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In the paper, the existence of soliton and periodic solutions is studied for the perturbed sine–cosine–Gordon equation. The corresponding traveling wave system is converted to a regularly Hamilton system via using bifurcation theory of differential equations and the geometric singular perturbation theory. Then by using Melnikov's method and symbolic computation, periodic wave solutions, solitary solutions, and kink and antikink solutions are obtained. The wave speed selection principles are given, respectively. Numerical simulations are performed to illustrate our theoretical analysis. [ABSTRACT FROM AUTHOR]

We consider the Paneitz type equation Δ 2 u − α Δ u + β (u − u q) = 0 on a closed Riemannian manifold (M n , g) of dimension n ≥ 3. We reduce the equation to a fourth order ordinary differential equation assuming that (M , g) admits a proper isoparametric function. Assuming that q > 1 , α and β are positive and α 2 > 4 β , we prove that the global nonconstant solutions of this ordinary differential equation only have nondegenerate critical points. Applying global bifurcation theory we then prove multiplicity results for positive solutions of the equation when q < p ⁎ , where p ⁎ = n + 4 n − 4 if n > 4 and p ⁎ = ∞ if n = 3 , 4. As an application and motivation we prove multiplicity results for conformal constant Q -curvature metrics. For example, consider closed positive Einstein manifolds (M n , g) and (X m , h) of dimensions n , m ≥ 3. Assuming that M admits a proper isoparametric function (with a symmetry condition) we prove that as δ > 0 gets close to 0, the number of constant Q -curvature metrics conformal to g δ = g + δ h goes to infinity. [ABSTRACT FROM AUTHOR]

In this paper, we study a system of reaction–diffusion equations arising from the competition of two competing species for a single limited nutrient with flocculation in an unstirred chemostat. By the conservation principle, we reduce the dimension of the system by eliminating the equation for the nutrient. Then the global structure of the reduced system is studied by the bifurcation theory in its feasible domain. Finally, we use numerical simulation to verify and supplement our theoretical results. [ABSTRACT FROM AUTHOR]

This study examines the modified Sardar sub-equation method (MSSEM) for deriving the novel solutions of the (3+1)-dimensional p-type model. This framework is commonly employed to explain the behavior of optical solitons in nonlinear media. The applications of MSSEM allows us to acquire the precise analytical solutions, which incorporate a diverse array of optical soliton solutions. We discuss the dynamical structure of the solitons, bifurcation and chaos theory to develop the multiple soliton solutions, including rational, hyperbolic, exponential, and trigonometric functions and depending on the principle of balancing equation. Moreover, by using bifurcation and chaos theory, we examine the governing model with and without the perturbation term and provide the three-dimensional, two-dimensional, and density profiles to improve the clarity of obtained results. The different aspects of the solutions are evident in our visual representations. These solutions are applicable to a wide range of domains, including fluid physics, oceanography, physics, engineering, and nonlinear optics. [ABSTRACT FROM AUTHOR]

This paper is motivated by study of long timescale variability of the climate system. We focus on a model of nonlinear behaviour that is used in climate modelling. This is the forced van der Pol oscillator, motivated by examination of the Pleistocene ice age oscillations forced by astronomical orbital variations. We discuss a forced van der Pol oscillator, following the analysis of Guckenheimer et al. for periodically cases. We use a geometric singular perturbation theory (GSPT) approach of Guckenheimer et al. to reduce to the dynamics of the return map and extend to their work to construct return maps for quasiperiodically forced cases. We note this return map can be noninvertible in various values to the parameters. [ABSTRACT FROM AUTHOR]

In this paper, a delayed predator–prey model with a square root functional response is structured and analyzed. Through a discussion of the time delay and an analysis of the characteristic equations, the local stability of the boundary equilibrium and the positive equilibrium and the existence of Hopf bifurcation are investigated. On this basis, the critical value of the Hopf bifurcation is derived. According to the central manifold theorem and normal form theory, the nature of the Hopf bifurcation is obtained. Finally, by conducting numerical simulations, it is observed that incorporating a time delay can influence the stability of the predator and prey populations, causing periodic oscillations in the number of two populations. [ABSTRACT FROM AUTHOR]

Mathematical model plays a significant role in describing the mutual interaction of various chemical compositions in chemistry. In this present work, we formulate a new plankton–oxygen dynamical model owning delay. By virtue of fixed point theorem, inequality techniques and construction of function, we set up the conditions on existence and uniqueness, non-negativeness and boundedness of the solution to the formulated plankton–oxygen model. Taking advantage of bifurcation and stability theory of delayed differential equation, we explore the existence of bifurcation and stability for the plankton–oxygen model and set up a novel delay-independent criterion ensuring the existence of bifurcation and stability of the model. Making use of two different extended hybrid controllers, we can successfully control the time of emergence of bifurcation and stability domain of this model. The impact of delay on bifurcation and stability of the model is explored. Software experiment results are provided to sustain the acquired key outcomes. The gained conclusions of this work are perfectly novel and possess immense theoretical significance in adjusting and balancing the concentrations of disparate chemical compositions. [ABSTRACT FROM AUTHOR]

This work introduces a discrete modified Leslie–Gower prey–predator system with Holling type-II functional response. The persistence of the discrete model under certain conditions is discussed. The conditions assuring the existence of fixed points are derived and nonlinear dynamics of system are explored at these fixed points. It has been shown that the system exhibits transcritical bifurcation and flip bifurcation at semi-trivial fixed point under certain bifurcation values. In addition, the center manifold and bifurcation theories are employed to attain the conditions for existence of flip and Neimark–Sacker bifurcations at coexistence fixed point. The system is found to exhibit periodic solutions along with bifurcations leading to wide range of chaotic dynamics. The numerical simulations are performed to confirm the analytical analysis. [ABSTRACT FROM AUTHOR]

This paper proposed two new generalizations of the logistic function, each drawing on non-extensive thermodynamics, the q-logistic Equation and the logistic Equation of arbitrary order, respectively. It demonstrated the impact of chaos theory by integrating it with logistics Equations and revealed how minor parameter variations will change system behavior from deterministic to non-deterministic behavior. Moreover, this work presented BifDraw – a Python program for drawing bifurcation diagrams using classical logistic function and its generalizations illustrating the diversity of the system’s response to the changes in the conditions. The research gave a pivotal role to the place of the logistic Equation in chaos theory by looking at its complicated dynamics and offering new generalizations that may be new in terms of thermodynamic basic states and entropy. Also, the paper investigated dynamics nature of the Equations and bifurcation diagrams in it which present complexity and the surprising dynamic systems features. The development of the BifDraw tool exemplifies the practical application of theoretical concepts, facilitating further exploration and understanding of logistic Equations within chaos theory. This study not only deepens the comprehension of logistic Equations and chaos theory, but also introduces practical tools for visualizing and analyzing their behaviors. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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

A deterministic dynamical system that slowly passes through a generic fold-type (saddle-node) bifurcation can be reduced to one-dimensional dynamics close to the bifurcation because of the center manifold theorem. It is often tacitly assumed that the same is true in the presence of stochasticity or noise so that, for example, critical slowing down (CSD) indicators can be applied as if the system were one-dimensional. In this work, we show that this is only true when given suitable system observables; specifically, we demonstrate that noise in other dimensions may interfere with indicators of CSD, also referred to as early warning signals (EWS). We point out a generic mechanism by which both variance and lag-1 autocorrelation (AC(1)), as well as other EWS, can fail to signal an approaching bifurcation. This can, in principle, occur whenever one noise source drives multiple system components simultaneously. Even under the favorable assumptions of uncoupled deterministic dynamics and stationary noise, some system observables can then exhibit false negative or false positive CSD indications. We isolate this phenomenon in an example that represents a generic two-dimensional fold-type bifurcation setting. [ABSTRACT FROM AUTHOR]

In the paper, a predator-prey model with the Allee effect and harvesting effort was proposed to explore the interaction mechanism between prey and predator. Under the framework of mathematical theory deduction, some conditions for the occurrence of transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations were derived with harvesting effort and the Allee effect as key parameters. Under the framework of bifurcation dynamics numerical simulation, the evolution process of specific bifurcation dynamics behavior was gradually visualized to reveal the influence mechanism of the Allee effect and harvesting effort. The research results indicated that the Allee effect and harvesting effort not only seriously affected the bifurcation dynamics essential characteristics of the model (1.3), but also could promote the formation of constant steady state and periodic oscillation persistent survival mode of prey and predator. Furthermore, it is worth noting that appropriate harvesting effort was beneficial for the formation of a sustainable survival cycle between prey and predator. In summary, we hoped that the research findings could contribute to the comprehensive promotion of bifurcation dynamics studies in the predator-prey model. [ABSTRACT FROM AUTHOR]

This study investigates a symmetric fractional-order epidemic model with time delays and non-monotonic incidence rates, considering two viral strains. By confirming the existence, uniqueness, and boundedness of the system's solutions, the research ensures the model's well-posedness, guaranteeing its mathematical soundness and practical relevance. The study calculates and evaluates the equilibrium points and the basic reproduction numbers R 0 1 and R 0 2 to understand the dynamic behavior of the model under different parameter settings. Through the application of the Lyapunov method, the research examines the asymptotic global stability of the system, determining whether it will converge to a particular equilibrium state over time. Furthermore, Hopf bifurcation theory is employed to investigate potential periodic solutions and bifurcation scenarios, highlighting how the system might shift from stability to periodic oscillations under certain conditions. By utilizing the Adams-Bashforth-Moulton numerical simulation method, the theoretical results are validated, reinforcing the conclusions and demonstrating the model's applicability in real-world contexts. It emphasizes the importance of fractional-order models in addressing epidemiological issues related to time delays (τ), individual heterogeneity (m, k), and memory effects (θ), offering greater accuracy compared with traditional integer-order models. In summary, this research provides a theoretical foundation and practical insights, enhancing the understanding and management of epidemic dynamics through fractional-order epidemic models. [ABSTRACT FROM AUTHOR]

In this paper, we consider the instability of a constant equilibrium solution in a general activator–inhibitor (depletion) model with passive diffusion, cross‐diffusion, and nonlocal terms. It is shown that nonlocal terms produce linear stability or instability, and the system may generate spatial patterns under the effect of passive diffusion and cross‐diffusion. Moreover, we analyze the existence of bifurcating solutions to the general model using the bifurcation theory. At last, the theoretical results are applied to the spatial water–biomass system combined with cross‐diffusion and nonlocal grazing and Holling–Tanner predator–prey model with nonlocal prey competition. [ABSTRACT FROM AUTHOR]

The exact solutions and qualitative analysis of the stochastic governing model for embedded solitons with χ (2) and χ (3) nonlinear susceptibilities are investigated in this study. The model introduces a stochastic term-white noise for the first time, bringing the model closer to reality. The trial equation method is used for mathematical analysis and the complete discriminant system for polynomial method is used for qualitative analysis. Using the bifurcation theory and the complete discriminant system for polynomial method, the existence of the soliton and periodic solutions is confirmed and the exact travelling wave solutions are generated to validate our findings. Furthermore, we explore the various sorts of exact solutions by illustrating the associated phase diagrams and providing two-dimensional diagrams to demonstrate the model’s dynamical behavior. The plethora of exact solutions shows that the effect of white noise exists only in the phase component of the solitons, providing insight into the optical solitons of stochastic nonlinear models. [ABSTRACT FROM AUTHOR]

Mathematical models play a crucial role in assisting public health authorities in timely disease control decision-making. For vector-borne diseases, integrating host and vector dynamics into models can be highly complex, particularly due to limited data availability, making system validation challenging. In this study, two compartmental models akin to the SIR type were developed to characterize vector-borne infectious disease dynamics. Motivated by dengue fever epidemiology, the models varied in their treatment of vector dynamics, one with implicit vector dynamics and the other explicitly modeling mosquito-host contact. Both considered temporary immunity after primary infection and disease enhancement in secondary infection, analogous to the temporary cross-immunity and the Antibody-dependent enhancement biological processes observed in dengue epidemiology. Qualitative analysis using bifurcation theory and numerical experiments revealed that the immunity period and disease enhancement outweighed the impact of explicit vector dynamics. Both models demonstrated similar bifurcation structures, indicating that explicit vector dynamics are only justified when assessing the effects of vector control methods. Otherwise, the extra equations are irrelevant, as both systems display similar dynamics scenarios. The study underscores the importance of using simple models for mathematical analysis, initiating crucial discussions among the modeling community in vector-borne diseases. [ABSTRACT FROM AUTHOR]

Recently, the research on the modified Leslie–Gower model has become an appealing topic. Due to economic efficiency and the complexity of discrete models, we investigate a discrete modified Leslie–Gower predator–prey model with prey harvesting in this paper. The stability of fixed points and bifurcations of the interior fixed points are studied. According to bifurcation theory and normal forms, we derived the conditions of codimension 2 bifurcations occurred, including 1:1 strong resonance bifurcation and fold-flip bifurcation. These two bifurcations are unusual in bifurcation analysis on discrete systems. In addition, the continuation curves, bifurcation diagrams, and phase diagrams are used to demonstrate theoretical results. Our study shows the interesting dynamics of this model that are very different from the continuous one. [ABSTRACT FROM AUTHOR]

Single abnormal gene structure of disorders, specifically the alpha (α) and beta (β) thalassemia recessive disorders are focused. From the NCBI website, the preferred DNA sequencing is downloaded. The objective is to study the structure of Single Abnormal Gene using modified Box counting principle and FFD-Fuzzy Fractal Dimension analysis. Initially the fractal dimension method is used and analyzed single abnormal gene structure with the help of box counting method where the grids are segmented into triangles. Further the analysis is enhanced through grid and triangular method of improved box counting methods named as Ruby Triangular dimension which is the novelty of the research. Comparison of Grid Dimension with Triangular Dimension based fractal and fuzzy fractal dimension in the severity of disease from its secondary structure of the disorder related genes structures are performed. Further the complexity of the Single Abnormal Gene structure evaluated to generate a unique Attractor for the prediction of the α-thalassemia and β-thalassemia disorder in earlier diagnosis, refer as bifurcation theory. The results shows that the triangular Ruby Dimension based improved box counting method facilitate quick with more exactitude. In grid method the size of the image should be 2n pixels and shrink to at most 2048 pixels, whereas the triangular pixels may be reduced to 23 times than grid method. Hence, this novel Fuzzy Fractal Ruby Triangular Dimension method shows better results and can be applied for image of higher dimensions with the same procedure. [ABSTRACT FROM AUTHOR]

In this contribution, the complicated dynamical behaviors and optimal harvesting policy of a discrete-time predator–prey model with fear and refuge effects are formulated. Both the fear and prey refuge effects refer to an interaction between predator and prey. In the first place, the existence and local stability of three fixed points of proposed model are investigated by virtue of our methodology, that is, the eigenvalues of the Jacobian matrix. One step further, it is worth mentioning that the model undergoes flip bifurcation (i.e., period–doubling bifurcation) and Neimark–Sacker bifurcation at the interior fixed point by the utilization of bifurcation theory and center manifold theory. Also, optimal harvesting strategy is investigated, and the expressions of optimal harvesting efforts are determined. Two examples, in the end, are put forward to prove that they are consistent with the previous theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study a discretized one‐predator two‐prey system along with prey refuge and Michaelis–Menten‐type prey harvesting. The interaction among the species is considered as Holling type III functional response. Firstly, existence and local stability of all the fixed points are derived under certain parametric conditions. Furthermore, a special consideration is made to global asymptotic stability of the interior fixed point. Then, we have shown that the system undergoes different types of bifurcations including transcritical bifurcation, flip bifurcation, and Neimark–Sacker bifurcation by using center manifold theorem, bifurcation theory, and normal form method. Also, Feigenbaum's constant of the system is calculated. It is observed that both harvesting and refuge have a stabilizing effect on the system, and the stabilizing effect of harvesting dominates the stabilizing effect of refuge. Of most interest is the finding of coexisting attractors and multistability. In particular, optimal harvesting policy has been obtained by extension of Pontryagin's maximum principle to discrete system. Finally, some intriguing numerical simulations are provided to verify our analytic findings and rich dynamics of the three species system. [ABSTRACT FROM AUTHOR]

This paper strives to investigate the time fractional system that characterizes the ion sound wave influenced by the ponderomotive force induced by a high‐frequency field, as well as the Langmuir wave in plasma. Initially, based on the qualitative theory for planar integrable systems, four‐phase portraits are found in the (u,y)$$ \left(u,y\right) $$ phase plane under certain conditions on the physical parameters. These conditions are used to prove analytically the existence of solitary, kink (anti‐kink), periodic, super‐periodic, and unbounded wave solutions. The correspondence between the energy levels, phase orbits, and consequently the type of the solution is announced. We derived the bounded wave solutions associated with the phase orbits, which are shown to be consistent with the qualitative analysis of the types of solutions. Moreover, we studied the consistency between the obtained solutions by investigating the degeneracy of the solutions through the transmission between the phase orbits, or equivalently, through the dependence on the initial conditions. With the presence of perturbed periodic terms, the quasi‐periodic behavior and chaotic patterns are investigated. [ABSTRACT FROM AUTHOR]

This paper consists of four parts. It begins by using the authors' generalized Schauder formula, [41] , and the algebraic multiplicity, χ , of Esquinas and López-Gómez [15,14,31] to package and sharpening all existing results in local and global bifurcation theory for Fredholm operators through the recent author's axiomatization of the Fitzpatrick–Pejsachowicz–Rabier degree, [42]. This facilitates reformulating and refining all existing results in a compact and unifying way. Then, the local structure of the solution set of analytic nonlinearities F (λ , u) = 0 at a simple degenerate eigenvalue is ascertained by means of some concepts and devices of Algebraic Geometry and Galois Theory, which establishes a bisociation between Bifurcation Theory and Algebraic Geometry. Finally, the unilateral theorems of [31,33] , as well as the refinement of Shi and Wang [53] , are substantially generalized. This paper also analyzes two important examples to illustrate and discuss the relevance of the abstract theory. The second one studies the regular positive solutions of a multidimensional quasilinear boundary value problem of mixed type related to the mean curvature operator. [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]

In this paper, we study Poincaré, Hopf and homoclinic bifurcations of limit cycles for planar near-Hamiltonian systems. Our main results establish Hopf and homoclinic bifurcation theories by higher order Melnikov functions, obtaining conditions on upper bounds and lower bounds of the maximum number of limit cycles. As an application, we concern a cubic near-Hamiltonian system, and study Hopf and homoclinic bifurcations in detail, finding more limit cycles than [26]. [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]

Incorporating a weak Allee effect in a two-dimensional biological model in ℜ 2 , the study delves into the application of bifurcation theory, including center manifold and Ljapunov–Schmidt reduction, normal form theory, and universal unfolding, to analyze nonlinear stability issues across various engineering domains. The focus lies on the qualitative dynamics of a discrete-time system describing the interaction between prey and predator. Unlike its continuous counterpart, the discrete-time model exhibits heightened chaotic behavior. By exploring a biological Mmdel with linear functional prey response, the research elucidates the local asymptotic properties of equilibria. Additionally, employing bifurcation theory and the center manifold theorem, the analysis reveals that, for all α 1 (i.e., intrinsic growth rate of prey), ð 1 ˙ (i.e., parameter that scales the terms y n), and m (i.e., Allee effect constant), the model exhibits boundary fixed points A 1 and A 2 , along with the unique positive fixed point A ∗ , given that the all parameters are positive. Additionally, stability theory is employed to explore the local dynamic characteristics, along with topological classifications, for the fixed points A 1 , A 2 , and A ∗ , considering the impact of the weak Allee effect on prey dynamics. A flip bifurcation is identified for the boundary fixed point A 2 , and a Neimark–Sacker bifurcation is observed in a small parameter neighborhood around the unique positive fixed point A ∗ = (m ð 1 ˙ − 1 , α 1 − 1 − α 1 m ð 1 ˙ − 1). Furthermore, it implements two chaos control strategies, namely, state feedback and a hybrid approach. The effectiveness of these methods is demonstrated through numerical simulations, providing concrete illustrations of the theoretical findings. The model incorporates essential elements of population dynamics, considering interactions such as predation, competition, and environmental factors, along with a weak Allee effect influencing the prey population. [ABSTRACT FROM AUTHOR]

Aiming to explore the subtle connection between the number of nonlinear terms in Lorenz-like systems and hidden attractors, this paper introduces a new simple sub-quadratic four-thirds-degree Lorenz-like system, where x ˙ = a (y − x) , y ˙ = c x − x 3 z , z ˙ = − b z + x 3 y , and uncovers the following property of these systems: decreasing the powers of the nonlinear terms in a quadratic Lorenz-like system where x ˙ = a (y − x) , y ˙ = c x − x z , z ˙ = − b z + x y , may narrow, or even eliminate the range of the parameter c for hidden attractors, but enlarge it for self-excited attractors. By combining numerical simulation, stability and bifurcation theory, most of the important dynamics of the Lorenz system family are revealed, including self-excited Lorenz-like attractors, Hopf bifurcation and generic pitchfork bifurcation at the origin, singularly degenerate heteroclinic cycles, degenerate pitchfork bifurcation at non-isolated equilibria, invariant algebraic surface, heteroclinic orbits and so on. The obtained results may verify the generalization of the second part of the celebrated Hilbert's sixteenth problem to some degree, showing that the number and mutual disposition of attractors and repellers may depend on the degree of chaotic multidimensional dynamical systems. [ABSTRACT FROM AUTHOR]

The production of oxygen through phytoplankton photosynthesis is a crucial phenomenon in the dynamics of marine ecosystems. A generic oxygen-phytoplankton interaction model is considered to comprehend its underlying mechanism. This paper investigates the discrete-time dynamics of oxygen and phytoplankton in aquatic ecosystems, incorporating factors that cause phytoplankton mortality due to external influences. We explore the conditions for the local stability of steady states concerning the oxygen content in dissolved water and phytoplankton density. The analysis reveals that the model undergoes a co-dimension one bifurcation, encompassing flip and Neimark–Sacker bifurcations, utilizing the center manifold theorem and bifurcation theory. To manage the chaos resulting from the Neimark–Sacker bifurcation, we apply the OGY feedback control method and a hybrid control methodology. Finally, we present numerical simulations to validate the theoretical discussion. [ABSTRACT FROM AUTHOR]

Solar radiation pressure can have a substantial long-term effect on the orbits of high area-to-mass ratio spacecraft, such as solar sails. We present a study of the coupling between radiation pressure and the gravitational perturbation due to polar flattening. Removing the short-period terms via perturbation theory yields a time-dependent two-degree-of-freedom Hamiltonian, depending on one physical and one dynamical parameter. While the reduced model is non-integrable in general, assuming coplanar orbits (i.e., both Spacecraft and Sun on the equator) results in an integrable invariant manifold. We discuss the qualitative features of the coplanar dynamics, and find three regions of the parameters space characterized by different regimes of the reduced flow. For each regime, we identify the fixed points and their character. The fixed points represent frozen orbits, configurations for which the long-term perturbations cancel out to the order of the theory. They are advantageous from the point of view of station keeping, allowing the orbit to be maintained with minimal propellant consumption. We complement existing studies of the coplanar dynamics with a more rigorous treatment, deriving the generating function of the canonical transformation that underpins the use of averaged equations. Furthermore, we obtain an analytical expression for the bifurcation lines that separate the regions with different qualitative flow. [ABSTRACT FROM AUTHOR]

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In this paper, by using the bifurcation theory for dynamical system, we construct traveling wave solutions of a high-order nonlinear Schrödinger equation with a quintic nonlinearity. Firstly, based on wave variables, the equation is transformed into an ordinary differential equation. Then, under the parameter conditions, we obtain the Hamiltonian system and phase portraits. Finally, traveling wave solutions which contains solitary, periodic and kink wave solutions are constructed by integrating along the homoclinic or heteroclinic orbits. In addition, by choosing appropriate values to parameters, different types of structures of solutions can be displayed graphically. Moreover, the computational work and it's figures show that this technique is influential and efficient. [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]

We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degree theory. [ABSTRACT FROM AUTHOR]

The present study has focused on the examination of a Holling type III nonlinear mathematical model that incorporates the influence of fear and Michaelis–Menten‐type predator harvesting. The incorporation of fear with respect to the prey species has been observed to result in a reduction in the survival probability of the prey population and a concurrent reduction in the reproduction rate of the prey species. The existence and stability of ecologically significant equilibria have been ascertained through mathematical analysis. Emphasis within the proposed model primarily centers on numerical bifurcations of codimensions 1 and 2. Numerical validation has been performed on all simulated outcomes within the feasible range of parametric values. Dynamical characteristics of the model have subsequently undergone investigation through a series of numerical simulations, successfully revealing various forms of local and global bifurcations. In addition to the identification of saddle‐node, Hopf, Bogdanov–Takens, transcritical, cusp, homoclinic, and limit point cycle (LPC) bifurcations, the model has also demonstrated bistability and global asymptotic stability. These bifurcation phenomena serve as illustrative examples of the intricate dynamical behavior inherent to the model. Numerical validation through graphical representations has been utilized to elucidate the effects of factors such as fear, nonlinear predator harvesting, and predation rate on the dynamics of the interacting species under different parametric conditions. [ABSTRACT FROM AUTHOR]

The global bifurcation diagrams for two different one-parametric perturbations ( + λ x and + λ x 2 ) of a dissipative scalar nonautonomous ordinary differential equation x ′ = f (t , x) are described assuming that 0 is a constant solution, that f is recurrent in t, and that its first derivative with respect to x is a strictly concave function. The use of the skewproduct formalism allows us to identify bifurcations with changes in the number of minimal sets and in the shape of the global attractor. In the case of perturbation + λ x , a so-called generalized pitchfork bifurcation may arise, with the particularity of lack of an analogue in autonomous dynamics. This new bifurcation pattern is extensively investigated in this work. [ABSTRACT FROM AUTHOR]

This paper concerns positive solutions to the boundary value problems of the scalar field equation in the half space with a Sobolev supercritical nonlinearity and an inhomogeneous Dirichlet boundary condition, admitting a nontrivial nonnegative Radon measure as the boundary data. Under a suitable integrability assumption on the boundary data and the Joseph–Lundgren subcritical condition on the nonlinear term, we give a complete classification of the existence/nonexistence of a positive solution with respect to the size of the boundary data. Furthermore, we give a result on the existence of multiple positive solutions via bifurcation theory. [ABSTRACT FROM AUTHOR]

In this paper we get the existence of nodal solutions and we will study some bifurcation properties for the following class of nonlocal problems P - Δ e - c Δ u + u = g (x , u) , in R N lim | x | → ∞ u (x) = 0 , where N ≥ 3 , c > 0 , g : R N × R → R is a C 1 - function, Δ is the euclidean Laplacian and the linear operator e - c Δ is defined by the Fourier transform on a certain Hilbert space. This class of problems arises as models in the mathematical physics literature. [ABSTRACT FROM AUTHOR]

In this paper, we study a generalized Schrödinger–Poisson system in a bounded domain of $\mathbb{R}^{3}$ and involving an asymptotically linear nonlinearity. We prove the existence of positive solutions using bifurcation theory.

Proper vertebrae formation relies on a tissue-wide oscillator called the segmentation clock. Individual cellular oscillators in the presomitic mesoderm are modulated by intercellular coupling and external signals, leading to the propagation of oscillatory waves of genetic expression eventually stabilizing into a static pattern. Here, we review 4 decades of biophysical models of this process, starting from the pioneering Clock and Wavefront model by Cooke and Zeeman, and the reaction–diffusion model by Meinhardt. We discuss how modern descriptions followed advances in molecular description and visualization of the process, reviewing phase models, delayed models, systems-level, and finally geometric models. We connect models to high-level aspects of embryonic development from embryonic scaling to wave propagation, up to reconstructed stem cell systems. We provide new analytical calculations and insights into classical and recent models, leading us to propose a geometric description of somitogenesis organized along two primary waves of differentiation. [ABSTRACT FROM AUTHOR]

In this study, the semi-discretization technique is employed to establish a discrete representation of a modified Leslie-Gower prey-predator system that includes a Holling II type functional response. The dynamics of this model are then analyzed through the application of center manifold theory and bifurcation theory. We present comprehensive results for the local stability of the fixed points across the entire parameter space. Additionally, we provide sufficient conditions for the occurrence of flip bifurcation and Neimark-Sacker bifurcation. Besides, the system has experienced a flip bifurcation to chaos controlled using the method of chaos control, viz., state feedback method, pole placement technique, and hybrid control strategy. Furthermore, we provide specific conditions to ensure that bifurcation and chaos can be stabilized. Finally, numerical simulations are conducted to validate theoretical analysis and illustrate several new complex dynamical behaviors between two species. [ABSTRACT FROM AUTHOR]

In high-power photovoltaic systems, the inverter with an LCL filter is widely used to reduce the value of output inductance at which a lower switching frequency is required. However, the effect on the stability of the system caused by an LCL filter due to its resonance characteristic cannot be ignored. This paper studies the stability of a single-phase voltage source full-bridge inverter with an LCL filter through the bifurcation theory as it is a nonlinear system. The simulation results show that low-frequency oscillation appears when the proportional coefficient of the system controller increases or the damping resistance decreases to a certain extent. The average model is derived to analyze the low-frequency oscillation; the theoretical analysis demonstrates that low-frequency oscillation is essentially a period in which doubling bifurcation occurs, which indicates the intrinsic mechanism of the instability of the full-bridge inverter with an LCL filter. Additionally, the limitation of the existing damping resistor design standards, which only considers the main circuit parameters but ignores the influence of the controller on system stability, is identified. To solve this problem, the analytical expression of the system stability boundary is provided, which can not only provide convenience for engineering design to protect the system from low-frequency oscillation but also expand the selection range of damping resistance in practice. The experiments are performed to verify the results of the simulation and theoretical analysis, demonstrating that the analysis method can facilitate the design of the inverter with an LCL filter. [ABSTRACT FROM AUTHOR]

The phenomena of leader extinction and restrike during lightning events, such as multiple strokes in ground flashes or recoil leaders in cloud flashes, present significant challenges. A key aspect of this issue involves the discussion of the channel's negative differential resistance and its instability. From the perspective of bifurcation theory in nonlinear dynamics, this paper posits an inherent consistency among the channel's negative differential resistance, channel instability, and the critical transition from insulation to conduction. This study examines the differential resistance characteristics of the leader-streamer system in lightning development. We correlate the differential resistance characteristics of the leader-streamer channel with the channel's state and instability transitions, investigating the critical current and potential difference conditions required for the stable transition of the leader-streamer channel. [ABSTRACT FROM AUTHOR]

Monod-Haldane functional response function is introduced into Leslie-Gower predator-prey model under the homogeneous Dirichlet boundary condition, the influence of the defensive ability of prey on the positive equilibrium solution of the predator prey system is studied. A priori estimate, sufficient and necessary conditions for the existence and local stability of the positive equilibrium solution are established by using the maximum principle, the super and sub-solution method, bifurcation theory and stabili- ty theory. Combined with numerical simulation, the positive equilibrium solution is quantitatively ana- lyzed. The research shows that as long as the intrinsic growth rate of prey and predator is greater than a certain constant the coexistence mode can be generated. At the same time, the defense ability of prey has an inhibitory effect on the predator; especially when the prey has a higher growth rate, the ability of prey to resist the predator is stronger. [ABSTRACT FROM AUTHOR]

Hybrid neural network models are effective in analyzing time-series data by combining the strengths of neural networks and differential equation models. Although most studies have focused on linear hybrid models, few have examined nonlinear problems. This work explores the potential of a hybrid nonlinear epidemic neural network in predicting the correct infection function of an epidemic model. We design a novel loss function by combining bifurcation theory and mean-squared error loss to ensure the trainability of the hybrid model. Additionally, we identify unique existence conditions that support ordinary differential equations for estimating the correct infection function. Moreover, numerical experiments using the Runge–Kutta method confirm our proposed model's soundness both on our synthetic data and the real COVID-19 data. [ABSTRACT FROM AUTHOR]

A simplified column-buckling model is developed to understand the buckling mechanism of thin-walled strips restrained by uniform lateral pressure in the milling process. The strip is simplified as two rigid columns connected by a rotation spring, resting on a smooth surface, restrained by a uniform pressure and loaded by an axial force. Two loading cases are considered, i.e., the dead load and the follower load. Analytical solutions for the post-buckling responses of the two cases are derived based on the energy method. The minimum buckling force, Maxwell force and stability conditions for the two cases are established. It is demonstrated that the application of higher uniform pressure increases the minimum buckling force for the column and thus makes the column less likely to buckle. For the same pressure level, the dead load is found to be more effective than the follower load in suppressing the buckling of the system. The effect of initial geometric imperfection is also investigated, and the imperfection amplitude and critical restraining pressure that prevent buckling are found to be linearly related. The analytical results are validated by finite element simulations. This analytical model reveals the buckling mechanism of strips under lateral pressure restraint, which cannot be explained by the conventional bifurcation buckling theory, and provides a theoretical foundation for buckling-prevention strategies during the milling process of thin-walled strips, plates and shells commonly encountered in aerospace or automotive industries. [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]

We analyze the spectral and dynamical stability of solitary wave solutions to the Lugiato–Lefever equation on R . Our interest lies in solutions that arise through bifurcations from the phase-shifted bright soliton of the nonlinear Schrödinger equation. These solutions are highly nonlinear, localized, far-from-equilibrium waves, and are the physical relevant solutions to model Kerr frequency combs. We show that bifurcating solitary waves are spectrally stable when the phase angle satisfies θ ∈ (0 , π) , while unstable waves are found for angles θ ∈ (π , 2 π) . Furthermore, we establish asymptotic orbital stability of spectrally stable solitary waves against localized perturbations. Our analysis exploits the Lyapunov–Schmidt reduction method, the instability index count developed for linear Hamiltonian systems, and resolvent estimates. [ABSTRACT FROM AUTHOR]

In this paper, we consider a diffusive–advective predator–prey system in a spatially heterogeneous environment subject to a hostile boundary condition, where the interaction term is governed by a Holling type II functional response. We investigate the existence and global attractivity of both trivial and semi-trivial steady-state solutions and the existence and local stability of coexistence steady-state solutions, depending on the size of a key principal eigenvalue. In addition, we show that the effect of advection on the principal eigenvalue is monotonic for small advection rates, depending on the concavity of the resource distribution. For arbitrary advection rates, we consider two explicit resource distributions for which we can say precisely the behaviour of the principal eigenvalue as it depends on advection, highlighting that advection can either improve or impair a population's ability to persist, depending on the characteristics of the resource distribution. We present some numerical simulations to demonstrate the outcomes as they depend on the advection rates for the full predator–prey system. These insights highlight the intimate relationship between environmental heterogeneity, directed movement, and the hostile boundary. The methods employed include upper and lower solution techniques, bifurcation theory, spectral analysis, and the comparison principle. [ABSTRACT FROM AUTHOR]