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Medical images make up for more than 25% of global attacks on privacy. Securing them is therefore of utmost importance. Chaos based image encryption is one of the most method suggested in the literature for image security due to their intrinsic characteristic, including ergodicity, aperiodicity, high sensitivity to initials conditions and system parameters. Dynamic systems such as bridge circuit, jerk circuit, Van der Pol circuit, Colpitts oscillator and many other pseudo-random numbers generators have been used in the process of encrypting images. Among them, are the jerk oscillators that have been used with different nonlinearities. In this paper, a new, simple, off-shell component of jerk oscillator (jerk quintic) with an interesting nonlinear function is proposed. Its dynamical behaviors are investigated using classical tools like bifurcation diagrams, Maximum Lyapunov exponent plot, basin of attraction, phase portraits. We showed that the nonlinear function is responsible of complex nonlinear behaviors displayed by the novel circuit, including symmetric/asymmetric bifurcation and coexisting bubbles, multistability just to name a few. The real implementation of the interesting circuit is embedded in a microcontroller verifies these dynamics. As an application of this contribution in multimedia, an encryption algorithm built on a new confusion-diffusion architecture using pseudo random number generated in high chaoticity regime of the new circuit is proposed. The cryptosystem underwent thorough security tests and proved to be fast thanks to the 3D map used, given its complex dynamical behaviors and large chaotic area. This approach yields a robust cipher that underwent thorough security tests better than the one in the literature like average NPCR=99.61, UACI=33.48, key space-sensitivity, entropy=7.9994, average correlation=0.0040. Furthermore, it proved to be robust in terms of noise and data loss in the transmission channel, offering a large key space of 10180 and an entropy close to the standard value, thus rendering the cryptosystem robust against various attacks, especially brute force and exhaustive attacks. [ABSTRACT FROM AUTHOR]

We investigate spatiotemporal pattern formation in cancer growth using discrete time and space variables. We first introduce the coupled map lattices (CMLs) model and provide a dynamical analysis of its fixed points along with stability results. We then offer parameter criteria for flip, Neimark–Sacker, and Turing bifurcations. In the presence of spatial diffusion, we find that stable homogeneous solutions can experience Turing instability under certain conditions. Numerical simulations reveal a variety of spatiotemporal patterns, including patches, spirals, and numerous other regular and irregular patterns. Compared to previous literature, our discrete model captures more complex and richer nonlinear dynamical behaviors, providing new insights into the formation of complex patterns in spatially extended discrete tumor models. These findings demonstrate the model's ability to capture complex dynamics and offer valuable insights for understanding and treating cancer growth, highlighting its potential applications in biomedical research. [ABSTRACT FROM AUTHOR]

A comprehensive theoretical investigation on the occurrence of secondary resonances in parametrically excited unbalanced spinning composite beams under the stretching effects is conducted numerically and analytically. Based on an optimal stacking sequence and Rayleigh’s beam theory, the governing equations of the system are derived using extended Hamilton’s principle. The system’s partial differential equations are then discretized using the Galerkin method. Numerical (Runge–Kutta technique) and analytical (multiple scales method) approaches are exploited to solve the reduced-order equations, and their results are compared and verified accordingly. Comparison and convergence investigations are performed to guarantee the validity of the outcomes. Stability and bifurcation analyses are accomplished, and resonance effects are thoroughly studied utilizing frequency-response diagrams, phase portraits, Poincaré maps and time-history responses. It is observed that among the various types of secondary resonance, only a combination resonance can be observed in the system dynamics. The outputs reveal that, in this resonance, the gyroscopic coupling results in the steady-state time response consisting of three main frequencies. By examining the effects of damping, eccentricity, and beam length, it is exhibited that this resonance does not occur in the system’s dynamics for any combination of these parameters. Therefore, these parameters can be adjusted in the design of asymmetric beams to prevent this type of resonance. [ABSTRACT FROM AUTHOR]

One way to characterize an orbit of a dynamical system is through its frequency content. Using a "spectral bifurcation diagram", it has been shown earlier how the frequency content changes when a system undergoes a period-doubling cascade. In this paper, we extend the scope of this technique to obtain newer insights into various bifurcations like pitchfork bifurcation, border-collision bifurcation, period-adding cascade, and various torus bifurcations. We show that applying this method can enrich our understanding of bifurcations by providing vital information about generating or annihilating frequency components in a bifurcation. [ABSTRACT FROM AUTHOR]

One main feature of a malignant tumor is its uncontrolled growth. In this paper, we propose a simple tumor-immune model to study the progressive characteristics of malignant and benign tumors, where the anti-tumor immunity can be described by the Michaelis–Menten function or the mass action law. The model includes only two state variables for the tumor cells and the effector cells representing the immune system. Three quantities with clear biological meanings are given to determine the asymptotic states of the tumor progression. Moreover, differences in asymptotic states between the two anti-tumor immunity descriptions are drawn. Differently from existing simple models, on the one hand, the model exhibits rich dynamical behaviors including super-critical and sub-critical Bogdanov–Takens bifurcations (consisting of Hopf bifurcation, saddle–node bifurcation, and homoclinic bifurcation) and saddle–node bifurcation of nonconstant periodic solutions (leading to the appearance of two periodic orbits) as the parameters vary; on the other hand, the malignant feature, dormancy, and immune escape of the tumor are revealed with numerical simulations. Furthermore, from the perspective of qualitative analysis and numerical simulations, how the obtained results can be applied to the treatment and control of tumors is illustrated. [ABSTRACT FROM AUTHOR]

This study investigates the behavior of the Izhikevich discrete neuron model across various parameter configurations. Bifurcation diagrams and Lyapunov exponents are utilized to examine the impact of these parameters on the behavior of the system. The study specifically identifies important parameter ranges in which the attractor undergoes a sudden expansion, displaying characteristics of extreme events. Within the system, two distinct categories of extreme events can be identified: rare occurrences of small probability events located in the tail of the probability distribution and Dragon-King (DK) events, which possess a high probability amplitude. DK events are verified through the use of the DK test. The research concludes by examining the practical ramifications of these findings. The significance of forecasting and controlling extreme events in intricate systems is underscored, along with the cruciality of identifying their happening. [ABSTRACT FROM AUTHOR]

In this paper, a two-patch model with additive Allee effect, nonlinear dispersal and commensalism is proposed and studied. The stability of equilibria and the existence of saddle-node bifurcation, transcritical bifurcation are discussed. Through qualitative analysis of the model, we know that the persistence and the extinction of population are influenced by the Allee effect, dispersal and commensalism. Combining with numerical simulation, the study shows that the total population density will increase when the Allee effect constant a increases or m decreases. In addition to suppress the Allee effect, nonlinear dispersal and commensalism are crucial to the survival of the species in the two patches. [ABSTRACT FROM AUTHOR]

This study delves into the intricate interplay between social media platforms, interpersonal word-of-mouth communication, and the transmission dynamics associated with non-communicable diseases, with a particular emphasis on type 2 diabetes. Leveraging advanced mathematical modeling and epidemiological methodologies, our objective is to furnish a comprehensive understanding of how information dissemination through digital and interpersonal networks can impact the proliferation of such diseases within populations. We conduct sensitivity analysis to discern the pivotal model parameters that can wield a substantial influence on the dynamics of disease transmission and control. Moreover, we endeavor to explore the capacity of these model parameters to elicit stability or instability within the system. Our focus lies in the rigorous examination of Hopf and transcritical bifurcations within the system. Furthermore, we consider the influence of seasonal fluctuations in the growth rate of social media advertisements with an aim to discern its role in potentially instigating chaotic dynamics within the context of disease progression. In sum, this research seeks to offer a comprehensive and scientifically robust understanding of the patterns of type 2 diabetes and associated communicable diseases within the context of evolving digital communication landscapes. [ABSTRACT FROM AUTHOR]

In this investigation, the analytical behavior of two prominent nonlinear wave equations, namely the doubly dispersive equation (DDE) and the Ablowitz-Kaup-Newell-Segur equation (AKNSE), have been scrutinized. Bifurcation analysis, sensitivity as well as chaotic phenomena are also performed for the earlier-mentioned dynamical systems. These analyzes have profuse applications in the field of electrical circuits and control systems, phase transitions in materials, climate patterns, chemical reaction networks, forecasting market trends, signal processing, and quantum mechanics. Using an advanced mathematical technique, the exact solutions of the mentioned two-wave equations with singular bell-shaped soliton, bell-shaped soliton, anti-bell-shaped soliton, singular soliton, and singular periodic soliton have been studied. The technique utilized in the study is dependable for solving complex nonlinear problems in various natural science and engineering disciplines employed by many researchers. This study identified ten general solutions and ten particular solutions for the two mentioned equations that are novel and precise wave solutions. The solutions obtained in our study are significant not only for understanding the mentioned field but also may be used in revealing other interesting phenomena, such as the analysis of seismology, the study of compulsive collapse, and the study of the material effects and inner construction of solids. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new tritrophic food chain model. We address the conditions for the coexistence of two different zooplankton species and one phytoplankton species. The results of the numerical and analytical studies of the model show that the most relevant parameters influencing the plankton ecosystem, to maintain a stable coexistence equilibrium, are the: carrying capacity and the constant intrinsic growth rate of the phytoplankton population, and the conversion rate with a mortality rate of carnivorous zooplankton. We prove the existence and direction of Hopf bifurcation in non delay system. To account for the time delay in the conversion of herbivorous consumption to carnivorous zooplankton, we incorporate a discrete delay into the consume response function. Furthermore, we establish some adequate conditions to prove the occurrence of Hopf bifurcation induced by delay in the delayed model. We also plot two-parameter bifurcation diagrams. The numerical results support the analytical findings. [ABSTRACT FROM AUTHOR]

Studying the magnetoelastic coupling vibration of moving plates under the influence of complex air-gap magnetic fields is of theoretical significance for modeling multi-field dynamics and solving nonlinear dynamic issues. This paper considers the scenario where an armature wall acts on one side of an axially moving ferromagnetic thin plate and investigates the multiple resonance of the plate subjected to a harmonic excitation in the air-gap magnetic field. First, based on electromagnetic theory combined with boundary conditions of the magnetic field, the distribution of the air-gap magnetic field is determined by solving the Laplace equation of the magnetic scalar potential function. Next, expressions for the magnetic and Lorentz forces acting on the moving ferromagnetic plate are derived. Then, the thin plate energy expression considering geometric nonlinearity and the virtual work expressions of the forces are obtained by applying the Kirchhoff plate theory and the principle of virtual work. Subsequently, the nonlinear vibration equation of the system with magnetoelastic coupling is established using the Hamiltonian variational principle. Utilizing the established mechanical model, the principal-internal resonance is analytically solved employing the Galerkin and multiple scales methods, with stability criteria for steady-state motion solutions determined using Lyapunov stability theory. Finally, the nonlinear dynamics of the system are analyzed through analytical and numerical examples, and the influences of different physical parameter variations on the dynamic behavior are discussed. Results indicate that variations in physical parameters affect the amplitude, resonance region, and regions of multiple solutions of the system. Moreover, the system may transition to chaos through quasiperiodic torus breakdown or period-doubling bifurcation due to changes in physical parameters. The results of this study provide theoretical references for research in magnetoelastic coupling nonlinear dynamics in fields of magnetic drive and electromagnetic control, as well as for product design and optimization. [ABSTRACT FROM AUTHOR]

This work investigates the global dynamics of the bottom hole assembly with non-uniformly distributed cutters on drill-bit which improves stability of steady drilling. In this work, the drill string with a bottom hole assembly is considered as a lumped parameter model in the axial and torsional directions to accommodate axial and torsional vibrations, respectively. The regenerative effect due to the variation in the depth of cut is considered the only source of instability in the system. The engagement of individual cutters with the cut surface is modeled by formulating separate cut surface profile between two simultaneous cutters which is essential in capturing the true global dynamics of the system. Linear stability analysis through eigenvalue analysis reveals the system's improved stability compared to the system equipped with uniformly distributed cutters. An optimal distribution of cutters on the drill-bit for different number of cutters leading to the most stable steady drilling is obtained. This optimal distribution has been found to be robust with respect to changes in the rock strength as well as drill-rig parameters. A detailed nonlinear numerical analysis of the system with the optimal distribution reveals the existence of subcritical and supercritical Hopf bifurcation at different lobes of stability boundaries. Further analysis unveils various complex phenomenon, such as the co-existence of the steady solutions with the chaotic solution, partial bit-bounce, complete bit-bounce, and stick–slip vibrations. [ABSTRACT FROM AUTHOR]

The study addresses the global impact of COVID-19 by developing a mathematical model that combines within-host and between-host factors to better understand the disease’s dynamics. It begins by describing SARS-CoV-2 dynamics within individual human hosts using fractional-order differential equations. The model is shown to be Ulam–Hyers stable, ensuring reliable predictions. The research then investigates virus transmission from infected to susceptible individuals using agent-based modeling (ABM). This approach allows us to capture the diversity and heterogeneity among individuals, including variations in internal state of individuals, immune response and responses to interventions, making the model more realistic compared to aggregate models. The agent-based model places individuals on a square lattice, assigns health states (susceptible, infectious, or recovered), and relies on infected individuals’ viral load for transmission. Parameter values are stochastically generated via Latin hypercube sampling. The study further explores the impact of viral mutation and control measures. Simulations demonstrate that vaccination substantially reduces transmission but may not eliminate it entirely. The strategy is more effective when vaccinated individuals are evenly distributed across the population, as opposed to concentrated on one side. The research further reveals that while reducing transmission probability decreases infections by implementing prevention protocols, it does not proportionally correlate with the reduction magnitude. This discrepancy is attributed to the intervention primarily addressing inter-host transmission dynamics without directly influencing intra-host viral dynamics. [ABSTRACT FROM AUTHOR]

The main result characterizes the global phenomena of components with one-sign solutions for the Kirchhoff type problem { − M (∫ Ω | ∇ u | 2 d x) Δ u = λ a (x) u (x) + f (x , u , λ) + g (x , u , λ) in Ω , u = 0 on ∂ Ω involving sign-changing weight function, where Ω is a smooth bounded domain in R N , λ ≠ 0 is a real parameter, a ∈ L ∞ (Ω) with a ≢ 0 , f , g ∈ C (Ω ‾ × R 2 , R). The method relies upon the bifurcation theory. According to the behaviors of f at 0, we determine the range of parameter λ for the one-sign solutions of the above problem with indefinite weight function a. Moreover, the global bifurcation phenomenon of this problem can be obtained mainly depending on f satisfying the signum condition s f (x , s , λ) < 0 for s ≠ 0. [ABSTRACT FROM AUTHOR]

In the face of ongoing challenges posed by the COVID-19 and the persistent threat of human immunodeficiency virus (HIV), the emergence of co-infections such as COVID-19 and HIV has heightened complexities in disease management. This study aims to identify effective control strategies to mitigate COVID-19 and HIV co-infection, which aggravates the existing challenges posed by these two diseases. To achieve this, we formulated a co-infection model that describes the transmission dynamics of COVID-19 and HIV. Under certain circumstances, we established that HIV infection may facilitate COVID-19 transmission, highlighting the need to identify and implement effective interventions to mitigate COVID-19 and HIV co-infection. As a result, we incorporated four time-dependent control strategies in the co-infection model: HIV prevention, HIV treatment, COVID-19 vaccination, and COVID-19 treatment. Numerical simulations were conducted to support and clarify the analytical results and to show how preventative efforts affect the co-infected population. Simulations confirm that applying any of the study's strategies will reduce the number of co-infection cases. However, the implementation of these strategies is constrained by limited resources. Therefore, a comprehensive cost-effectiveness analysis was conducted to identify the most economically viable strategy. The analysis concludes that implementing a combined approach of vaccination and treatment for COVID-19 emerges as the most cost-effective measure for preventing the spread of COVID-19 and HIV. These findings provide crucial guidance for decision-makers in adopting precise preventive strategies, ultimately aiming to reduce mortality rates among HIV patients. [ABSTRACT FROM AUTHOR]

The use of plastic is very widespread in the world and the spread of plastic waste has also reached the oceans. Observing marine debris is a serious threat to the management system of this pollution. Because it takes years to recycle the current wastes, while their amount increases every day. The importance of mathematical models for plastic waste management is that it provides a framework for understanding the dynamics of this waste in the ocean and helps to identify effective strategies for its management. A mathematical model consisting of three compartments plastic waste, marine debris, and recycle is studied in the form of a system of ordinary differential equations. After describing the formulation of the model, some properties of the model are given. Then the equilibria of the model and the basic reproduction number are obtained by the next generation matrix method. In addition, the global stability of the model are proved at the equilibria. The bifurcations of the model and sensitivity analysis are also used for better understanding of the dynamics of the model. Finally, the numerical simulations of discussed models are given and the model is examined in several aspects. It is proven that the solutions of the system are positive if initial values are positive. It is shown that there are two equilibria E 0 and E ∗ and if B R < 1 , it is proven that E 0 is globally stable, while when B R > 1 , the equilibrium E ∗ exists and it is globally stable. Also, at B R = 1 the model exhibits a forward bifurcation. The sensitivity analysis of B R concludes that the rates of waste to marine, new waste, and the recycle rate have most effect on the amount of marine debris. [ABSTRACT FROM AUTHOR]

Several diseases like Sickle Cell Anemia, Thalassemia, Hereditary Spherocytosis, Malaria, and Micro-angiopathic Hemolytic Anemia can alter the normal shape of red blood cells (RBCs). The objective of this study is to gain insight into how a change in RBC deformability can affect blood heat transfer. The heat sink effect in a bifurcated vessel with two asymptotic cases (case 1: deformable and case 2: nondeformable RBCs) is being studied during hyperthermia treatment in a three-dimensional bifurcated vessel, whose wall is being subjected to constant heat flux boundary condition. Euler–Euler multiphase method along with the granular model and Kinetic theory is used to include the particle nature of RBCs during blood flow in the current model. To enhance the efficiency of the numerical model, user-defined functions (UDFs) are imported into the model from the C++ interface. The numerical model used is verified with the experimental results from (Carr and Tiruvaloor, 1989, “Enhancement of Heat Transfer in Red Cell Suspensions In Vitro Experiments,” ASME J. Biomech. Eng., 111(2), pp. 152–156; Yeleswarapu et al. 1998, “The Flow of Blood in Tubes: Theory and Experiment,” Mech. Res. Commun., 25(3), pp. 257–262). The results indicate that the deformability of RBCs can change both the flow dynamics and heat sink effect in a bifurcated vessel, which subsequently affects the efficacy and efficiency of the thermal ablation procedure. Both spatial and transient Nusselt numbers of blood flow with deformable RBCs are slightly higher compared to the one with nondeformable RBCs. [ABSTRACT FROM AUTHOR]

p53 is a star protein in cancer biology, and its oscillatory dynamics have received much attention. However, most studies do not consider spatial effects. For this reason, we introduce the diffusion term in a classical p53-Mdm2 autoregulatory loop model. First, the equation is linearized at the positive equilibrium so that we can discuss the local asymptotic stability of this equilibrium. By taking the delay as a bifurcation parameter, the positive equilibrium transitions from stable to unstable and occurs a Hopf bifurcation. Second, we determine the stability of the bifurcating period solution by using a multiple time scales approach. Finally, the theoretical analysis is verified by the numerical simulations. Our study contributes to providing new insight in the controlling of p53 dynamics. [ABSTRACT FROM AUTHOR]

This paper studies an Ebola epidemic model with an exponential nonlinear incidence function that considers the efficacy and the behaviour change. The current model also incorporates a new density-dependent treatment that catches the impact of the disease transmission on the treatment. Firstly, we provide a theoretical study of the nonlinear differential equations model obtained. More precisely, we derive the effective reproduction number and, under suitable conditions, prove the stability of equilibria. Afterwards, we show that the model exhibits the phenomenon of backward-bifurcation whenever the bifurcation parameter and the reproduction number are less than one. We find that the bistability and backward-bifurcation are not automatically connected in epidemic models. In fact, when a backward-bifurcation occurs, the disease-free equilibrium may be globally stable. Numerically, we use well-known standard tools to fit the model to the data reported for the 2018e2020 Kivu Ebola outbreak, and perform the sensitivity analysis. To control Ebola epidemics, our findings recommend a combination of a rapid behaviour change and the implementation of a proper treatment strategy with a high level of efficacy. Secondly, we propose and analyze a fractional-order Ebola epidemic model, which is an extension of the first model studied. We use the Caputo operator and construct the Grünwald-Letnikov nonstandard finite difference scheme, and show its advantages. [ABSTRACT FROM AUTHOR]

Parental care is a behavioral strategy observed in many living organisms, where parents devote time and resources to ensuring their offspring's survival, growth, and well-being. Such care can be in various forms and involve different levels of investment, depending on the species and its ecological context. This study examines how parental care among prey contributes to the coexistence of prey and predator in an ecosystem. For this, we proposed a three-dimensional prey-predator system with the following factors–(i) stage structure and parental care in prey population, (ii) linear functional response for the interaction between immature prey and predators, whereas Holling type II functional response for the case of mature prey. We examined the conditions for the existence of equilibrium points, and their local and global stability for our proposed model. We observed the switching of stability of equilibria via transcritical bifurcation concerning the maturation rate and attack rate on adult prey. We examined all three possible existence scenarios for the maturation rate, e.g., extinction of both populations, prey-only existence, and coexistence of both prey and predator. We also investigated the appearance and disappearance of the limit cycle via Hopf-bifurcation concerning the predation rate and saturation constant of adult prey. We observed that parental care has a stabilizing effect on the system dynamics. Numerical simulations and graphical representations have been done to verify our theoretical findings. Sensitivity analysis is conducted to illustrate the importance of parental care in a stage-structure model. With the introduction of parental care parameter in prey species, we observe a more realistic dynamics of stage structured model. [ABSTRACT FROM AUTHOR]

In this paper we study the stability of a Killing cylinder in hyperbolic 3-space when regarded as a capillary surface for the partitioning problem. In contrast with the Euclidean case, we consider a variety of totally umbilical support surfaces, including horospheres, totally geodesic planes, equidistant surfaces and round spheres. In all of them, we explicitly compute the Morse index of the corresponding eigenvalue problem for the Jacobi operator. We also address the stability of compact pieces of Killing cylinders with Dirichlet boundary conditions when the boundary is formed by two fixed circles, exhibiting an analogous to the Plateau–Rayleigh instability criterion for Killing cylinders in the Euclidean space. Finally, we prove that the Delaunay surfaces can be obtained by bifurcating Killing cylinders supported on geodesic planes. [ABSTRACT FROM AUTHOR]

In 1959 Crawford S. Holling formulated a classification to model the action of the predators over their prey, doing empirical works. In this taxonomy, he introduced only three types of functional responses dependent only on the prey population, which are described by saturated functions. Later, various other types have been proposed, including the functional responses dependent on both populations. This work concerns the study of the Leslie–Gower type predator-prey model, incorporating the Rosenzweig functional response described by a power law. The elected function does not conform to the types proposed by Holling since it is unbounded, being, besides, non-differentiable for x = 0 ; nonetheless, the obtained system is Lipschitzian. Moreover, the existence of a separatrix curve Σ in the phase plane is proven, which is divided into two complementary sectors. According to the position of the initial conditions with respect to the curve, the trajectories can have different ω -limit sets, which can be the equilibrium 0 , 0 , or a positive equilibrium, or a heteroclinic curve, or a stable limit cycle. These properties show the great difference of this model with the original Leslie–Gower model, in which a unique positive equilibrium exists, which is globally asymptotically stable, when it exists. Then, the analyzed system has a richer dynamic than the original system in which a linear functional response is considered, also unbounded. Numerical simulations and bifurcation diagrams are given to endorse our analytical results. [ABSTRACT FROM AUTHOR]

The bifurcations and monotonicity of the period function of the Lakshmanan–Porsezian–Daniel equation with Kerr law of nonlinearity are discussed. Firstly, by the traveling wave transformations, the Lakshmanan–Porsezian–Daniel equation is reduced to the planar Hamiltonian system whose Hamiltonian function includes a 6-th degree polynomial. Then we give the phase portraits of the Hamiltonian system, and some traveling waves including dark wave solutions, kink and anti-kink solutions and periodic solutions are constructed by using the bifurcation method of dynamical systems. Furthermore, we discuss the monotonicity of the period function of periodic wave solutions by using some Lemmas proposed by Yang and Zeng (Bull Sci Math 133(6):555-557, 2009). Finally, some numerical simulations are presented. [ABSTRACT FROM AUTHOR]

This paper studies the steady states to a diffusive Lotka–Volterra cooperative model with population flux by attractive transition. The first result gives many bifurcation points on the branch of the positive constant solution under the weak cooperative condition. The second result shows every steady state approaches a solution of the scalar field equation as the coefficients of the flux tend to infinity. Indeed, the numerical simulation using pde2path exhibits the global bifurcation branch of the cooperative model with large population flux is near that of the scalar field equation. [ABSTRACT FROM AUTHOR]

The impact of predator‐driven fear on ecosystems is significant and can encompass both trophic (direct) and nontrophic (indirect) effects. Previous studies have shown that nontrophic fear effects have an important role in predator–prey dynamics. This study investigates the nontrophic fear effect on prey caused by generalist predators and explores optimal harvesting. We assume that the reproduction rate of prey is reduced due to the fear effect, and generalist predator follows Holling type II foraging strategy for predation. Additionally, we assume that predators are commercially valuable and harvested proportionately to their density. We demonstrate the existence of equilibrium points, their local and global stability, and bifurcation analysis. We observe that the model system undergoes a sequence of codimension one and codimension two bifurcations. Our results show that in the absence of predator harvesting, increasing levels of fear destabilize the predator–prey system, and controlled harvesting is beneficial for the coexistence of both populations. Also, the harvesting of predators may produce hydra and multiple hydra effects. We identify different types of bistability phenomena, which emphasize the importance of initial population size. Further, an optimal harvesting policy is also explored by formulating an optimal control problem (OCP). The harvesting cost‐functional is designed by incorporating the bionomic equilibrium state. We use Pontryagin's maximum principle and solve the OCP numerically. It is observed that implementing optimal harvesting not only contributes to the ecological benefits by maintaining a sustainable balance of predator–prey evolution but also plays a significant role in maximizing the economic benefits. [ABSTRACT FROM AUTHOR]

This study seeks to explore the integrable dynamics of induced curves through the utilization of the complex-coupled Kuralay system. The importance of the coupled complex Kuralay equation lies in its role as a fundamental model that contributes to the understanding of intricate physical and mathematical concepts, making it a valuable tool in scientific research and applications. The soliton solutions originating from the Kuralay equations are believed to encapsulate cutting-edge research in various important domains such as optical fibers, nonlinear optics, and ferromagnetic materials. Analytical techniques are employed to derive traveling wave solutions for this model, given that the Cauchy problem cannot be resolved using the inverse scattering transform. In the quest for solitary wave solutions, the extended modified auxiliary equation mapping (EMAEM) method is employed. We derive several novel families of precise traveling wave solutions, encompassing trigonometric, hyperbolic, and exponential forms. Moreover, the planar dynamical system of the concerned equation is created, 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 incorporated alongside the analytical results to enhance comprehension of the dynamic characteristics of the solutions obtained. This study’s outcomes can offer valuable insights for addressing other nonlinear partial differential equations (NLPDEs). The soliton solutions obtained in this study offer important insights into the intricate nonlinear equation being examined. [ABSTRACT FROM AUTHOR]

For this study, a three-species food chain model that considers interactions between small fish, remora fish, and large fish has been developed. It is assumed that large fish consume small fish through the functional response of Holling type II. Remora fish are considered to benefit from eating the remaining parts of small fish hunted by larger fish. Remora fish may benefit from eating the parasites found in the bodies of large fish. The non-negativity and boundedness of the solution of the model are investigated. In addition, each potential stable state for the model has been determined. We also investigated the local stability of the model, which is close to the steady state. The Lyapunov function and geometric technique are used to investigate the global stability of the model. It has been found that the increase in the refuge rate for small fish contributes to the instability of the model. It has been observed that the model may show unstable behaviour due to an increase in the rate of consumption of small fish by big fish. It has been found that the density of remora fish may rise because of their mutualistic relationship with large fishes. [ABSTRACT FROM AUTHOR]

A nonlinear parabolic equation with transformation of the spatial variable and periodic conditions on the circle is considered. Using the method of separation of variables, we obtain properties of eigenfunctions and eigenvalues of the corresponding linearized problem. Using the method of central manifolds, we prove the existence and stability of spatially inhomogeneous stationary solutions. Based on the Galerkin method, we analyze approximate solutions of the original problem. [ABSTRACT FROM AUTHOR]

This study explores the dynamics of sequentially arranged coupled oscillator loops, focusing on inducing steadily rotating phase waves. Notably, when interconnected loops of differing sizes are analyzed, the resulting phase waves exhibit a synchronization pattern, either in-phase or out-of-phase, with pulse counts proportional to their size ratios. This synchronization is further investigated through the introduction of subharmonic injection locking, achieved by applying an external frequency signal matching one of the pulses within the rotating phase waves. Our research delineates the core characteristics of phase waves across oscillator loops of varying sizes, offering a novel approach to generate multiple rotary pulses and amplify the subharmonic response by fine-tuning the oscillators' bias voltage. This paper sheds light on the underlying mechanisms of phase wave synchronization and subharmonic injection locking, with potential applications in enhancing the functionality of coupled oscillator systems. [ABSTRACT FROM AUTHOR]

We propose a Neurosimilator, novel analog neuron circuit and mathematical model of its dynamics that both simulate with high precision the spiking behavior of known types of excitable cells. The analog circuit is compact and can be built using off-the-shelf components. This could facilitate its use in teaching neuroscience and biophysics. The circuit is scalable down to the pF-valued capacitors, presenting an advantage in research on the analog nerve fiber networks. The equations of circuit dynamics contain exponential non-linearities and Heaviside functions, so that the model combines features from the generalized adapting exponential integrate-and-fire neuron model and from the intermittent feedback Hindmarsh-Rose model, but it is not directly related to them. Our four-dimensional system (4D-Neurosimilator) simulates most of excitable cells' spiking, bursting and chaotic behavior depending on only fewer predefined parameters. In bursting and chaotic oscillatory patterns, the model demonstrates self-adaptive energy flow redistribution. The energy expenditure amounts to ≈36 µJ per one spiking event in original model and to ≈0.63 pJ in its down-scaled version. The model has computational cost comparable to that of the Hodgkin–Huxley model, but it tends to handle noisy input stimulations more efficiently. Our work provides novel insights to the simulation of neuron's non-linear dynamics and may constitute another choice of available model in computational neuroscience research that expands the limits of a tradeoff between accuracy, biological explainability, noise-resistance and computing time. [ABSTRACT FROM AUTHOR]

Small finite-size particles suspended in fluid flow through an enclosed curved duct can focus to points or periodic orbits in the two-dimensional duct cross-section. This particle focusing is due to a balance between inertial lift forces arising from axial flow and drag forces arising from cross-sectional vortices. The inertial particle focusing phenomenon has been exploited in various industrial and medical applications to passively separate particles by size using purely hydrodynamic effects. A fixed size particle in a circular duct with a uniform rectangular cross-section can have a variety of particle attractors, such as stable nodes/spirals or limit cycles, depending on the radius of curvature of the duct. Bifurcations occur at different radii of curvature, such as pitchfork, saddle-node and saddle-node infinite period (SNIPER), which result in variations in the location, number and nature of these particle attractors. By using a quasi-steady approximation, we extend the theoretical model of Harding et al. (J. Fluid Mech. , vol. 875, 2019, pp. 1–43) developed for the particle dynamics in circular ducts to spiral duct geometries with slowly varying curvature, and numerically explore the particle dynamics within. Bifurcations of particle attractors with respect to radius of curvature can be traversed within spiral ducts and give rise to a rich nonlinear particle dynamics and various types of tipping phenomena, such as bifurcation-induced tipping (B-tipping), rate-induced tipping (R-tipping) and a combination of both, which we explore in detail. We discuss implications of these unsteady dynamical behaviours for particle separation and propose novel mechanisms to separate particles by size in a non-equilibrium manner. [ABSTRACT FROM AUTHOR]

The goal of this work is to look at how a nonlinear model describes hematopoiesis and its complexities utilizing commonly used techniques with historical and material links. Based on time delay, the Mackey–Glass model is explored in two instances. To offer a range, the relevance of the parameter impacting stability (bifurcation) is recorded. The power spectrum of the considered model is collected in order to analyze the periodic behavior of a solution in a differential equation. The complex nature of the system is relayed on a parameter which is illustrated in the bifurcation plot. Due to the fact that the considered model is associated with blood‐related diseases, the effect coefficients are effectively captured. The corresponding parameters‐based consequences of the generalized model in different order are deduced. The parametric charts for both examples reveal intriguing results. The current work enables investigations into complex real‐world problems as well as forecasts of essential techniques. [ABSTRACT FROM AUTHOR]

Llibre et al. in [ J. Nonlinear Sci. , 2019] studied the number of limit cycles of piecewise linear Liénard system with one anti-saddle at the central zone in some parameter regions. However, the exact number and hyperbolicity of limit cycles were still unclear as in these cases of parameter. The goal of this paper is to give the exact number of limit cycles and the hyperbolicity of limit cycles in these parameter cases, where the maximum number of limit cycles is two. Moreover, all corresponding global phase portraits in the Poincaré disc are obtained. [ABSTRACT FROM AUTHOR]

This article explores the dynamic interplay between human and non-human entities, focusing on how embodied data representation is distributed. It examines how predictive coding, which utilizes preconceived knowledge, interacts with tangible experiences to shape our understanding of the world. Emphasizing this, I propose the concept of co-Ability as a deep underlying explanatory framework for understanding adaptive behaviors within a networked world. A nonverbal dialog between humans and a data-saturated environment is analyzed through an action-oriented perspective and the predictive coding framework in cognition, utilizing digital craft and rapid prototyping. This transformative approach augments human interaction with digital landscapes through tangible prototypes, bridging physical experience with abstract information, and identifying potential ways to conceptualize data materially. The article discusses the various aspects of connectivity among network agents and the evolving nature of these connections as they adapt to real-world conditions and dynamic shifts in data, highlighting that information exchange in an interconnected network is more than bilateral; it generates ripple effects that extend beyond immediate connections. These reciprocal exchanges simultaneously alter both the digital and analog domains, with data constantly bifurcating into multiple pathways and outcomes. A significant challenge addressed in this article is the question of how to frame information materially, inviting further exploration. [ABSTRACT FROM AUTHOR]

A four-dimensional mathematical model is formulated to explore the fear effect exerted by large carnivore in the grassland ecosystem. The model depicts the interactions among herbage, domestic herbivore, wild herbivore and large carnivore, which incorporates both direct predation and anti-predator mechanisms. The dynamic properties of the model are analytically investigated, including the dissipativity of solutions, and the existence and stability of different equilibria. Some numerical simulations are also presented to exhibit rich dynamical behaviors, such as various types of bistabilities, periodic oscillation and chaotic oscillation. The study reveals that the appropriate level of fear factors can stabilize the system and increase the density of herbage and domestic herbivore. The fear effect plays an important role in maintaining the balance of the grassland ecosystem and promoting the economy of human society. [ABSTRACT FROM AUTHOR]

Malaria is known globally as the foremost cause of death in children and adults. Several intervention strategies and controls have been implemented and proposed, among which is Long Lasting Insecticide Treated Nets (LLINs). Malaria cases have been reported to reduce by 5 0 % due to the use of LLINs. However, the laid-back behaviors of humans negatively impact its effective use through improper handling and exposure to direct sunlight. To this end, a mathematical model is formulated to investigate the influence of individual response to information on the transmission dynamics of malaria infection in both human and mosquito populations. The threshold for the effective reduction of the prevalence of malaria through a behavioral change in the use of LLINs in response to the created awareness is determined in this study. The point of departure of this study from other studies is that it incorporates information/awareness as a dynamic variable to investigate the impact of LLINs on malaria. The equilibrium points of the model are analyzed using stability theory and the associated basic reproduction number R 0 . The implicit function theorem is used to determine the direction of bifurcation. We use Latin hypercube sampling (LHS) to test the most sensitive parameters in the basic reproduction number. The correlation between the R 0 and parameters is displayed using Scatter plots. The mathematical analysis shows that the direction of bifurcation is supercritical, which suggests that we only need to bring R 0 below unity for disease eradication. The study also revealed that an increase in awareness of the use of LLINs has an impact on reducing malaria transmission. [ABSTRACT FROM AUTHOR]

This paper examines a three-dimensional delayed discrete neural network model analytically and numerically to determine the existence of different types of bifurcations of the involved fixed points. The model exhibits different bifurcations such as pitchforks, flips, Neimark–Sackers, and flip-Neimark–Sackers. The critical coefficients are used to determine the structure of each bifurcation. The curves are calculated and plotted for each bifurcation when the parameters are changed. Further, these bifurcations are theoretically analyzed and numerically verified. From the obtained results, we observed that by drawing the curves associated with each bifurcation, the numerical simulations are consistent with the analytical results. [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$$ {\mathrm{\mathbb{R}}}&#x0005E;{n_1&#x0002B;{n}_2&#x0002B;m} $$ to ℝn1+n2+δ1+δ2$$ {\mathrm{\mathbb{R}}}&#x0005E;{n_1&#x0002B;{n}_2&#x0002B;{\delta}_1&#x0002B;{\delta}_2} $$. 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]

This research delves into the generalized Beddington host–parasitoid model, which includes an arbitrary parasitism escape function. Our analysis reveals three types of equilibria: extinction, boundary, and interior. Upon examining the parameters, we discover that the first two equilibria can be globally asymptotically stable. The boundary equilibrium undergoes period-doubling bifurcation with a stable two-cycle and a transcritical bifurcation, creating a threshold for parasitoids to invade. Furthermore, we determine the interior equilibrium's local stability and analytically demonstrate the period-doubling and Neimark–Sacker bifurcations. We also prove the permanence of the system within a specific parameter space. The numerical simulations we conduct reveal a diverse range of dynamics for the system. Our research extends the results in [Kapçak et al., 2013] and applies to a broad class of the generalized Beddington host–parasitoid model. [ABSTRACT FROM AUTHOR]

The objective of this work is to investigate the complex dynamics of a discrete predator–prey system using the method of piecewise constant argument for discretization. An analysis is conducted to examine the presence and stability of fixed points. Furthermore, the system is shown to undergo period-doubling (PD) and Neimark–Sacker (NS) bifurcations by the use of center manifold and bifurcation theories. The feedback and hybrid control strategies are used to regulate the system's bifurcating and chaotic behaviors. Both strategies seem to be effective in managing bifurcation and chaos inside the system. Finally, the main results are validated by numerical evidence. Parameters of the system are varied to produce time graphs, phase portraits, bifurcation diagrams, and maximum Lyapunov exponent (MLE) graphs. The discrete model displays rich dynamics, as seen in the numerical simulations and graphs, indicating a complex and chaotic system. [ABSTRACT FROM AUTHOR]

This research investigates the interaction between a generalist predator and a prey, where the predator exhibits cooperative behavior during hunting, inducing fear into the prey population. Additionally, both the prey and predator populations are subject to harvesting. The study establishes the positivity and boundedness of the model's solutions, ensuring the existence of the population. Analyzing the system, we explore its feasible steady states and their stability, along with various types of bifurcations, including Hopf with direction of stability, Saddle-node, Transcritical, Homoclinic, Bogdanov–Takens, and Cusp bifurcation. We also demonstrate the stability and bifurcation behavior of a delayed system. These findings are verified through one-parameter and two-parameter bifurcation structures, complemented by respective phase portraits. Notably, the system displays transition between different equilibria and bistability. Furthermore, numerical investigations reveal the impact of gestation delay, indicating chaotic behavior in the system due to this time delay. [ABSTRACT FROM AUTHOR]

In this paper, we investigate population agglomeration in a long narrow economy, in which an odd number of places are evenly distributed over a line segment. The bifurcation analysis of this economy elucidates the mechanism of the emergence of twin cities around the central city. The validity and usefulness of this analysis are confirmed using several well-known economic geography models that display various kinds of bifurcation behaviors. By this analysis, we investigate the historical change in the population distribution in a chain of cities on Japan's Main Island. [ABSTRACT FROM AUTHOR]

It is widely acknowledged that fluidelastic instability (FEI), among other mechanisms, is of the greatest concern in the flow-induced vibration (FIV) of tube bundles in steam generators and heat exchangers. A range of theoretical models have been developed for FEI analysis, and, in addition to the earliest semi-empirical Connors' model, the unsteady model, the quasi-steady model and the semi-analytical model are believed to be three advanced models predominant in the literature. The unsteady and the quasi-static models share the merits of having explicit fluid force expressions and ease of being implemented but require more experimental inputs, whereas the semi-analytical model requires fewer parameters due to its analytical nature but is hard, if not prohibitive, to derive explicit fluid force expressions. Since the fluid force formulations set in the core of development of FEI models, the understanding and in particular the implementation of the semi-analytical model has been impaired by the nonexistence of explicit fluid force expression. This issue becomes more profound in time-domain analysis whereby the simple harmonic assumption is discarded. Here we report a new semi-analytical time-domain (SATD) FEI model with explicit fluid force expressions. The new model allows a consistent frequency-domain stability analysis and more importantly a truly time-domain response analysis. The theory was validated by calculating linear stability thresholds of two typical tube array patterns and comparing against reported experimental data. We then present a nonlinear time-domain analysis of a single loosely-supported tube with piece-wise linear stiffness. The nonlinear and nonsmooth dynamics was probed in details by utilizing various techniques, playing an emphasis on characterizing and distinguishing the chaotic vibration. We found that the system follows a quasi-periodic route to chaos. Such an in-depth study of the nonlinear dynamics of tubes in crossflow has never been reported in the context of SATD model. Our results enrich the theory and provide a different approach for linear and nonlinear dynamics of tube bundles, which are essential for the subsequent fretting wear analysis. • A new formulation on semi-analytical time-domain model for fluidelastic instability of tube bundles. • The formulation has explicit fluid force expressions. • Linear frequency- and time-domain analysis are validated against available experimental and theoretical results. • Nonlinear dynamics behavior of a loosely-supported tube is probed based on the new formulation by using various techniques. • The route to chaos of the system is found through quasi-periodic to chaos. [ABSTRACT FROM AUTHOR]

This paper conducts an in-depth study on the dynamic stability and complex vibration behavior of a 12-pole active magnetic bearing (AMB) system considering gravitational effects under a PD controller. Firstly, based on electromagnetic theory and Newton's second law, a two-degree-of-freedom control equation of the system, including PD control terms and gravitational effects, is constructed. This equation involves not only parametric excitation, quadratic nonlinearity, and cubic nonlinearity but also a more pronounced coupling effect between the magnetic poles due to the presence of gravity. Secondly, using the multi-scale method, a four-dimensional averaged equation of the system in Cartesian and polar coordinates is derived. Finally, through numerical analysis, the system's amplitude–frequency response, motion trajectory, the relationship between energy and amplitude, and global dynamic behaviors such as bifurcation and chaos are discussed in detail. The results show that the PD controller significantly affects the system's spring hardening/softening characteristics, excitation, amplitude, energy, and stability. Specifically, increasing the proportional gain can quickly suppress the rotor's motion, but it also increases the system's instability. Adjusting the differential gain can transition the system from a chaotic state to a stable periodic motion. [ABSTRACT FROM AUTHOR]

We study bifurcations of nonlinear waves (spatially inhomogeneous solutions) emerging from homogeneous equilibrium states of an initial boundary value problem, arising in nonlinear optics, for a nonlinear parabolic equation on a disk with a spatial argument rescaling operator and with time delay. In the plane of the main parameters of the equation, we construct stability (instability) domains of homogeneous equilibrium states and study the dynamics of the stability domains depending on the rescaling coefficient. We investigate the mechanisms of stability loss by homogeneous equilibrium states, the possible bifurcations of spatially inhomogeneous self-oscillatory solutions, and their stability. We demonstrate the possibility of bifurcation of stable rotational and spiral waves. [ABSTRACT FROM AUTHOR]

We study the dynamics of a system of differential equations with the diffusion interaction and an additional internal coupling. Such systems are interesting because a slight variation in the coefficient at the additional coupling allows obtaining intricate scenarios of phase rearrangements. For the system under study, we find the critical dependence of the parameters such that zero equilibrium loses stability as two spatially inhomogeneous states appear in one case and a cycle in another case. With the parameter values close to the critical ones, asymptotic formulas are obtained for the regimes that branch off from the zero solution. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of a fourth-order Newtonian iterative method for finding roots of polynomials of degrees three and four. Unlike traditional fourth-order methods requiring third derivatives, this technique avoids them by using the same derivative order in each of its three steps per iteration. When applied to differentiable functions, the method generates chaotic dynamics, as shown for quartic polynomials. Specifically, we apply this root-finding approach to the bifurcation diagram of the logistic map over an interval. Our findings demonstrate the potential for complex behavior even in simple iterative methods, and highlight the usefulness of this approach for exploring polynomial system dynamics. The paper identifies examples of fourth-degree polynomials, explains bifurcation, and chaos. [ABSTRACT FROM AUTHOR]

Background: Treating wide-neck bifurcated cerebral aneurysms (WNBAs) using various techniques and new devices has shown favorable outcomes. However, endovascular coiling can be technically challenging when the aneurysm neck is incorporated into the parent vessel. Furthermore, although recent research has reported favorable outcomes of Neuroform Atlas stent (NAS)-assisted coiling, broad inclusion criteria have hampered precise evaluations of their effectiveness and safety for treating complex WNBAs. Therefore, this study evaluated whether the use of a single NAS is a safe and effective approach for treating complex WNBAs. Methods: We treated 76 complex WNBAs (unruptured, n = 49; ruptured, n = 27) using single NAS-assisted coil embolization and retrospectively analyzed the clinical and angiographic outcomes. Results: In a cohort of 68 patients (mean age, 58.3 ± 11.6 years; males n = 20, 29.4%; females, n = 48, 70.6%), 76 stents were successfully delivered to the target aneurysms, yielding a technical success rate of 98.6%. Complete occlusion was evident in 59 (77.6%) of 76 aneurysms, with neck remnants found in 16 (21.1%) and partial occlusion in 1 (1.3%). Treatment-related morbidities comprised one branch occlusion and one parenchymal hemorrhage. However, no new neurological symptoms of unruptured aneurysms were evident at discharge. The outcomes of 20 of the 27 ruptured aneurysms were favorable (Glasgow Outcome Scale scores of 4 or 5) at the final follow-up assessment (mean 12.2 [6-29] months), except for one initial subarachnoid hemorrhage. Post- treatment angiography revealed complete occlusion in 89.1%, neck remnants in 7.8%, and incomplete occlusion in 3.1% of the aneurysms. Approximately 88.2% of the patients were assessed at least once by follow-up diagnostic or magnetic resonance angiography (mean, 12.5 ± 4.3 [range, 6-29] months), with five (7.8%) minor and two (3.1%) major recurrences. Conclusion: A single NAS is safe and effective for treating WNBAs incorporated into parent vessels. [ABSTRACT FROM AUTHOR]

In this article, the dynamic behavior and solitary wave solutions of the Akbota equation were studied based on the analysis method of planar dynamic system. This method can not only analyze the dynamic behavior of a given equation, but also construct its solitary wave solution. Through traveling wave transformation, the Akbota equation can easily be transformed into an ordinary differential equation, and then into a two-dimensional dynamical system. By analyzing the two-dimensional dynamic system and its periodic disturbance system, planar phase portraits, three-dimensional phase portraits, Poincar'e sections, and sensitivity analysis diagrams were drawn. Additionally, Lyapunov exponent portrait of a dynamical system with periodic disturbances was drawn using mathematical software. According to the maximum Lyapunov exponent portrait, it can be deduced whether the system is chaotic or stable. Solitary wave solutions of the Akbota equation are presented. Moreover, a visualization diagram and contour graphs of the solitary wave solutions are presented. [ABSTRACT FROM AUTHOR]