Search

This paper is devoted to the bifurcations and exact traveling wave solutions in a model of slightly dispersive quasi-incompressible hyperelastic materials by introducing a very general response function for the Cauchy stress tensor of dispersive hyperelastic solids, which was recently established by Saccomandi and Vergor [1]. By using the method of dynamical systems, we study the bifurcations of phase portraits and dynamical behaviors of solutions for the corresponding traveling wave system, which is a planar dynamical system. In addition to the results on the existence of solitary wave and kink wave solutions that has been proved, we not only find more traveling wave solutions, such as periodic wave and anti-kink wave solutions, but also obtain the exact explicit parametric representations and bifurcations of all possible traveling wave solutions under various parameter conditions. [ABSTRACT FROM AUTHOR]

Toward the understanding of bifurcation phenomena of dynamics on the Berkovich projective line $ \mathbb{P}^{1,an} $ over non-archimedean fields, we study the stability (or passivity) of critical points of families of rational functions parametrized by analytic curves. We construct the activity measure of a critical point of a family of rational functions, and study its properties. For a family of polynomials, we analyze the support of the activity measure, for example its relation to boundedness locus, i.e., the Mandelbrot set, and to the normality of the sequence of the forward orbit. [ABSTRACT FROM AUTHOR]

In this paper, we study a free boundary problem modeling the growth of 3-dimensional tumor cords. Since tumor cells grow freely in both the longitudinal and cross-sectional directions of blood vessels, the investigation of symmetry-breaking phenomena in both directions is biologically very reasonable. This forces the possible bifurcation value γ m , n to be dependent on two variables m and n. Some monotonicity properties of the possible bifurcation value μ n or μ j obtained in Friedman and Hu (2008) [1] and He and Xing (2023) [2] no longer hold here, which brings a great challenge to the bifurcation analysis. The novelty of this paper lies in determining the order of γ m , n for m 2 + n 2 . Together with periodicity and symmetry, we propose an effective method to avoid the need for the monotonicity of γ m , n. We give symmetry-breaking bifurcation results for every γ m , n > 0. [ABSTRACT FROM AUTHOR]

The overgrowth of algae in lakes often stems from an influx of nutrients from various sources, such as run-off from agricultural areas, anthropogenic and industrial drainage. Phosphorus and nitrogen play a crucial role as catalysts for algae growth, driving their rapid proliferation and leading to the formation of algal blooms. Both herbivorous and carnivorous fish play vital roles in the aquatic food web, and their presence can significantly affect the dynamics of algae within the aquatic ecosystem. Thus, a mathematical model is proposed to investigate the influence of fish on algae-nutrient interactions. For the model formulation, herbivorous fish are considered to depend on algae as their primary food source, while carnivorous fish rely on herbivorous fish for their survival and growth. Our analytical results confirm the existence of one parametric bifurcation, including saddle-node and Hopf bifurcations. Additionally, when the model is transformed into discrete-time intervals, it undergoes a Neimark-Sacker bifurcation. The existence of one parametric bifurcation is shown by considering the maximum uptake rate of nutrients by algae as a bifurcation parameter. Numerical simulations further demonstrate that the proposed model system exhibits two-parametric bifurcations, such as cusp, Bogdanov-Takens, generalized Hopf, Chenciner, and zero-Hopf bifurcations. The basin of stability is used to assess how the initial conditions and parameter values influence the bistability of the proposed mathematical model. This comprehensive analysis of algae-nutrient-fish interactions provides valuable insights into the complex dynamics of aquatic ecosystems, offering a foundation for better understanding and potentially managing algal blooms in aquatic ecosystem. [ABSTRACT FROM AUTHOR]

Nonlinear dynamical systems in such applications as design and control often depend on a set of parameters, resulting in parameterized dynamical systems. Establishing mathematical models of such parameterized systems is essential for numerical simulations, where first-principle models are often not affordable although they are desired whenever possible. Data-driven modeling is an important alternative with a trade-off among modeling difficulty, model complexity, and computational efficiency. However, data-driven modeling of such parameterized systems is challenging because not only nonlinear dynamics but also their parametric dependence need to be identified from data; especially for bifurcation systems where small changes in the parameters may cause drastic, qualitative changes of dynamical behaviors. Thus, data-driven modeling of bifurcation systems can be intractable. This work presents a novel method for data-driven modeling of bifurcation systems by transforming the intractable modeling problem into two tractable steps: first, learning an intermediate meta-model that is general for a wide range of bifurcation parameter values; subsequently, using this meta-model to perform efficient adaptation to target/new bifurcation values. Particularly, we leverage the meta-learning to guide the intermediate model to learn to generalize over the bifurcation parameter values, yielding a meta-model which allows a fast and data-efficient adaptation (generalization) to any new bifurcation parameter values. We conduct numerical experiments to validate the presented method on three classic bifurcation systems. It is observed that the adaptation (generalization) to new parameter values enabled by the obtained meta-model is faster than the direct modeling for the new parameter values from scratch. Furthermore, the meta-model-based adaptation yields more accurate models that allow long-term future-state prediction. Finally, we discuss the limitations of this work and potential future studies needed. [ABSTRACT FROM AUTHOR]

The present study explores the nonlinear dynamics of a self-excited smooth and discontinuous (SD) oscillator with geometric nonlinearity at the switching surface. Using a novel framework called hidden dynamics, introduced by Jeffrey, this work addresses the challenge posed by dry friction oscillators where the static friction coefficient is greater than the kinetic friction coefficient, which is ignored in Filippov's theory. By modelling the belt friction in the SD oscillator as Coulomb friction, the consequence of discontinuity in the friction model is investigated. The sliding regions were determined analytically using blow-up analysis and validated with numerical simulations. The system's behaviour is analyzed through the examination of phase portraits and bifurcations, and a comparison is conducted with Filippov's theory. Some interesting bifurcation phenomena are highlighted, including a novel phenomenon involving the collision and merging of two degenerate boundaries and the bifurcation of a sliding homoclinic orbit to a saddle. Furthermore, the system's response to harmonic excitation is analyzed, wherein the oscillator displays stick-slip limit cycles, pure slip limit cycles and the emergence of chaotic solutions through periodic doubling bifurcation. [ABSTRACT FROM AUTHOR]

Systems designed for nonlinear vibration isolation that incorporate chaotic states demonstrate superior capabilities in vibration attenuation, adeptly modulating the spectral constituents of vibrational noise. Yet, the challenge of eliciting low-amplitude chaotic dynamics and perpetuating these states across a diverse array of parameters remains formidable. This study proposes a pioneering strategy and technique for modulating the chaos band by incorporating a time-delayed feedback control mechanism within the framework of nonlinear vibration isolation systems.The investigation commences with an exhaustive analysis of the nonlinear dynamics, shedding light on the principles dictating the evolution of chaos. The study then advances to scrutinize the dynamics of systems with delays to elucidate the chaos-inducing processes engendered by feedback with temporal lags. Building upon the system's responses, the chaotic performance and the effectiveness of the vibration isolation are crafted. Consequently, the time-delayed feedback control parameters are identified as pivotal design variables, which are then employed to dissect the control mechanisms influenced by the time-delayed feedback on the chaos band. Utilizing the delineated control mechanism, the nonlinear vibration isolation system is precipitously transitioned from a state of stable periodicity to one of chaos, fostering low-amplitude chaotic dynamics across an expansive parameter space, and in turn, resolving the previously stated challenge. Perhaps most significantly, the mechanism for attaining low-amplitude chaos introduced here paves the way for innovative methodologies in the active vibration isolation design of similar systems. Furthermore, it is anticipated to yield theoretical guidance for the manipulation of chaos bands and the formulation of active vibration isolation strategies within the domain of nonlinear vibration isolation systems. [ABSTRACT FROM AUTHOR]

A thermal tooth backlash model of a planetary gear system, which includes the effects of thermal deformation and thermal elastohydrodynamic lubrication film, was established. The variation laws of tooth backlash under variable speed, variable torque, and constant power conditions, as well as the nonlinear dynamic characteristics of the system were analyzed. The results showed that the change in tooth backlash is greatly influenced by thermal deformation numerically, while the thermal elastohydrodynamic lubrication film affects the trend of tooth backlash along the meshing line direction. Combining bifurcation diagrams, Largest Lyapunov exponent plots, poincaré sections, phase portraits, and frequency spectra analysis reveals that under variable speed and constant power conditions, the presence of thermal tooth backlash reduces the chaotic range of the system and transforms some unstable motion states into more stable ones. However, for variable torque conditions, the influence of thermal tooth backlash on the system is more complex with both stable and unstable situations coexisting. This study provides a theoretical basis for selecting backlash parameters in planetary gear system design and avoiding chaotic responses. [ABSTRACT FROM AUTHOR]

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

Abstract 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 $$E^0$$ E 0 and $$E^*$$ E ∗ and if $${{\mathcal {B}}}{{\mathcal {R}}}1$$ B R > 1 , the equilibrium $$E^*$$ E ∗ exists and it is globally stable. Also, at $${{\mathcal {B}}}{{\mathcal {R}}}=1$$ B R = 1 the model exhibits a forward bifurcation. The sensitivity analysis of $${{\mathcal {B}}}{{\mathcal {R}}}$$ 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.

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.

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 bi-stability 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 2018–2020 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.

We are concerned with the principal eigenvalue of \begin{equation*} \begin{cases} -\Delta_p u= \lambda\theta_1\varphi_p(v), &x\in \Omega,\\ -\Delta_p v= \lambda\theta_2\varphi_p(u), &x\in \Omega,\\ u=0=v,\ &x\in\partial\Omega \end{cases} \tag{P} \end{equation*} and the global structure of positive solutions for the system \begin{equation*} \begin{cases} -\Delta_p u= \lambda f(v), &x\in \Omega,\\ -\Delta_p v= \lambda g(u), &x\in \Omega,\\ u=0=v,\ &x\in\partial\Omega, \end{cases} \tag{Q} \end{equation*} where $\varphi_p(s)=|s|^{p-2}s$, $\Delta_p s=\text{div}(|\nabla s|^{p-2}\nabla s)$, $\lambda>0$ is a parameter, $\Omega\subset\mathbb{R}^N$, $N> 2$, is a bounded domain with smooth boundary $\partial\Omega$, $f,g:\mathbb{R}\to(0,\infty)$ are continuous functions with $p$-superlinear growth at infinity. We obtain the principal eigenvalue of $(P)$ by using a nonlinear Krein–Rutman theorem and the unbounded branch of positive solutions for $(Q)$ via bifurcation technology.

The pursuit of solitary wave solutions to complex nonlinear partial differential equations is gaining significance across various disciplines of nonlinear science. This study seeks to uncover the solutions to the perturbed nonlinear Schrödinger equation using a robust and efficient analytical method, namely, the generalized exponential rational function technique. This equation is a fundamental tool used in various fields, including fluid mechanics, nonlinear optics, plasma physics, and optical communication systems, and has numerous practical applications across multiple disciplines. The employed method in this study stands out from existing approaches by being more comprehensive and straightforward. It offers a broader range of symbolic structures, surpassing the capabilities of some previously known methods. By applying this method to the perturbed nonlinear Schrödinger equation, we obtain a variety of exact solutions that significantly expand the existing literature and provide a fresh understanding of the model’s properties. Through numerical simulations, we demonstrate the dynamic characteristics of the system, including bifurcation and chaos analysis, and validate our findings by adjusting parameter settings to match expected behaviors.

Bursting is a crucial form of firing in neurons, laden with substantial information. Studying it can aid in understanding the neural coding to identify human behavioral characteristics conducted by these neurons. However, the high-dimensionality of many neuron models imposes a difficult challenge in studying the generative mechanisms of bursting. On account of the high complexity and nonlinearity characteristic of these models, it becomes nearly impossible to theoretically study and analyze them. Thus, this paper proposed to address these issues by focusing on the midbrain dopamine neurons, serving as the central neuron model for the investigation of the bursting mechanisms and bifurcation behaviors exhibited by the neuron. In this study, we considered the dimensionality reduction of a high-dimensional neuronal model and analyzed the dynamical properties of the reduced system. To begin, for the original thirteen-dimensional model, using the correlation between variables, we reduced its dimensionality and obtained a simplified three-dimensional system. Then, we discussed the changing characteristics of the number of spikes within a burst by simultaneously varying two parameters. Finally, we studied the co-dimension-2 bifurcation in the reduced system and presented the bifurcation behavior near the Bogdanov-Takens bifurcation.

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, Poincaré 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.

Molecular biological networks are highly nonlinear systems that exhibit limit point singularities. Bifurcation analysis and multiobjective nonlinear model predictive control (MNLMPC) of a molecular network problem represented by the Pettigrew model were performed. The Matlab program MATCONT (Matlab continuation) was used for the bifurcation analysis and the optimization language PYOMO (python optimization modeling objects) was used for performing the multiobjective nonlinear model predictive control. MATCONT identified the limit points, branch points, and Hopf bifurcation points using appropriate test functions. The multiobjective nonlinear model predictive control was performed by first performing single objective optimal control calculations and then minimizing the distance from the Utopia point, which was the coordinate of minimized values of each objective function. The presence of limit points (albeit in an infeasible region) enabled the MNLMPC calculations to result in the Utopia solution. MNLMPC of the partial models also resulted in Utopia solutions.

Abstract This research paper investigates discrete predator-prey dynamics with two logistic maps. The study extensively examines various aspects of the system’s behavior. Firstly, it thoroughly investigates the existence and stability of fixed points within the system. We explores the emergence of transcritical bifurcations, period-doubling bifurcations, and Neimark-Sacker bifurcations that arise from coexisting positive fixed points. By employing central bifurcation theory and bifurcation theory techniques. Chaotic behavior is analyzed using Marotto’s approach. The OGY feedback control method is implemented to control chaos. Theoretical findings are validated through numerical simulations.

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.

For the cantilever beam vibration model with small damping and forced terms, under some special parameter conditions, the corresponding unperturbed system can be described by a planar dynamical system. In this paper, by using the dynamical system method to analyze the unperturbed differential system and find bifurcations from the corresponding phase portraits, the dynamical behavior can be derived. Exact explicit homoclinic and heteroclinic solutions, and periodic solution families can be found. For the periodic perturbation system, the chaotic behavior of system is revealed. Numerical simulations are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This research delves into simple Jerk-type dynamical networks, constructed using a specific bidirectional coupling scheme that influences the sub-oscillators and compound nonlinearity gradient functions that perturb each sub-unit of the network. The novelty of this work is to demonstrate a new pedagogical method able to stimulate the formation of higher-order multiscroll dynamics in the Jerk sub-oscillator. This is done by modeling a simple dynamic network built by applying a special bidirectional coupling between the Jerk sub-systems. Another novelty of this approach is that it allows us to study the collective dynamics of Jerk system networks, which has not yet been done in the literature. Multiscroll systems are exceedingly complex dynamically, making them valuable in chaos-based applications. The pedagogical approach used in this study is exceptional because it produces many additional equilibrium points (from 5 to 25 in dynamical network 1, and from 3 to 15 in dynamical network 2) in each Jerk sub-unit of the network. These equilibria elevate the complexity of Jerk systems by emulating higher-order multiscroll dynamics. The methodology used in this study is efficient and differs from those used in the literature, which mainly uses nonlinear multizero functions in dissipative systems. This research further explores the dynamic system characterization tools and conducts an experimental investigation on a microcontroller (ATMEGA2560) to confirm the predictions. [ABSTRACT FROM AUTHOR]

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

In this paper, a three-dimensional (3D) Filippov SEIS epidemic model is proposed to characterize the impact of limited medical resources on disease transmission with discontinuous treatment functions. Qualitative analysis of non-smooth dynamical behaviors are performed on two subsystems and sliding modes. Criteria on the stability of various kinds of feasible equilibria and bifurcations, e.g., saddle-node bifurcation, transcritical bifurcation, and boundary equilibrium bifurcation, are established. The theoretical results are illustrated by numerical simulation, from which we find there could exist bistable phenomena, e.g., endemic and pseudo-equilibria, endemic equilibria of the two subsystems, or endemic and disease-free equilibria, even the basic reproduction numbers of two subsystems are less than 1. The disease spread is dependent both on the limited medical resources and latent compartment, which are more beneficial to effective disease control than planar Filippov and smooth models. [ABSTRACT FROM AUTHOR]

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]

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]

This work addresses the significant issue of plasma waves interacting with non-linear dynamical systems in both perturbed and unperturbed states, as modeled by the generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt (WBK-BK) Equations. We investigate analytical solutions and the subsequent emergence of chaos within these systems. Initially, we apply advanced mathematical techniques, including the transform method and the G ′ G 2 method. These methods allow us to derive new precise solutions and enhance our understanding of the non-linear processes dominating plasma wave dynamics. Through a systematic analysis, we identify the conditions under which the system transitions from orderly patterns to chaotic behavior. This investigation provides valuable insights into the fundamental mechanisms of non-linear wave propagation in plasmas. Our results highlight the dynamic interplay between non-linearity and variation, leading to chaos, which may be useful in predicting and potentially controlling similar phenomena in practical applications. [ABSTRACT FROM AUTHOR]

This paper delves into the dynamics of a discrete-time predator–prey system. Initially, it presents the existence and stability conditions of the fixed points. Subsequently, by employing the center manifold theorem and bifurcation theory, the conditions for the occurrence of four types of codimension 1 bifurcations (transcritical bifurcation, fold bifurcation, flip bifurcation, and Neimark–Sacker bifurcation) are examined. Then, through several variable substitutions and the introduction of new parameters, the conditions for the existence of codimension 2 bifurcations (fold–flip bifurcation, 1:2 and 1:4 strong resonances) are derived. Finally, some numerical analyses of two-parameter planes are provided. The two-parameter plane plots showcase interesting dynamical behaviors of the discrete system as the integral step size and other parameters vary. These results unveil much richer dynamics of the discrete-time model in comparison to the continuous model. [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 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 this work, the influence of magneto-rheological fluid embedded on journal bearings in the dynamic behavior of rotors is considered. The modified Reynolds equations for Bingham viscoplastic materials are used for calculation of the nonlinear hydrodynamic forces. Flexible rotors are modeled by the finite element method. The static weight of the rotor, unbalance and bearing hydrodynamic forces are included in the equations of motion. Non-linear hydrodynamic forces calculation depends on the relative positions of the journal bearings. The dynamic system response is computed by the Newmark method modified to obtain the calculation of the differential displacements and velocities for each time step. By incorporating the Newton-Raphson method the necessary corrections are included in the equations of motion. Time and frequency responses are presented for two of the case studies. The sudden elevation in oscillation magnitudes due to the oil whip phenomenon is not observed in the 1-un-up test after the application of electromagnetic induction on the MR fluid. Furthermore, the controlled variation in the viscosity of the MR fluid causes significant changes in the bearing movements, as demonstrated by the orbit graphs. [ABSTRACT FROM AUTHOR]

We investigate the existence of nodal solutions for the nonlinear Lidstone boundary value problem { (− 1) m (u (2 m) (t) + c u (2 m − 2) (t)) = λ a (t) f (u) , t ∈ (0 , r) , u (2 i) (0) = u (2 i) (r) = 0 , i = 0 , 1 , ⋯ , m − 1 , (P) where λ > 0 is a parameter, c is a constant, m ≥ 1 is an integer, a : [ 0 , r ] → [ 0 , ∞) is continuous with a ≢ 0 on the subinterval within [ 0 , r ] , and f : R → R is a continuous function. We analyze the spectrum structure of the corresponding linear eigenvalue problem via the disconjugacy theory and Elias's spectrum theory. As applications of our spectrum results, we show that problem (P) has nodal solutions under some suitable conditions. The bifurcation technique is used to obtain the main results of this paper. [ABSTRACT FROM AUTHOR]

In this paper, two classes of near-Hamiltonian systems with a nilpotent center are considered: the coexistence of algebraic limit cycles and small limit cycles. For the first class of systems, there exist 2 n + 1 limit cycles, which include an algebraic limit cycle and 2 n small limit cycles. For the second class of systems, there exist n 2 + 3 n + 2 2 limit cycles, including an algebraic limit cycle and n 2 + 3 n 2 small limit cycles. [ABSTRACT FROM AUTHOR]

This study aims to analyze how the parameter flow rate and amplitude of walling waves affect the peristaltic flow of Jeffrey's fluid through an irregular channel. The movement of the fluid is described by a set of non-linear partial differential equations that consider the influential parameters. These equations are transformed into non-dimensional forms with appropriate boundary conditions. The study also utilizes dynamic systems theory to analyze the effects of the parameters on the streamline and to investigate the position of critical points and their local and global bifurcation of flow. The research presents numerical and analytical methods to illustrate the impact of flow rate and amplitude changes on fluid transport. It identifies three types of streamline patterns that occur: backwards, trapping, and augmented flow resulting from changes in the value of flow rate parameters. [ABSTRACT FROM AUTHOR]

The problem of density wave propagation is considered for a logistic equation with delay and diffusion. This equation, called the Kolmogorov–Petrovsky–Piscounov–Fisher equation with delay, is investigated by asymptotic and numerical methods. Local properties of solutions corresponding to this equation with periodic boundary conditions are studied. It is shown that an increase in the period leads to the emergence of stable solutions with a more complex spatial structure. The process of wave propagation from one and from two initial perturbations is analyzed numerically, which allows tracing the process of wave interaction in the second case. The complex spatially inhomogeneous structure arising during the wave propagation and interaction can be explained by the properties of the corresponding solutions of a periodic boundary value problem with an increasing range of the spatial variable. [ABSTRACT FROM AUTHOR]

In the spread of infectious diseases, intervention levels play a crucial role in shaping interactions between healthy and infected individuals, leading to a nonlinear transmission process. Additionally, the availability of medical resources limits the recovery rate of infected patients, adding further nonlinear dynamics to the healing process. Our research introduces novelty by combining nonlinear incidence and recovery rates alongside waning immunity in an epidemic model. We present a modified SIRW-type model, examining the epidemic problem with these factors. Through analysis, we explore conditions for non-endemic and co-existing cases based on the basic reproduction ratio. The local stability of equilibria is verified using the Routh-Hurwitz criteria, while global stability is assessed using Lyapunov functions for each equilibrium. Furthermore, we investigate bifurcations around both non-endemic and co-existing equilibria. Numerically, we give some simulations to support our analytical findings. [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]