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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]

This research studies the robustness of permanence and the continuous dependence of the stationary distribution on the parameters for a stochastic predator–prey model with Beddington–DeAngelis functional response. We show that if the model is extinct (resp. permanent) for a parameter, it is still extinct (resp. permanent) in a neighbourhood of this parameter. In the case of extinction, the Lyapunov exponent of predator quantity is negative and the prey quantity converges almost to the saturated situation, where the predator is absent at an exponential rate. Under the condition of permanence, the unique stationary distribution converges weakly to the degenerate measure concentrated at the unique limit cycle or at the globally asymptotic equilibrium when the diffusion term tends to 0. [ABSTRACT FROM AUTHOR]

Consideration of generalist predators leads to relatively complex dynamics due to alternative food sources. Here, we propose and analyze a prey-predator model with a generalist predator. The availability of alternative food sources for the predator and a density-dependent growth rate induces not only bistability and tristability, but also more complicated dynamical behaviors. We have studied the possible number and geometric configurations of positive equilibria in detail. A systematic bifurcation analysis has revealed the existence of the degenerate Bogdanov-Takens bifurcation of codimension four and degenerate Hopf bifurcation of codimension three. We found that degenerate local bifurcations with a higher codimension are responsible for three limit cycles. Derivation of the analytical conditions for three limit cycles for a suitable range of parameters is a crucial finding of this work. [ABSTRACT FROM AUTHOR]

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

In this paper, we study a class of non-autonomous piecewise differential equations defined as follows: d x / d t = a 0 (t) + ∑ i = 1 n a i (t) | x | i , where n ∈ N + and each a i (t) is real, 1-periodic, and smooth function. We deal with two basic problems related to their limit cycles (isolated solutions satisfying x (0) = x (1)) . First, we prove that, for any given n ∈ N + , there is no upper bound on the number of limit cycles of such equations. Second, we demonstrate that if a 1 (t) , … , a n (t) do not change sign and have the same sign in the interval [0, 1], then the equation has at most two limit cycles. We provide a comprehensive analysis of all possible configurations of these limit cycles. In addition, we extend the result of at most two limit cycles to a broader class of general non-autonomous piecewise polynomial differential equations and offer a criterion for determining the uniqueness of the limit cycle within this class of equations. [ABSTRACT FROM AUTHOR]

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

Swarmalator model attracts the attention of numerous scholars because of its rich dynamical behaviors. We investigate the reason for the phase transition that occurs in a 2D swarmalator model, and conclude that the limit cycle, a reflection of the coexistence of the forces of maintaining and disrupting orders, results in the various clustering phenomena. Through novel mean-field approximation and using the self-consistency argument method, we prove the clustering conditions and the influence of the number of clusters on the existence of clusters, and provide estimates of the cluster size of the splintered phase wave state and the phase transition threshold between the splintered phase wave state and active phase wave state. Due to the widespread presence of nonlinearity, our study is essential to the analysis of clustering phenomena in real physical models. [ABSTRACT FROM AUTHOR]

We study the bacterial respiration through the numerical solution of the Fairen–Velarde coupled nonlinear differential equations. The instantaneous concentrations of the oxygen and the nutrients are computed. The fixed point solution and the stable limit cycle are found in different parameter ranges as predicted by the linearized differential equations. In a specified range of parameters, it is observed that the system spends some time near the stable limit cycle and eventually reaches the stable fixed point. This metastability has been investigated systematically. Interestingly, it is observed that the system exhibits two distinctly different time scales in reaching the stable fixed points. The slow time scale of the metastable lifetime near the stable limit cycle and a fast time scale (after leaving the zone of limit cycle) in rushing towards the stable fixed point. The gross residence time, near the limit cycle (described by a slow time scale), can be reduced by varying the concentrations of nutrients. This idea can be used to control the harmful metastable lifespan of active bacteria. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of Boolean networks by using their matrices of support. We derive properties about the transients, fixed points, and limit cycles. We also study the graph properties of the phase spaces of Boolean networks, and show how to use matrices of support to construct Boolean networks with prescribed dynamics. [ABSTRACT FROM AUTHOR]

This paper investigates a mathematical model of a star formation system, including in-fall of gas from the local environment and out-flow of gas due to supernovae explosions and perturbations. In particular, the objective is to study how the variable density of interstellar components of the system namely atomic, molecular, and stellar components, interact in the star formation cycle. Under special parametric conditions, both limit cycle and stationary state behavior are observed. This indicates a stable star formation cycle in discrete episodes and an unstable star formation cycle converging to a stationary state. Observations of duty cycles under various parameters with varying intensity of supernovae shockwaves for dwarf galaxies, showed that the system adopts a self-regulatory oscillation state beyond a particular value. Analysis of giant galaxies showed decreased oscillatory periods for higher values. This is implied by the low production rate of supernovae in dwarf galaxies, increased production of cold gas, continuation of the cyclic behavior for a longer time. Stronger shockwaves increase the rate of dissipation of gases, resulting in shredding of mass, to get transferred into a precipitous lower mass. Star-forming rate (SFR) of high-mass star systems was found to vary against higher-order perturbations of supernovae shockwave but remained the same in the case of low-mass star systems. [ABSTRACT FROM AUTHOR]

The subject of this paper concerns with the bifurcation of limit cycles for a predator-prey model with small immigration. Since, in general, the biological systems are not isolated, taking into account immigration in the model becomes more realistic. In this context, we deal with a model with a Holling type Ⅰ function response and study, using averaging theory of second order, the Hopf bifurcation that can emerge under small perturbation of the biological parameters. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of stability discrimination of limit cycles for piecewise smooth systems. We first establish the Poincaré map near a periodic orbit, and deduce the first order derivative of the map for general piecewise smooth systems on the plane. Then, we obtain a sufficient condition for determining the stability of limit cycles for these systems. [ABSTRACT FROM AUTHOR]

In this work, we study the monotonicity of the ratio of two line integrals I i (h) = ∫ Γ h + f i (x) y d x + ∫ Γ h - f i (x) y d x , i = 0 , 1 , with Γ h + = { (x , y) ∈ R 2 | H + (x , y) = h , x > 0 } and Γ h - = { (x , y) ∈ R 2 | H - (x , y) = h ~ , x ≤ 0 } , where H + (x , y) and H - (x , y) have the form 1 2 y 2 + Ψ 1 (x) and 1 2 y 2 + Ψ 2 (x) , respectively. We first present a criterion function defined directly by the functions which appear in the above integrals and prove the monotonicity of this criterion function implying the monotonicity of the ratio of these line integrals which is very important to find the number of limit cycles. Then we give several examples to illustrate the application of this criterion. [ABSTRACT FROM AUTHOR]

In this paper we classify the phase portraits in the Poincaré disc of a class of cubic polynomial differential systems having an invariant ellipse and an invariant straight line. We prove that such a class of cubic polynomial differential systems have exactly 43 topologically different phase portraits in the Poincaré disc. Also we obtain that the invariant ellipse in two of these phase portraits is a limit cycle. [ABSTRACT FROM AUTHOR]

In this paper, the existence and number of non-contractible limit cycles of the Josephson equation $ \beta \frac{d^{2}\Phi}{dt^{2}}+(1+\gamma \cos \Phi)\frac{d\Phi}{dt}+\sin \Phi = \alpha $ are studied, where $ \phi\in \mathbb S^{1} $ and $ (\alpha,\beta,\gamma)\in \mathbb R^{3} $. Concretely, by using some appropriate transformations, we prove that such type of limit cycles are changed to limit cycles of some Abel equation. By developing the methods on limit cycles of Abel equation, we prove that there are at most two non-contractible limit cycles, and the upper bound is sharp. At last, combining with the results of the paper (Chen and Tang, J. Differential Equations, 2020), we show that the sum of the number of contractible and non-contractible limit cycles of the Josephson equation is also at most two, and give the possible configurations of limit cycles when two limit cycles appear. [ABSTRACT FROM AUTHOR]

In this paper, we study limit cycle bifurcations near homoclinic and heteroclinic loops in piecewise smooth systems with three zones separated by two parallel straight lines. By introducing suitable Poincaré map near a homoclinic loop, we derive some stability criteria and establish bifurcation theory of limit cycles via stability-changing method near homoclinic and heteroclinic loops, respectively. As applications, we give two examples, which consider a class of Liénard piecewise linear differential systems. [ABSTRACT FROM AUTHOR]

For family x ′ = (a 0 + a 1 cos t + a 2 sin t) | x | + b 0 + b 1 cos t + b 2 sin t , we solve three basic problems related with its dynamics. First, we characterize when it has a center (Poincaré center focus problem). Second, we show that each equation has a finite number of limit cycles (finiteness problem), and finally we give a uniform upper bound for the number of limit cycles (Hilbert problem 16). [ABSTRACT FROM AUTHOR]

This paper studies the bifurcations of limit cycles from the system x ˙ = y , y ˙ = - x with the switching curve y = x 3 / 3 - x under the perturbations of arbitrary polynomials of x and y with degree n. We obtain the lower bound and upper bound of the maximum number of limit cycles bifurcating from h ∈ (0 , 3 / 2) if the first order Melnikov function is not identically 0. When the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first order Melnikov function. We also give an example to illustrate our result. [ABSTRACT FROM AUTHOR]

В работе предложена математическая модель нелинейного осциллятора Ван дер Поля-Эйри с учетом наследственности. Нелинейность осциллятора обусловлена наличием зависимости коэффициента трения от квадрата функции смещения, что характерно для осциллятора Ван дер Поля. Также собственная частота колебаний представляет собой функцию от времени, которая линейно возрастает при его возрастании. Последнее характерно для осциллятора Эйри. Эффекты наследственности вводятся в модельное уравнение посредством дробных производных в смысле Герасимова-Капуто. Они указывают на то, что колебательная система может обладать эффектами памяти, которые проявляются в зависимости текущего ее состояния от предыдущих. Для предложенной математической модели был разработан численный алгоритм, основанный на явной конечно-разностной схемы первого порядка. Численный алгоритм был реализован в компьютерной программе на языке Maple, с помощью которой была произведена визуализация результатов моделирования. Были построены осциллограммы и фазовые траектории при различных значениях параметров модели. Показано, что дробная математическая модель может обладать различными колебательными режимами: от автоколебательных, затухающих и хаотических. Дается интерпретация результатов моделирования.

Despite widespread and striking examples of physiological oscillations, their functional role is often unclear. Even glycolysis, the paradigm example of oscillatory biochemistry, has seen questions about its oscillatory function. Here, we take a systems approach to argue that oscillations play critical physiological roles, such as enabling systems to avoid desensitization, to avoid chronically high and therefore toxic levels of chemicals, and to become more resistant to noise. Oscillation also enables complex physiological systems to reconcile incompatible conditions such as oxidation and reduction, by cycling between them, and to synchronize the oscillations of many small units into one large effect. In pancreatic β-cells, glycolytic oscillations synchronize with calcium and mitochondrial oscillations to drive pulsatile insulin release, critical for liver regulation of glucose. In addition, oscillation can keep biological time, essential for embryonic development in promoting cell diversity and pattern formation. The functional importance of oscillatory processes requires a re-thinking of the traditional doctrine of homeostasis, holding that physiological quantities are maintained at constant equilibrium values, a view that has largely failed in the clinic. A more dynamic approach will initiate a paradigm shift in our view of health and disease. A deeper look into the mechanisms that create, sustain and abolish oscillatory processes requires the language of nonlinear dynamics, well beyond the linearization techniques of equilibrium control theory. Nonlinear dynamics enables us to identify oscillatory ('pacemaking') mechanisms at the cellular, tissue and system levels.

In this paper, we study the number of limit cycles H(n) bifurcating from the piecewise smooth system formed by the quadratic reversible system (r22) for y ≥ 0 and the cubic system x ˙ = y (1 + x ¯ 2 + y 2) , y ˙ = - x ¯ (1 + x ¯ 2 + y 2) for y < 0 under the perturbations of polynomials with degree n, where x ¯ = x - 1 . By using the first-order Melnikov function, it is proved that 2 n + 3 ≤ H (n) ≤ 2 n + 7 for n ≥ 3 and the results are sharp for n = 0 , 1 , 2 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the number of limit cycles bifurcating from a kind of quadratic Hamiltonian system having a homoclinic loop under perturbations of piecewise smooth polynomials with degree $ n $. It is proved that at most $ 10n+[\frac {n-1}2]+7 (n\geq 1) $ limit cycles bifurcating from the periodic annulus if the first order Melnikov function is not identically zero, and it is also proved that there exist piecewise smooth polynomials with degree $ n $ such that the corresponding system has at least $ n+[\frac{n+1}{2}] $ limit cycles. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate limit cycles of the generalized Rayleigh–Liénard oscillator x ˙ = φ (y) , y ˙ = - g (x) - f (x , y) y . A criterion on the uniqueness and stability of limit cycles is given. We apply the criterion to two generalized Rayleigh–Liénard systems and get the uniqueness of limit cycles. Moreover, the stability and locations of limit cycles are obtained if they exist. [ABSTRACT FROM AUTHOR]

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We study the bifurcation problem of limit cycles in near-Hamiltonian systems near a double homoclinic loop on the cylinder. We obtain a sufficient condition to find a lower bound of the maximal number of limit cycles near the loop by the coefficients of the expansions of the three Melnikov functions corresponding to the three families of periodic orbits near the double homoclinic loop. We also provide an application of our main results to a class of cylindrical systems. [ABSTRACT FROM AUTHOR]

We consider the planar family of rigid systems of the form x ′ = - y + x P (x , y) , y ′ = x + y P (x , y) , where P is any polynomial with monomials of degree one and three. This is the simplest non-trivial family of rigid systems with no rotatory parameters. The family can be compactified to the Poincaré sphere such that the vector field along the equator is not identically null. We study the centers, singular points and limit cycles of that family on the plane and on the sphere. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the small-amplitude limit cycles of two classes of Liénard systems of the form x ˙ = y - F (x , μ) , y ˙ = - g (x) , where the damping term F (x , μ) is either a polynomial or a rational function q n (x) p m (x) , ( q n (x) and p m (x) are polynomials in x with degrees n and m, respectively), μ represents coefficients and the cubic restoring term g (x) = x - x 2 - 1 k x 3 with a non-zero constant k. By utilizing Picard-Fuchs equation, we gain the upper bounds of the number of small-amplitude limit cycles of these two systems for any k ≠ 0 . Moreover, the smaller upper bounds are obtained for k < - 4 , k ≠ - 9 2 , and the upper bound is sharp if the damping term is polynomial. [ABSTRACT FROM AUTHOR]

Consider the number of limit cycles of a family of systems with homogeneous components: x ˙ = y , y ˙ = - x 3 + α x 2 y + y 3. We show that there is an α ∗ < 0 such that the system has exactly one limit cycle for α ∈ (α ∗ , 0) , while no limit cycle for the else region. This completes a previous result and also gives a positive answer to the second part of Gasull’s 3rd problem listed in the paper (SeMA J 78(3):233–269, 2021). To obtain this result, we mainly analyse the behavior of the heteroclinic separatrices at infinity. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear mathematical model for tuberculosis transmission, which incorporates multiple saturated exogenous reinfections, is proposed and explored. The existence of disease-free and endemic steady states is investigated. Disease-free equilibrium (DFE) is locally asymptotically stable (LAS) but not globally asymptotically stable (GAS) when the basic reproduction number, ℛ 0 < 1. However, it is GAS only when there is no exogenous reinfection. The local asymptotic stability and global asymptotic stability of the unique endemic equilibrium point (EEP) are established under certain conditions when ℛ 0 > 1. Further, the EEP is GAS when ℛ 0 > 1 , provided there is no exogenous reinfection. When ℛ 0 is below unity, the presence of multiple endemic equilibria is found which leads to backward bifurcation. It is demonstrated that the system encounters a Hopf-bifurcation when the transmission rate β crosses a critical value, resulting in the formation of limit cycles, i.e. periodic solutions bifurcate around the EEP when β passes a critical value. The stability and direction of Hopf-bifurcation are also studied. The results of the analytical work are validated through numerical simulations. A numerical simulation illustrates that EEP losses its stability via Hopf-bifurcation for specific parameters. However, when the bifurcation parameter β is increased further, the EEP regains its stability. In addition, Hopf-bifurcation occurs due to exogenous reinfection rates p and . Thus, our model shows some important nonlinear dynamical behaviors. [ABSTRACT FROM AUTHOR]

Characterizing existence or not of periodic orbit is a classical problem, and it has both theoretical importance and many real applications. Here, several new criterions on nonexistence of periodic orbits of the planar dynamical system x ˙ = y , y ˙ = - g (x) - f (x , y) y are obtained and by examples shows that these criterions are applicable, but the known ones are invalid to them. Based on these criterions, we further characterize the local topological structures of its equilibrium, which also show that one of the classical results by Andreev (Am Math Soc Transl 8:183–207, 1958) on local topological classification of the degenerate equilibrium is incomplete. Finally, as another application of these results, we classify the global phase portraits of a planar differential system, which comes from the third question in the list of the 33 questions posed by A. Gasull and also from a mechanical oscillator under suitable restriction to its parameters. [ABSTRACT FROM AUTHOR]

A new mathematical model describing the dynamics of sales of goods on the market has been proposed. This model takes into account the following characteristics: the average welfare of buyers, the maximum welfare of buyers, the number of buyers who know about the incoming product and the level of market saturation with this product. Examples demonstrating the features of the constructed model are given.

This paper is concerned with a two‐variable Oregonator model for Belousov–Zhabotinsky Reaction. Llibre and Oliveira (2022) proved that Oregonator model with an unstable node or focus has at least one limit cycle in the positive quadrant, and the system has a unique stable limit cycle for some values of the parameters. It is shown in the present paper that the positive equilibrium is not a center, and that if the system has a positive equilibrium which is a weak focus, then its order is at most 2. There exist some parameter values such that system has i(i=1,2)$$ i\left(i&#x0003D;1,2\right) $$ small limit cycle around the positive equilibrium. [ABSTRACT FROM AUTHOR]

For the quadratic reversible systems of genus one, all of their periodic orbits are higher-order algebraic curves. When they are perturbed by polynomials of degree n , the numbers of zeros of their Abelian integrals will change and we study the upper bounds of these numbers by using the methods of Riccati equation and Picard–Fuchs equation. We consider both the highest and lowest degrees of polynomials, and more importantly, we consider the law of polynomials and the range of values for their variables. Consequently, some laws of the polynomials are discovered and many upper bounds are obtained, and these upper bounds are sharper than the results obtained by other techniques. [ABSTRACT FROM AUTHOR]

This study presents the path-following control design for a group of autonomous underwater vehicles in a plane with obstacle and collision avoidance. A distributed backstepping controller is exploited for an autonomous vehicle to design a path-following controller. Then, the controller is applied for the path-following of a group of autonomous vehicle systems in the formation setting. The graph theory is used to model the interconnection between the vehicles. To avoid obstacles and collisions, a dynamic limit-cycle-based algorithm is proposed. It is assumed that each obstacle is involved an elliptical shape and if the collision or conflict of one vehicle to the surrounded environmental obstacles is detected, the obstacle avoidance algorithm will be active to prevent collisions. Finally, the introduced algorithm is implemented and simulated for a group of underwater autonomous vehicles in MATLAB software. The simulation results illustrate the effectiveness and performance of the proposed distributed controller apparently. [ABSTRACT FROM AUTHOR]

In this paper, we study the number of limit cycles bifurcated from the periodic orbits of a cubic uniform isochronous center with continuous and discontinuous quartic polynomial perturbations. Using the averaging theory of first order for continuous and discontinuous differential systems and comparing the obtained results, we show that the discontinuous systems have at least 6 more limit cycles than the continuous ones. This study needs some computations that have been verified using Maple. [ABSTRACT FROM AUTHOR]

In this paper, we consider the influence of a nonlinear contact rate caused by multiple contacts in classical SIR model. In this paper, we unversal unfolding a nilpotent cusp singularity in such systems through normal form theory, we reveal that the system undergoes a Bogdanov-Takens bifurcation with codimension 2. During the bifurcation process, numerous lower codimension bifurcations may emerge simultaneously, such as saddle-node and Hopf bifurcations with codimension 1. Finally, employing the Matcont and Phase Plane software, we construct bifurcation diagrams and topological phase portraits. Additionally, we emphasize the role of symmetry in our analysis. By considering the inherent symmetries in the system, we provide a more comprehensive understanding of the dynamical behavior. Our findings suggest that if this occurrence rate is applied to the SIR model, it would yield different dynamical phenomena compared to those obtained by reducing a 3-dimensional dynamical model to a planar system by neglecting the disease mortality rate, which results in a stable nilpotent cusp singularity with codimension 2. We found that in SIR models with the same occurrence rate, both stable and unstable Bogdanov-Takens bifurcations occur, meaning both stable and unstable limit cycles appear in this system. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of compatimental epidemiological models consisting of Susceptible, Infected, and Removed (SIR) individuals with a perturbation factor or exterior effects such as noise, climate change, pollution, etc. We prove the existence and uniqueness of a limit cycle confined in a nonempty closed and convex set by relying on a recent result of Lobanova and Sadovskii. Moreover, we study the existence of Hopf and Bogdanov-Takens bifurcations by applying respectively Poincare-Andronov-Hopf bifurcation theorem and Bogdanov-Takens theorem. Eventually, using Scilab, we illustrate the validity of our results with numerical simulations and also interpret them. [ABSTRACT FROM AUTHOR]

In this paper, we present a new criterion function for investigating the monotonicity of the ratio of two Abelian integrals in piecewise-smooth differential systems, and then, apply it to deal with some examples. More precisely, we consider the Abelian integrals of the form ..., with Γh = ΓhL + ΓhR, where ... and .... We prove that the monotonicity of the presented criterion function implies the monotonicity of the ratio ... and provide a few examples to explain the application of this criterion. [ABSTRACT FROM AUTHOR]

This paper is devoted to analyzing the codimension-2 bifurcations of a generalized three-dimensional (3D) jerk system which is jerk function with all quadratic and cubic terms. First, the locally stability of Bagdanov–Takens (double-zero) equilibrium is systematically investigated using the center manifold and normal form theory combined with some bifurcation theories, and under some certain parameter conditions, the Bagdanov–Takens equilibrium of this jerk system could be a saddle, a focus, a degenerate equilibrium or an equilibrium which neighborhood is formed from an elliptic sector and a hyperbolic sector on its two-dimensional center manifold. Second, some adequate parameter conditions for guaranteeing the occurrence of Hopf bifurcation, Bautin (degenerate Hopf) bifurcation and the stability of the Hopf equilibrium of this jerk system are given. Finally, numerical simulation is used to verify the above theoretical analyses. [ABSTRACT FROM AUTHOR]

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

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

In this paper, we discuss a wide family of dynamical systems whose characteristic feature is a polynomial right-hand side containing coprime forms of the phase variables of the system. One of the equations of the system contains a third-degree polynomial (cubic form), the other equation contains a quadratic form. We consider the problem of constructing all possible phase portraits in the Poincaré disk for systems from the family considered and establish criteria for the implementation of each portrait that are close to coefficient criteria. This problem is solved by using the central and orthogonal Poincaré methods of sequential mappings and a number of other methods developed by the authors for the purposes of this study. We obtained rigorous qualitative and quantitative results. More than 250 topologically distinct phase portraits of various systems were constructed. The absence of limit cycles of systems of this family is proved. Methods developed can be useful for the further study of systems with polynomial right-hand sides of other forms. [ABSTRACT FROM AUTHOR]

In this work, a new glucose–insulin model incorporating time delay and obesity is developed to gain insights of its dynamical mechanisms. Through the method of multiple scales, we theoretically demonstrate that time delay can drive the system to yield Hopf bifurcation, thereby producing oscillating solutions that are consistent with the simulation results. Moreover, obesity changes the level of glucose, but cannot induce oscillations. In particular, it is found that under the combined effect of obesity and time delay, obesity delays the appearance of Hopf bifurcation which is induced by time delay. Results show that a low calorie diet can achieve therapeutic effects including reducing blood glucose fluctuations and insulin resistance, which can be used as an adjuvant for the treatment of diabetes. In addition, our results indicate that the delay, together with an optimal rate of model parameters can cause a variety of dynamics and induce glucose oscillations. The result obtained in this paper may help to better understand the obesity, diabetes, and the interaction between glucose and insulin, so that control strategies can be designed to better regulate blood glucose levels and fluctuations and mitigate the occurrence of type-2 diabetes. [ABSTRACT FROM AUTHOR]