Narrow Results

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.

The perturbed nonlinear Schrödinger equation plays a crucial role in various scientific and technological fields. This equation, an extension of the classical nonlinear Schrödinger equation, incorporates perturbative effects that are essential for modeling real-world phenomena more accurately. In this paper, we investigate the traveling wave solutions of the perturbed nonlinear Schrödinger equation using the bifurcation theory of dynamical systems. Graphical presentations of the phase portrait are provided, revealing the traveling wave solutions under various conditions. By employing the auxiliary equation method, we derive a variety of solutions including periodic, dark, singular and bright optical solitons. To provide comprehensive and clearer depiction of the model’s behavior 2D, contour and 3D graphical representations are offered. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. This study expands the scope of known exact solutions and their stability qualities which is offering an extensive analytical technique which enhances previous research. The novelty of our research lies in its examination of bifurcation analysis and the auxiliary equation method within the context of a perturbed nonlinear Schrödinger wave equation for the first time. By integrating these two perspectives, this paper contributes to establishing the complex dynamics and stability characteristics of soliton solutions under perturbations.

This paper discusses a fractional-order prey–predator system with Gompertz growth of prey population in terms of the Caputo fractional derivative. The non-negativity and boundedness of the solutions of the considered model are successfully analyzed. We utilize the Mittag-Leffler function and the Laplace transform to prove the boundedness of the solutions of this model. We describe the topological categories of the fixed points of the model. It is theoretically demonstrated that under certain parametric conditions, the fractional-order prey–predator model can undergo both Neimark–Sacker and period-doubling bifurcations. The piecewise constant argument approach is invoked to discretize the considered model. We also formulate some necessary conditions under which the stability of the fixed points occurs. We find that there are two fixed points for the considered model which are semi-trivial and coexistence fixed points. These points are stable under some specific constraints. Using the bifurcation theory, we establish the Neimark–Sacker and period-doubling bifurcations under certain constraints. We also control the emergence of chaos using the OGY method. In order to guarantee the accuracy of the theoretical study, some numerical investigations are performed. In particular, we present some phase portraits for the stability and the emergence of the Neimark–Sacker and period-doubling bifurcations. The biological meaning of the given bifurcations is successfully discussed. The used techniques can be successfully employed for other models.

The fractional Klein–Gordon equation (fKGE) holds a crucial position in various fields of theoretical and applied physics, with wide applications covering different areas such as nonlinear optics, condensed matter physics, and quantum mechanics. In this paper, we carry out analytical investigation to the fKGE with beta fractional derivative by using the Bernoulli $(G^{\prime }∕G)$-expansion method and improved $tan(\varphi ∕2)$-expansion method. In order to better comprehend the physical structure of the obtained solutions, three-dimensional visualizations, contour diagrams, and line graphs of the exponent function solutions are depicted with the aid of Matlab. Moreover, the phase portraits and bifurcation behaviors of the fKGE under transformation are studied. Sensitivity and chaotic behaviors are analyzed in specific conditions. The phase plots and time series map are exhibited through sensitivity analysis and perturbation factors. These studies enhance our understanding of practical phenomena governed by the model of fKGE, and are crucial for examining the dynamic behaviors and phase portraits of the fKGE system. The strategies utilized here are more direct and effective, which can be effortlessly utilized to various fractional-order differential equations arising in nonlinear optics and quantum mechanics.

The alteration in concentration of atmospheric pollutants is influencing the functionality and growth of forests. Also, the growing human demand for forestry resources is detrimentally affecting the sustainability of these valuable resources. In this study, we present a mathematical model that incorporates the influence of atmospheric pollutants on the intrinsic growth rate of forests, while concurrently addressing the utilization of forestry land by the human population for diverse purposes which diminishes the carrying capacity of forestry resources. We establish sufficient conditions under which all relevant dynamic variables stabilize at their equilibria. Upon scrutinizing the model system, we observe multiple bifurcations concerning certain key parameters. Additionally, numerical simulations have been conducted to corroborate the analytically derived findings. Moreover, we fortify the proposed model through the integration of a time delay in the impact of pollutants on the intrinsic growth rate of forestry resources. Despite the conventional belief that introducing a time delay tends to destabilize systems, our resolute delayed model system showcases that a time delay in the effect of pollutants on intrinsic growth rate of forestry resources can, in fact, stabilize the unstable interior equilibrium.

In this paper, we use the techniques from dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] to investigate the exact explicit solutions for the perturbed Gerdjikov–Ivanov (GI) equation. By considering the corresponding dynamical system and finding the bifurcations of phase portraits for the amplitude component of the traveling solutions, the dynamical behavior can be revealed. Under different parameter conditions, exact explicit solutions of the perturbed GI equation are found.

Analyzing tumor growth dynamics improves treatment strategies for cancer. Many models have been proposed to analyze cancer development dynamics, which do not always exhibit chaotic behavior. This research aims to create and examine a unique dynamical cancer model that exhibits chaotic behavior for certain parameters. Introducing chaos into the model allows for the exploration of irregular tumor growth patterns and the identification of critical thresholds that can influence treatment outcomes. The model is examined, and each system parameter’s impact on the model’s dynamics is evaluated. The analysis of the bifurcation and Lyapunov diagrams demonstrates chaos in three populations of tumor, healthy, and immune system cells. By suppressing the immunological response, the cancer cell gains control of the chaotic attractor and establishes a stable state. This might lower the cancer condition by altering the appropriate parameter range assisting in tumor treatment.

In this paper, we consider a class of near-Hamiltonian systems with a nilpotent center, and study the number of limit cycles including algebraic limit cycles. We prove that there are at most $n(m+1)+1$ large amplitude limit cycles if the first-order Melnikov function is not zero identically, including an algebraic limit cycle. Moreover, it can have $\frac{n(n+3)}{2}$ when $m\ge n$ and $\frac{m(2n-m+1)}{2}+n$ when $m<n$ small limit cycles. We also provide two examples as applications of our main results.

The paper is devoted to analyzing the mechanisms of spread and prevention of epidemics, based on a discrete model that takes into account the infection spread because of contacts among infected and susceptible individuals, disease-induced mortality, and the factor of treatment. Pathways leading to the complete extinction, complete recovery, and nontrivial coexistence of susceptible and infected individuals are revealed by bifurcation analysis. Parametric conditions of nontrivial modes of coexistence in the form of equilibrium, discrete cycle, quasiperiodic closed invariant curve and chaos are found. An extended analysis on transformation scenarios of these regimes in dependence of the variation of the rate of the infection spread and the treatment intensity is performed. Phenomena of infection-induced chaos and its suppression by treatment are discussed.

In this paper, we consider the dynamics of a prey–predator model with logistic growth incorporating refuge in the prey and cooperation among predators population. Furthermore, multiplicative Allee effect in the prey growth is added to account from biological and mathematical perspectives. First, the existence and stability of equilibria of the model are discussed. Next, the existence of several kinds of bifurcation is provided and also studied the direction and stability of Hopf bifurcation. In addition, we study the influence of hunting cooperation on the model analytically and numerically, and find that the hunting cooperation cannot only reduce the density of prey population, but also destabilize the system dynamics irrespective of Allee effect. We choose also the impact of refuge on the model numerically, and explore that refuge stabilize the system dynamics. Moreover, the comparison between the dynamics of strong and weak Allee effect is taken into consideration.

In this paper, we investigate the dynamical behaviors of a modified Bazykin-type two predator-one prey model involving the intra-specific and inter-specific competition among predators. A Caputo fractional-order derivative is utilized to include the influence of the memory on the constructed mathematical model. The mathematical validity is ensured by showing the model always has a unique, non-negative and bounded solution. Four kinds of equilibria are well identified which represent the condition when all populations are extinct, both two predators are extinct, only the first predator is extinct, only the second predator is extinct, and all populations are extinct. The Matignon condition is given to identify the dynamics around equilibrium points. The Lyapunov direct method, the Lyapunov function, and the generalized LaSalle invariant principle are also provided to show the global stability condition of the model. To explore the dynamics of the model more deeply, we utilize the predictor–corrector numerical scheme. Numerically, we find the forward bifurcation and the bistability conditions by showing the bifurcation diagram, phase portraits, and the time series. The biological interpretation of the analytical and numerical results is described explicitly when an interesting phenomenon occurs.

The nonlinear energy sink (NES) system mainly dissipates energy through damping elements, and changing the position of the damping element will also change the performance of the NES. In this paper, a grounded damping NES is proposed by grounding the damping element of the traditional cubic stiffness NES. The complex dynamics of this two-degree-of-freedom system are investigated. The slow-varying equations of the system under 1:1 internal resonance are derived by using the complexification-averaging (C×A) method, based on which the influence of the primary structure’s damping on a bifurcation is analyzed. The conditions for the existence of strongly modulated response (SMR) are studied, and the accuracy of the results is verified using the slow invariant manifold (SIM), Poincare mapping, and time-history diagrams. This provides a means of verifying the analytical findings. The vibration suppression effects of the grounded damping NES, as compared to the cubic stiffness NES, are thoroughly studied under both pulse and harmonic excitations. The results indicate that the main structure damping affects the stability of the system and the occurrence of the SMR. Most previous studies of the NESs have overlooked the effect of main structure damping, which may influence the selection of structural parameters. Moreover, under relatively large pulse or harmonic excitations, the vibration suppression effectiveness of the grounded damping NES surpasses that of traditional cubic stiffness NES. This finding has important practical significance for improving the vibration suppression effect and robustness of the NES, and provides a reference for the structural design of the NES.

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.

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.

This work examines a discrete Leslie–Gower predator–prey system with herd behavior, focusing on the influence of the slow–fast effect on predator dynamics. We analyze the presence and stability of fixed points, investigating period-doubling and Neimark–Sacker bifurcations at the positive fixed point. A hybrid control technique is implemented to successfully handle chaotic dynamics. Numerical simulations validate the theoretical findings, supporting our analysis. The parameter $𝜖$ in the predator–prey system has a significant impact on system dynamics due to the slow–fast effect it represents. A stable positive fixed point signifies a lasting and balanced coexistence of predator and prey populations within a certain $𝜖$ interval. Deviation from this range results in period-doubling and Neimark–Sacker bifurcations, increasing both unpredictability and instability in the system.

The SOS/ERK cascades are key signaling pathways that regulate cellular processes ranging from cellular proliferation, differentiation and apoptosis to tumor formation. However, the properties of these signaling pathways are not well understood. More importantly, how stochastic perturbations of internal and external cellular environment affect these pathways remains unanswered. To answer these questions, we, in this paper, propose a stochastic model according to the biochemical reaction processes of the SOS/ERK pathways, and, respectively, research the dynamical behaviors of this model under the four kinds of noises: Gaussian noise, colored noise, Lévy noise and fraction Brown noise. Some important results are found that Gaussian and colored noises have less effect on the stability of the system when the strength of the noise is small; Lévy and fractional Brownian noises significantly change the trajectories of the system. Power spectrum analysis shows that Lévy noise induces a system with quasi-periodic trajectories. Our results not only provide an understanding of the SOS/ERK pathway, but also show generalized rules for stochastic dynamical systems.

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.

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.

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.

In an ecosystem, harvesting infected prey can assist in managing and containing the spread of the illness within the prey species. On the other hand, the harvesting of predators can be beneficial as it regulates their numbers, preventing them from over-consuming prey and subsequently preserving existence of the prey population. This study introduces a predator–prey model that encompasses prey infection, predator–prey interactions influenced by fear, switching and harvesting. We derive an analytic expression for the basic reproduction number, a critical determinant of disease spread. We investigate the global stability of disease-free and endemic equilibria contingent on the basic reproduction number’s value, highlighting the potential for disease eradication by maintaining it below unity. In-depth analysis of the deterministic model is undertaken, with a focus on Hopf bifurcations that delineate thresholds for disease-free and endemic states. Furthermore, the deterministic model is extended to incorporate environmental stochasticity. We obtain the conditions under which population extinction occurs. Our findings elucidate how the intensity of environmental noise influences population dynamics, providing valuable insights into extinction risks under varying noise levels.