative hunting can be a stable or unstable force in the system. The fear can increase the density of the prey, which is different from the results of all previous scholars, and is a new discovery in our study of the fear effect. Another new finding is that fear has an opposite effect on the densities of two species, which is different from the results of most other scholars in that fear synchronously reduces the densities of both species. Numerical simulations have also revealed that the fear effect extends the time required for the population to reach its survival state and accelerates the process of population extinction. [ABSTRACT FROM AUTHOR]

This work develops the dynamic behaviour of the two-species model in the context of phytoplankton and zooplankton interactions with the consequences of global warming. Without zooplankton and without global warming, phytoplankton is thought to be growing logistically. Growing global warming is considered to be causing phytoplankton growth to decline. It is proposed that the eating of phytoplankton causes an increase in zooplankton. The combination of rising water temperatures and natural mortality is considered to have reduced the amount of zooplankton. It is speculated that a variety of natural and human activities contribute to the slow growth of global warming. Along with the natural rate at which global warming decays, a rise in phytoplankton is thought to be another factor contributing to a potential reduction in global warming. Various potential equilibrium points of the model have been found. Furthermore, the model's stability is examined in close proximity to every equilibrium point. It can be seen from the numerical simulation findings that, even in the presence of global warming, phytoplankton and zooplankton can coexist in the ecological system. Unpredictable or unstable behaviour of the model is attributed to how rising global temperatures influence phytoplankton growth rates. The increased absorption of atmospheric CO2 by phytoplankton for photosynthesis might enable the ecological system to maintain its stable behaviour. It has been noted that the biological system may become unstable due to the rise in zooplankton conservation rates. It is seen that the increase in water temperature or global warming may make the system unstable. Increasing rate of consumption of phytoplankton by zooplankton may show the oscillatory or unstable dynamics of the ecosystem due to global warming. Lastly, it is evident that the planktonic ecosystem may become unstable because of the continuous rise in global warming. [ABSTRACT FROM AUTHOR]

Following a change in the Danish gang milieu in 2008, where ethnic minority street gangs challenged the established outlaw motorcycle gangs, the Danish government has formulated three anti-gang policy 'packages'. To unfold the development they represent to Nordic penology, this article analyses elements of both penal exceptionalism and excess. In this article, it is shown how the packages are based on the notion of gang membership as a choice, which legitimated the development of a parallel justice system for gang members. This foundation is built upon a gang-specific subsection that allows for the doubling of gang-related sentences and for restricting prisoner rights and traditional rehabilitative treatment for gang-related convicts. The packages, however, maintained 'a way out' for gang members who voluntarily entered a formal EXIT program, and thus gained access to traditional penal treatment and also support for leaving the gang milieu. It is argued that the packages represent a development of intended bifurcation based on status differentiation between citizen groups, a process also observed in regard to Danish anti-ghetto policies. Thus, rather than resembling a general turn to punitiveness, the packages indicate a penological development based on penal differentiation, which raises questions about access to justice for those found wanting. [ABSTRACT FROM AUTHOR]

The Sasa-Satsuma (SS) equation is studied in this research study using ideas from planar dynamical theory and the beta differential operator. The SS equation is converted into two ordinary differential equations by applying the Galilean transformation. The work is since concentrated on examining the system's bifurcation points and equilibrium points. The sensitivity of the linked system to its initial values is demonstrated via graphical representations. In order to examine chaos and phase transitions, the system is changed by adding the periodic function cos (ω t) . This modification is done as part of this study. Specific optical soliton solutions are illustrated using the first integral technique. Additionally, for various combinations of frequency and amplitude values, numerical simulations are demonstrated the existence of unusual chaotic attractors, such as candy-type, torus-type, and multiscroll chaotic structures. The impact of the beta differential operator on the amplitude of various optical solitons, such as bright, dark, W-shaped, and breather solitons, are also studied. [ABSTRACT FROM AUTHOR]

In this paper, we study the polynomial Liénard differential system of arbitrary degree: x ˙ = y , y ˙ = a 1 x + a 2 x 2 m + (a 3 + a 4 x 2 n) y applying to a Duffing–Van der Pol oscillator. We prove that it has abundant dynamics, such as the generalized transcritical bifurcation, Hopf bifurcation, heteroclinic bifurcation, homoclinic bifurcation and double limit cycle bifurcation. The associated global bifurcation diagram in the parameter space and the corresponding global phase portraits in the Poincaré disc are presented. These analytic results are also demonstrated by concrete examples via numerical simulations. [ABSTRACT FROM AUTHOR]

We are concerned with the existence of nodal solutions for a third order boundary value problem u ′ ′ ′ (x) = g (u (x)) + p (x , u (x) , u ′ (x) , u ′ ′ (x)) , x ∈ (0 , 1) , u (0) = u (1) = u ′ (1) = 0 , where g : R → R is continuous and satisfies lim | ξ | → ∞ g (ξ) / ξ = ∞ (g is superlinear as | ξ | → ∞ ), p : [ 0 , 1 ] × R 3 → R is continuous and satisfies | p (x , ξ 0 , ξ 1 , ξ 2) | ≤ C + | ξ 0 | / 3 , x ∈ [ 0 , 1 ] , (ξ 0 , ξ 1 , ξ 2) ∈ R 3 , for some C > 0 . We obtain infinitely many solutions having specified nodal properties. The proof of our main result is based upon bifurcation techniques. [ABSTRACT FROM AUTHOR]

The aim of this study is to analyze and investigate the COVID-19 transmission with effect of symptomatic and asymptomatic in the community. Mathematical model is converted into fractional order with the help of fractal fractional definition. The proposed fractional order system is investigated qualitatively as well as quantitatively to identify its stable position. Local stability of the COVID-19 system is verified and test the system is tested with flip bifurcation. Also the system is investigated for global stability using Lyapunov first and second derivative functions to see its rate of spread after recovery. The existence, boundedness and positivity of the COVID-19 are checked which are the key properties for such of type of epidemic problem to identify reliable findings. Effect of global derivative is demonstrated to verify its rate of effects according to their sub compartments to identify in which rate the symptomatic and asymptomatic transmission occurs. Solutions for fractional order system are derived with the help of advanced tool fractal fractional operator with generalized mittag-leffler kernel for different fractional values. Simulations are carried out to see symptomatic as well as asymptomatic effects of COVID-19 in the worldwide using MATLAB Coding. They show the actual behavior of COVID-19 especially for asymptomatic measures which will be helpful in early detection, also which will be helpful to understand the outbreak of COVID-19 as well as for future prediction and better control strategies. [ABSTRACT FROM AUTHOR]

In this paper, we apply Lin's method to study the existence of heteroclinic orbits near the degenerate heteroclinic chain under m$$ m $$‐dimensional periodic perturbations. The heteroclinic chain consists of two degenerate heteroclinic orbits γ1$$ {\gamma}_1 $$ and γ2$$ {\gamma}_2 $$ connected by three hyperbolic saddle points q1,q2,q3$$ {q}_1,{q}_2,{q}_3 $$. Assume that the degeneracy of the unperturbed heteroclinic orbit γi$$ {\gamma}_i $$ is ni$$ {n}_i $$, the splitting index is δi$$ {\delta}_i $$. By applying Lin's method, we construct heteroclinic orbits connected q1$$ {q}_1 $$ and q3$$ {q}_3 $$ near the unperturbed heteroclinic chain. The existence of these orbits is equivalent to finding zeros of the corresponding bifurcation function. The lower order terms of the bifurcation function is the map from ℝn1+n2+m$$ {\mathrm{\mathbb{R}}}^{n_1+{n}_2+m} $$ to ℝn1+n2+δ1+δ2$$ {\mathrm{\mathbb{R}}}^{n_1+{n}_2+{\delta}_1+{\delta}_2} $$. Using the contraction mapping principle, we provide a detailed analysis on how zeros can exist based on different cases of splitting indices δ1$$ {\delta}_1 $$, δ2$$ {\delta}_2 $$ and then obtain the existence of the heteroclinic orbits which backward asymptotic to q1$$ {q}_1 $$ and forward asymptotic to q3$$ {q}_3 $$. [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]

Using experiments and a depth-averaged numerical model, we study instabilities of two-phase flows in a Hele-Shaw channel with an elastic upper boundary and a non-uniform cross-section prescribed by initial collapse. Experimentally, we find increasingly complex and unsteady modes of air-finger propagation as the dimensionless bubble speed $Ca$ and level of collapse are increased, including pointed fingers, indented fingers and the feathered modes first identified by Cuttle et al. (J. Fluid Mech. , vol. 886, 2020, A20). By introducing a measure of the viscous contribution to finger propagation, we identify a $Ca$ threshold beyond which viscous forces are superseded by elastic effects. Quantitative prediction of this transition between 'viscous' and 'elastic' reopening regimes across levels of collapse establishes the fidelity of the numerical model. In the viscous regime, we recover the non-monotonic dependence on $Ca$ of the finger pressure, which is characteristic of benchtop models of airway reopening. To explore the elastic regime numerically, we extend the depth-averaged model introduced by Fontana et al. (J. Fluid Mech. , vol. 916, 2021, A27) to include an artificial disjoining pressure that prevents the unphysical self-intersection of the interface. Using time simulations, we capture for the first time the majority of experimental finger dynamics, including feathered modes. We show that these disordered states evolve continually, with no evidence of convergence to steady or periodic states. We find that the steady bifurcation structure satisfactorily predicts the bubble pressure as a function of $Ca$ , but that it does not provide sufficient information to predict the transition to unsteady dynamics that appears strongly nonlinear. [ABSTRACT FROM AUTHOR]

Mixed-mode oscillations (abbreviated as MMOs) belong to a typical kind of fast/slow dynamical behavior, and how to investigate the mechanism is an important problem in nonlinear dynamics. In this paper, we explore the MMOs induced by the bifurcation delay phenomenon and twist of the trajectories in space based on a coupled system consisting of a Van der Pol system and a Duffing oscillator with two potential wells. Regarding the low-frequency external excitation as a generalized state variable, we obtain the traditional fast and slow subsystems. Appling the equilibrium analysis and bifurcation theory, the stability critical conditions of the equilibrium and the generation conditions of fold and Hopf bifurcation are also presented. To analyze the critical conditions clearly, the two-parameter bifurcation and one-parameter bifurcation diagrams are performed by using numerical simulation method. The bifurcation characteristics are studied, especially the effects of parameter δ on the bifurcation structures. We find that the fast subsystem performs different dynamical behaviors such as fold bifurcation of limit cycles, period-doubling bifurcations, inverse-period-doubling bifurcations and chaos, when parameter δ is taken at different values. By using phase diagrams, time series, maximum Lyapunov exponent diagrams, three-dimensional phase diagrams and superimposed diagrams, the mechanisms of the MMOs are investigated numerically in detail. The Hopf bifurcation delay can lead the trajectories to arrive at the vector fields of the equilibrium point and limit cycles. In addition, the chaotic behaviors can be found on the route of period doubling, which lead to the chaotic spiking-state-oscillations types. Our findings are helpful to understand the generation of the MMOs and intensify the understanding of some special dynamical behaviors on the MMOs. [ABSTRACT FROM AUTHOR]

In the present paper, the sloshing flow in a cuboid tank forced to oscillate horizontally is investigated with both experimental and numerical approaches. The filling depth chosen is $h/L=0.35$ (with h the water depth and L the tank height), which is close to the critical depth. According to Tadjbakhsh & Keller (J. Fluid Mech. , vol. 8, issue 3, 1960, pp. 442–451), as the depth passes through this critical value the response of the resonant sloshing dynamics changes from 'hard spring' to 'soft spring'. The experimental tank has a thickness of $0.1L$ , reducing three-dimensional effects. High-resolution digital camera and capacitance wave probes are used for time recording of the surface elevation. By varying the oscillation period and the amplitude of the motion imposed on the tank, different scenarios are identified in terms of free-surface evolution. Periodic and quasi-periodic regimes are found in most of the frequencies analysed but, among these, sub-harmonic regimes are also identified. Chaotic energetic regimes are found with motions of greater amplitude. Typical tools of dynamical systems, such as Fourier spectra and phase maps, are used for the regime identification, while the Hilbert–Huang transform is used for further insight into doubling-frequency and tripling-period bifurcations. For the numerical investigation, an advanced and well-established smoothed particle hydrodynamics method is used to aid the understanding of the physical phenomena involved and to extend the range of frequencies investigated experimentally. [ABSTRACT FROM AUTHOR]

Solid particles trapped in an acoustic standing wave have been observed to undergo propulsion. This phenomenon has been attributed to the generation of a steady streaming flow, with a reversal in the propulsion direction at a distinct frequency. We explain the mechanism underlying this reversal by considering the canonical problem of a sphere executing oscillatory rotation in an unbounded fluid that undergoes rectilinear oscillation; these two oscillations occur at identical frequency but with an arbitrary phase difference. Two distinct bifurcations in the flow field occur: (1) a stagnation point first forms with increasing frequency, which (2) splits into a saddle node and a vortex centre. Reversal in the propulsion direction is driven by reversal in the flow far from the sphere, which coincides with the second bifurcation. This flow is identified with that of a Stokeslet whose strength is the net force exerted on the particle, which has implications for studying the flow field around particles of non-spherical geometries and for modelling suspensions of particles in acoustic fields. [ABSTRACT FROM AUTHOR]

We conducted a systematic numerical investigation of spherical, prolate and oblate particles in an inertial shear flow between two parallel walls, using smoothed particle hydrodynamics (SPH). It was previously shown that above a critical Reynolds number, spherical particles experience a supercritical pitchfork bifurcation of the equilibrium position in shear flow between two parallel walls, namely that the central equilibrium position becomes unstable, leading to the emergence of two new off-centre stable positions (Fox et al. , J. Fluid Mech. , vol. 915, 2021). This phenomenon was unexpected given the symmetry of the system. In addition to confirming this finding, we found, surprisingly, that ellipsoidal particles can also return to the centre position from the off-centre positions when the particle Reynolds number is further increased, while spherical particles become unstable under this increased Reynolds number. By utilizing both SPH and the finite element method for flow visualization, we explained the underlining mechanism of this reverse of bifurcation by altered streamwise vorticity and symmetry breaking of pressure. Furthermore, we expanded our investigation to include asymmetric particles, a novel aspect that had not been previously modelled, and we observed similar trends in particle dynamics for both symmetric and asymmetric ellipsoidal particles. While further validation through laboratory experiments is necessary, our research paves the road for development of new focusing and separation methods for shaped particles. [ABSTRACT FROM AUTHOR]

For the optical soliton model in fifth-order weakly nonlocal nonlinear media, to find its exact explicit solutions, the corresponding traveling wave system is formulated as a planar dynamical system with a singular straight line. Then, by using techniques from dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] to analyze the planar system and find the corresponding phase portraits, the dynamical behavior of the amplitude component can be assessed. Under different parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, kink, and anti-kink wave solutions, compacton solutions, as well as peakons and periodic peakons are found with precise formulations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the spatiotemporal dynamics in a diffusive two-species system with taxis term and general functional response, which means the directional movement of one species upward or downward the other one. The stability of positive equilibrium and the existences of Turing bifurcation, Turing–Hopf bifurcation and Turing–Turing bifurcation are investigated. An algorithm for calculating the normal form of the Turing–Hopf bifurcation induced by the taxis term and another parameter is derived. Furthermore, we apply our theoretical results to a cooperative Lotka–Volterra system and a predator–prey system with prey-taxis. For the cooperative system, stable equilibrium becomes unstable by taxis-driven Turing instability, which is impossible for the cooperative system without taxis. For a predator–prey system with prey-taxis, the dynamical classification near the Turing–Hopf bifurcation point is clearly described. Near the Turing–Hopf point, there are spatially inhomogeneous steady-state solution, spatially homogeneous/nonhomogeneous periodic solution and pattern transitions from one spatiotemporal state to another one. [ABSTRACT FROM AUTHOR]

Disorder in parameters appears to influence the collective behavior of complex adaptive networks in ways that might seem unconventional. For instance, heterogeneities may, unexpectedly, lead to enhanced regions of existence of stable synchronization states. This behavior is unexpected because synchronization appears, generically, in symmetric networks with homogeneous components. Related works have, however, misidentified cases where disorder seems to play a critical role in enhancing synchronization, where it is actually not the case. Thus, in order to clarify the role of disorder in adaptive networks, we use normal forms to study, mathematically, when and how the presence of disorder can facilitate the emergence of collective patterns. We employ parameter symmetry breaking to study the interplay between disorder and the underlying bifurcations that determine the conditions for the existence and stability of collective behavior. This work provides a rigorous justification for a certain barycentric condition to be imposed on the heterogeneity of the parameters while studying the synchronization state. Theoretical results are accompanied by numerical simulations, which help clarify incorrect claims of disorder purportedly enhancing synchronization states. [ABSTRACT FROM AUTHOR]

To research the special transmission system type of the multistage closed-loop gear system of the 3D braiding machine, a multi-degree of freedom nonlinear dynamic model was established by using the lumped mass method, taking into account parameters such as meshing stiffness, static transmission error, backlash, and meshing damping. To reduce the computational complexity, a gear system with 12 gears was selected and solved using the Runge-Kutta method. Using meshing frequency, backlash, and meshing damping ratio as control parameters, we analyzed the bifurcation and chaos of the system under different conditions through bifurcation diagrams. Simultaneously using sequence diagram, phase plane diagrams, Poincare maps, and FFT diagrams to accurately display the motion state of the system. The analysis results indicate that with the increase of meshing frequency and backlash, the system exhibits a range of motion states, such as period-one motion, multi-periodic motion, quasi-periodic motion, and chaotic motion. Meanwhile, increasing the damping ratio can effectively suppress the chaotic state of the gear system and also reduce the maximum vibration displacement of the gear system. [ABSTRACT FROM AUTHOR]

Dielectric elastomers have attracted considerable attention both from academia and industry alike over the last two decades due to their superior mechanical properties. In parallel, research on the mechanical properties of dielectrics has been steadily advancing, including the theoretical, experimental, and numerical aspects. It has been recognized that the electromechanical coupling property of dielectric materials can be utilized to drive deformations in functional devices in a more controllable and intelligent manner. This paper reviews recent advances in the theory of dielectrics, with specific attention focused on the theory proposed by Dorfmann and Ogden. Additionally, we provide examples illustrating the application of this theory to analyze the electromechanical deformations and the associated bifurcations in soft dielectrics. We compared the bifurcations in elastic and dielectric materials and found that only compressive bifurcation modes exist in elastic structures, whereas both compressive and tensile modes coexist in dielectric structures. We summarize two proposed ways to suppress and prevent the tensile bifurcations in dielectric materials. We hope that this literature survey will foster further advancements in the field of the electroelastic theory of soft dielectrics. [ABSTRACT FROM AUTHOR]

The current article focuses on the examination of nonlinear instability and dynamic transitions in a double-diffusive rotating couple-stress fluid layer. The analysis was based on the newly developed dynamic transition theory by T. Ma and S. Wang. Through a comprehensive linear spectrum analysis and investigation of the principle of exchange of stability (PES) as the thermal Rayleigh number crosses a threshold, the nonlinear orbital changes during the transition were rigorously elucidated utilizing reduction methods. For both single real and complex eigenvalue crossings, local pitch-fork and Hopf bifurcations were discovered, and directions of these bifurcations were identified along with transition types. Furthermore, nondimensional transition numbers that signify crucial factors during the transition were calculated and the orbital structures were illustrated. Numerical studies were performed to validate the theoretical results, revealing the relations between key parameters in the system and the types of transition. The findings indicated that the presence of couple stress and a slow diffusion rate of solvent and temperature led to smoother nonlinear transitions during convection. [ABSTRACT FROM AUTHOR]

The posterior cerebral arteries (PCAs) are terminal branches of the basilar artery (BA) and are responsible for the primary supply of the occipital lobe. Saccular aneurysm is most commonly seen close to the bifurcation of the BA. Various surgical interventions are performed for aneurysms. Therefore, the anatomy and localization of the BA and PCA are crucial. The aim of this study was to determine the characteristics of these arteries in a large Anatolian population. The study included 170 Anatolian fresh cadavers. The diameters of the BA and PCA were measured. Correlations according to sex and age groups were analyzed. The Q1, Q2, and Q3 angles between the right and left PCA, between the right PCA and BA, and between the left PCA and BA, respectively, were measured. The location of the PCA relative to the sulcus pontocruralis (pontocrural groove) was also evaluated. The diameter of the artery increased with age and was higher in males than in females. Q1 and Q2 diameters were larger in males, while the Q3 diameter was larger in females. The Q1 angle between the right and left PCAs was found to be higher in age range 40–59 years with a mean of 87.33 ± 17.91 mm. Finally, the bifurcation point of the PCA was most frequently located above the sulcus pontocruralis (pontocrural groove) and least frequently located on the sulcus pontocruralis (pontocrural groove). The findings of our study will contribute to the planning of surgical approaches, the development of endovascular devices, the success of invasive procedures, and the reduction of complications. [ABSTRACT FROM AUTHOR]

In response to external changes, ecosystems can undergo catastrophic transitions. Early warning indicators aim to predict such transitions based on the phenomenon of critical slowing down at bifurcation points found under a constant environment. When an explicit rate of environmental change is considered, catastrophic transitions can become distinct phenomena from bifurcations, and result from a delayed response to noncatastrophic bifurcations. We use a trophic metacommunity model where transitions in time series and bifurcations of the system are distinct phenomena. We calculate early warning indicators from the time series of the continually changing system and show that they predict not the bifurcation of the underlying system but the actual catastrophic transition driven by the explicit rate of change. Predictions based on the bifurcation structure could miss catastrophic transitions that can still be captured by early warning signals calculated from time series. Our results expand the repertoire of mechanistic models used to anticipate catastrophic transitions to nonequilibrium ecological systems exposed to a constant rate of environmental change. [ABSTRACT FROM AUTHOR]