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In this paper, we apply Lin's method to study the existence of heteroclinic orbits near the degenerate heteroclinic chain under m$$ m $$‐dimensional periodic perturbations. The heteroclinic chain consists of two degenerate heteroclinic orbits γ1$$ {\gamma}_1 $$ and γ2$$ {\gamma}_2 $$ connected by three hyperbolic saddle points q1,q2,q3$$ {q}_1,{q}_2,{q}_3 $$. Assume that the degeneracy of the unperturbed heteroclinic orbit γi$$ {\gamma}_i $$ is ni$$ {n}_i $$, the splitting index is δi$$ {\delta}_i $$. By applying Lin's method, we construct heteroclinic orbits connected q1$$ {q}_1 $$ and q3$$ {q}_3 $$ near the unperturbed heteroclinic chain. The existence of these orbits is equivalent to finding zeros of the corresponding bifurcation function. The lower order terms of the bifurcation function is the map from ℝn1+n2+m$$ {\mathrm{\mathbb{R}}}^{n_1+{n}_2+m} $$ to ℝn1+n2+δ1+δ2$$ {\mathrm{\mathbb{R}}}^{n_1+{n}_2+{\delta}_1+{\delta}_2} $$. Using the contraction mapping principle, we provide a detailed analysis on how zeros can exist based on different cases of splitting indices δ1$$ {\delta}_1 $$, δ2$$ {\delta}_2 $$ and then obtain the existence of the heteroclinic orbits which backward asymptotic to q1$$ {q}_1 $$ and forward asymptotic to q3$$ {q}_3 $$. [ABSTRACT FROM AUTHOR]

Understanding and accounting for the slow–fast effect are crucial for accurately modeling and predicting the dynamics of predator–prey models, emphasizing the importance of considering the relative speeds of interacting populations in ecological research. This paper examines a predator–prey interaction to study its complex dynamics due to its slow–fast effect on predator populations. The occurrence and stability of equilibria are analyzed. The stability of positive fixed point is dependent on the slow–fast effect parameter ϵ$$ \epsilon $$, which must fall within a specific range when the generation gap is larger. The positive fixed point becomes unstable for bigger values of ϵ$$ \epsilon $$ because the growth of predators is faster, resulting in the extinction of all prey. Smaller values of ϵ$$ \epsilon $$ cause the positive fixed point to become unstable since the prey grows more quickly while the predator grows more slowly, ultimately causing the extinction of the predator. Moreover, it is shown that Leslie–Gower model experiences Neimark–Sacker and period‐doubling bifurcations at positive equilibrium point. In order to control bifurcation, hybrid control and feedback control methods are employed. Finally, analytical results are confirmed by numerical examples. [ABSTRACT FROM AUTHOR]

This paper deals with a model of chemostat with two cross‐feeding species and involving two inhibitions. The excess of hydrogen inhibits the growth of acetogenic bacteria which, in turn, inhibit the growth of methanogenic hydrogenotrophic bacteria. The model is described by a system of four ordinary differential equations. We established the conditions of existence and stability of equilibria with respect to the operating parameters which are the dilution rate and the input substrate concentrations of the species of the model, then we illustrate the operating and bifurcation diagrams. This technique makes it easy to interpret different regions of operating diagrams. Inhibitions let some regions of stability rise and others vanish. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence, nonexistence, and multiplicity of positive solutions for a impulsive periodic boundary value problem with positive parameter λ$$ \lambda $$. Using the critical point theorem, we obtain a bifurcation type result about positive solutions; that is, there exists 0<Λ<∞$$ 0<\Lambda <\infty $$ such that our problem has at least two positive solutions for all λ∈(0,Λ)$$ \lambda \in \left(0,\Lambda \right) $$, one positive solution for λ=Λ$$ \lambda =\Lambda $$, and no positive solution for λ>Λ$$ \lambda >\Lambda $$ under the appropriate conditions. [ABSTRACT FROM AUTHOR]

In this paper, an improved railway wheelset system is presented. The dynamical behaviors of the system are investigated, including dissipativity and invariance of the system, stability of zero‐equilibrium point, and the bifurcation characteristics of railway wheelset system at zero equilibrium point. Furthermore, the chaotic behaviors of the motion of railway wheelset and the dynamical behaviors of the railway wheelset system under geometric constraint are studied. It shows that the railway wheelset system has complex dynamical phenomena owing to nonlinear factor and the railway wheelset system may have different complex dynamical behaviors with different nonlinear parameters. In addition, the motion of railway wheelset also has chaotic behaviors under geometric constraint. Finally, the chaos control of the railway wheelset system is achieved by using linear feedback control method. The numerical simulations are carried out in order to analyze the complex phenomena of the railway wheelset system. [ABSTRACT FROM AUTHOR]

In this paper, a discrete‐time seasonally forced SIR epidemic model with a nonstandard discretization scheme is investigated for different types of bifurcations. Although many researchers have already suggested numerically that this model can exhibit chaotic dynamics, not much focus is given to the bifurcation theory of the model. We prove analytically and numerically the existence of different types of bifurcations in the model. First, one and two parameters bifurcations of this model are investigated by computing their critical normal form coefficients. Second, the flip, Neimark–Sacker, and strong resonance bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complete complex dynamical behavior of the model is investigated. The model is discretized by a novel technique, namely a nonstandard finite difference discretization scheme (NSFD). Some graphical representations of the model are presented to verify the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, resonance and bifurcation of a nonlinear damped fractional‐order Duffing system are studied. The amplitude and phase of the steady‐state response of system are obtained by means of the average method, the stability is analyzed, and then the amplitude‐frequency characteristic curves of the system with different parameters are drawn based on the implicit function equation of amplitude. Grunwald–Letnikov fractional derivative is used to discretize the system numerically, the response curve and phase trajectory of the system under different parameters are obtained; meanwhile, the dynamic behavior is analyzed. The bifurcation and saddle bifurcation behavior of the system is studied through numerical simulation with different parameters. [ABSTRACT FROM AUTHOR]

In this paper, we explore local dynamics with topological classifications, bifurcation analysis, and chaos control in a discrete‐time COVID‐19 epidemic model in the interior of ℝ+4$$ {\mathbb{R}}_{+}^4 $$. It is explored that for all involved parametric values, discrete‐time COVID‐19 epidemic model has boundary equilibrium solution and also it has an interior equilibrium solution under definite parametric condition. We have explored the local dynamics with topological classifications about boundary and interior equilibrium solutions of the discrete‐time COVID‐19 epidemic model by linear stability theory. Further, for the discrete‐time COVID‐19 epidemic model, existence of periodic points and convergence rate are also investigated. It is also studied the existence of possible bifurcations about boundary and interior equilibrium solutions and proved that there exists no flip bifurcation about boundary equilibrium solution. Moreover, it is proved that about interior equilibrium solution, there exist Hopf and flip bifurcations, and we have studied these bifurcations by utilizing explicit criterion. Moreover, by feedback control strategy, chaos in the discrete COVID‐19 epidemic model is also explored. Finally, theoretical results are verified numerically. [ABSTRACT FROM AUTHOR]

This paper is devoted to the Schnakenberg model with crucial reversible reactions, under Neumann boundary conditions. Existence and uniqueness of the strong solution are obtained on basis of semigroup theory. It is found that the proposed system admits four possible positive constant steady states, and explicit conditions of the stability, Turing instability, steady‐state bifurcation, and Hopf bifurcation are determined. Besides, numerical simulations are given to show the theoretical results and depict the spatiotemporal patterns. [ABSTRACT FROM AUTHOR]

In this paper, we present an improved wheelset motion model with two degrees of freedom and study the dynamic behaviors of the system including the symmetry, the existence and uniqueness of the solution, continuous dependence on initial conditions, and Hopf bifurcation. The dynamic characteristics of the wheelset motion system under a nonholonomic constraint are investigated. These results generalize and improve some known results about the wheelset motion system. Meanwhile, based on multiple equilibrium analysis, calculation of Lyapunov exponents and Poincaré section, the chaotic behaviors of the wheelset system are discussed, which indicates that there are more complex dynamic behaviors in the railway wheelset system with higher order terms of Taylor series of trigonometric functions. This paper has also realized the chaos control and bifurcation control for the wheelset motion system by adaptive feedback control method and linear feedback control. The results show that the chaotic wheelset system and bifurcation wheelset system are all well controlled, whether by controlling the yaw angle and the lateral displacement or only by controlling the yaw angle. Numerical simulations are carried out to further verify theoretical analyses. [ABSTRACT FROM AUTHOR]

We study a free boundary problem modeling the growth of multilayer tumors. This model describes the invasion of tumors: the tumor cells produce matrix degrading enzymes (MDEs) to degrade the extracellular matrix (ECM) which provides structural support of the surrounding tissue. As in Pan and Xing, the influence of ECM and MDE interactions is considered in this paper. The model equations include two diffusion equations for the nutrient concentration and MDE concentration and an ordinary differential equation for ECM concentration. The tumor cell proliferation is at the rate λ, which characterizes the aggressiveness of the tumor. In contrast to Pan and Xing, we consider "flat stationary solution" in this paper. We first show that there exists a unique flat stationary solution for any λ>0. Then, we prove that there are infinite branches of bifurcation solutions from the flat stationary solutions at the bifurcation points λ=λk(ρ*) (k≥1). [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical transition and chaos for a five‐dimensional Lorenz system. Based on the eigenvalue analysis, the principle of exchange of stabilities conditions is obtained. By using the dynamical transition theory, three different types of dynamical transition for the five‐dimensional Lorenz system are derived. More precisely, when the control parameter r=1, the system has a continuous transition and bifurcates to two stable steady states. As r further increases, the system undergoes two successive transitions. That is, under some condition, the transition is continuous and a stable limit cycle is bifurcated; if not, the system undergoes a jump transition and an unstable periodic orbit occurs. Especially, the chaotic orbits occur when r=36.91. Finally, numerical results are given to illustrate our theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the discrete Sturm–Liouville problem −Δ(p(k)Δu(k−1))+q(k)u(k)=λm(k)u(k)+f1(k,u(k),λ)+f2(k,u(k),λ),k∈[1,T]Z,a0u(0)−b0Δu(0)=0,a1u(T+1)+b1Δu(T)=0,where λ∈ℝ is a parameter, [1,T]Z:={1,2,...,T},f1,f2∈C([1,T]Z×ℝ2,ℝ),f1 is not differentiable at the origin and infinity. Under some suitable assumptions on nonlinear terms, we prove the existence of unbounded continua of positive and negative solutions of this problem which bifurcate from intervals of the line of trivial solutions or from infinity, respectively. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study the evolution of the transmembrane potential on the cardiac cell under different rates and amplitudes of stimulation. For modeling this potential, the modification of the Fenton‐Karma model was applied. It is a phenomenological model with 3 degrees of freedom that corresponds to nondimensional transmembrane potential and gating variables for regulation of inward and outward ion currents that can better reproduce the shape of the transmembrane potential than the original Fenton‐Karma model. The model was newly forced by stimulus with the shape of the half‐sine period. As the main goal of the paper is to show that this model is showing regular as well as irregular motion; periodic and chaotic patterns are detected using bifurcation diagrams, the Fourier spectra, Poincaré sections, and 0‐1 test for chaos. [ABSTRACT FROM AUTHOR]

This paper focuses on the different bifurcations of fixed points of a delayed discrete neural network model analytically and numerically. The conditions and critical values of different bifurcations including the pitchfork, flip, Neimark–Sacker, and flip–Neimark–Sacker are analyzed. By using the critical coefficients, the structure for each bifurcation are determined. By taking one and two parameters, the critical coefficients are calculated and the curves associated with each bifurcation are plotted. The numerical simulation results demonstrate the effectiveness and feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fractional‐order delay differential model for tuberculosis (TB) transmission with the effects of endogenous reactivation and exogenous reinfections. We investigate the qualitative behaviors of the model throughout the local stability of the steady states and bifurcation analysis. A discrete time delay is introduced in the model to justify the time taken for diagnosis of the disease. Existence and positivity of the solutions are investigated. Some interesting sufficient conditions that ensure the local asymptotic stability of infection‐free and endemic steady states are studied. The fractional‐order TB model undergoes Hopf bifurcation with respect to time delay and disease transmission rate. The presence of fractional order and time delay in the model improves the model behaviors and develops the stability results. A numerical example is provided to support our theoretical results. [ABSTRACT FROM AUTHOR]

The main aim of the paper is to research dynamic properties of a mechanical system consisting of a ball jumping between a movable baseplate and a fixed upper stop. The model is constructed with one degree of freedom in the mechanical oscillating part. The ball movement is generated by the gravity force and non-harmonic oscillation of the baseplate in the vertical direction. The impact forces acting between the ball and plate and the stop are described by the nonlinear Hertz contact law. The ball motion is then governed by a set of two nonlinear ordinary differential equations. To perform their solving, the Runge-Kutta method of the fourth order with adaptable time step was applied. As the main result, it is shown that the systems exhibit regular, irregular, and chaotic pattern for different choices of parameters using the newly introduced 0-1 test for chaos, detecting bifurcation diagram, and researching Fourier spectra. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper aims to study the deterministic and stochastic features of a modified Leslie–Gower predator–prey model with Holling‐type II functional response. We first investigate the dynamical properties of the deterministic model, including existence and stability of the equilibrium, and different types of bifurcations. For the stochastic model, a phenomenon of noise‐induced state transition is found. By applying the stochastic sensitivity functions technique, we construct the confidence domain of stochastic attractor and then estimate the critical value of the intensity of noise generating this transition. Numerical simulations are performed to validate the analytical findings. [ABSTRACT FROM AUTHOR]

This paper studies a fractional‐order modelling chronic wasting disease (CWD). The basic results on existence, uniqueness, non‐negativity, and boundedness of the solutions are investigated for the considered model. The criterion for local as well as global stability of the equilibrium points is derived. A numerical analysis for Hopf‐type bifurcation is presented. Finally, numerical simulations are provided to justify the results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we employed the bifurcation method and qualitative theory of dynamical systems to study the peakons and periodic cusp waves of the generalization of the Camassa-Holm equation, which may be viewed as an extension of peaked waves of the same equation. Through the bifurcation phase portraits of traveling wave system, we obtained the explicit peakons and periodic cusp wave solutions. Further, we exploited the numerical simulation to confirmthe qualitative analysis, and indeed, the simulation results are in accord with the qualitative analysis. Compared with the previous works, several new nonlinear wave solutions are obtained. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Human immunodeficiency virus (HIV)/AIDS and cancer coexistence both in vivo and in vitro in a cancer‐immune environment leads to specific cytokines being produced by various immune cells and the cancer cells. Most of the studies have suggested that specific cytokines produced by the immune system cells and the tumor play an important role in the dynamics of non‐Hodgkin lymphomas (NHLs). In this paper, a mathematical model describing the NHL‐immune system interaction in the presence of the HIV, HIV treatment, and chemotherapy is developed. The formulated model, described by nonlinear ODEs, shows existence of multiple equilibria whose stability and bifurcation analysis are presented. From the bifurcation analysis, bistability regions are evident. We observe that with and without HIV treatment, the system results in a nonaggressive tumor size or aggressive tumor (full‐blown tumor) depending on the initial conditions. The results further suggest that at a low endemic state, patients can live for longer period with the tumor, which might explain why some patients can live with cancer for many years. However, initiation of HIV treatment in patients with NHL is observed to lower these endemic states of the tumor. Our results explain why late initiation of HIV treatment might not be helpful to NHL patients. We further investigated the effect of chemotherapy on the dynamics of the tumor. Our simulation results might explain why a few of these chemotherapeutic drugs are more effective when given at a slow continuous rate. The model provides a unique opportunity to influence policy on HIV‐related cancer treatment and management. [ABSTRACT FROM AUTHOR]

In this paper, using the methods of killing tumors and impulsive differential equations, two immunotherapy antitumor models for describing therapies of general tumors and advanced solid tumors are established. By using the theories of impulsive equations, small amplitude perturbation techniques, and the comparison technique, we obtain the conditions which guarantee the global asymptotical stability of the tumor‐eliminated periodic solution and system permanence, when immunotherapy alone is performed. The numerical results of the influences of the impulsive perturbation on the inherent oscillation show rich dynamics, such as period‐doubling bifurcation and chaos. Moreover, the effects of the combination of radiotherapy with immunotherapy on antitumor are obtained, including the threshold value of stability conditions of tumor‐eradication periodic solution when the mixed combination treatment of immunotherapy and radiotherapy is performed. Some numerical simulations for the effects of the timing of radiotherapy application and the timing of injection T cells on the threshold value are performed. Finally, we present some theoretical methods for suppressing the growth of tumors. [ABSTRACT FROM AUTHOR]

We study bifurcations from radial solution of a free boundary problem modeling the dormant state of nonnecrotic solid tumors in the presence of external inhibitors. This problem consists in three linear elliptic equations with two Dirichlet and one Neumann boundary conditions and a fourth boundary condition coupling surface tension effects on free boundary. In this paper, surface tension coefficient γ plays the role of bifurcation parameter. We prove that in certain situations there exists a positive null point sequence for γ where bifurcation occurs from radial solution, while in the other situations, either bifurcation occurs at only finite many points of γ or even it does not occur for any γ > 0. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we investigate local and global asymptotic stability of a positive equilibrium point of system of differential equations [ABSTRACT FROM AUTHOR]

In this paper, a stage-structured SI epidemic model with time delay and nonlinear incidence rate is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium, and the existence of Hopf bifurcations are established. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A new fractional difference equation two‐dimensional triangle function combination discrete chaotic map (2D‐TFCDM) based on Caputo derivative is proposed. Using the bifurcation diagram, the maximum Lyapunov exponent, and the phase diagram, the numerical solutions of the fractional difference equations are obtained, and the chaotic behavior is observed numerically. After encrypting the key with elliptic curve cryptosystem, the fractional map is developed as an encryption algorithm and applied to color image encryption. Finally, the proposed encryption system is systematically analyzed from five main aspects, and the results show that the proposed encryption system has a good encryption effect. [ABSTRACT FROM AUTHOR]

The Shilnikov-type single-pulse homoclinic orbits and chaotic dynamics of a simply supported truss core sandwich plate subjected to the transverse and the in-plane excitations are investigated in detail. The resonant case considered here is principal parametric resonance and 1:2 internal resonance. Based on the normal form theory, the desired form for the global perturbation method is obtained. By using the global perturbation method developed by Kovacic and Wiggins, explicit sufficient conditions for the existence of a Shilnikov-type homoclinic orbit are obtained, which implies that chaotic motions may occur for this class of truss core sandwich plate in the sense of Smale horseshoes. Numerical results obtained by using the fourth-order Runge-Kutta method agree with theoretical analysis at least qualitatively. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this study, a mathematical, fractional‐order model was developed for B cell chronic lymphocytic leukemia, with immune system, and then analyzed. Interactions between B leukemia cells, natural killer cells, cytotoxic T cells, and T‐helper cells are considered to be incorporated into a system consisting of four fractional differential equations. For estimation of the parameters, clinical data of six patients were used. By numerical solution of the system, the interactions between the leukemia cell population and the immune system cell populations for values of α ∈ (0,1) at different times were explained. By determining points of equilibrium and stability of the system were met. Bifurcation analysis showed that use of the fractional‐order model, figure out unpredictable behaviors of the system such as saddle‐node, bistability and hysteresis phenomenon occurred in the system by changing the values of some of the parameters, it was predictable. [ABSTRACT FROM AUTHOR]

We explore the local dynamics, N‐S bifurcation, and hybrid control in a discrete‐time Lotka‐Volterra predator‐prey model in R+2. It is shown that ∀ parametric values, model has two boundary equilibria: P00(0,0) and Px0(1,0), and a unique positive equilibrium point: Pxy+dc,rc−dbc if c>d. We explored the local dynamics along with different topological classifications about equilibria: P00(0,0), Px0(1,0), and Pxy+dc,rc−dbc of the model. It is proved that model cannot undergo any bifurcation about P00(0,0) and Px0(1,0) but it undergoes an N‐S bifurcation when parameters vary in a small neighborhood of Pxy+dc,rc−dbc by using a center manifold theorem and bifurcation theory and meanwhile, invariant close curves appears. The appearance of these curves implies that there exist a periodic or quasiperiodic oscillations between predator and prey populations. Further, theoretical results are verified numerically. Finally, the hybrid control strategy is applied to control N‐S bifurcation in the discrete‐time model. [ABSTRACT FROM AUTHOR]

The dynamics of a Leslie‐Gower prey‐predator system with ratio‐dependent Holling IV functional response and constant harvesting rate of prey are taken into account. The results developed in this article reveal far richer dynamics compared with the system without harvesting. We first make qualitative and bifurcation analysis of the system without harvesting and show that the system has a weak focus of multiplicity at most 2, at which a Hopf bifurcation occurs. However, the system with harvesting has four nonhyperbolic equilibria for some parameter values, such as two saddle‐node, a cusp, and a weak focus of multiplicity at most 4, and exhibits two saddle‐node bifurcations, a Bogdanov‐Takens bifurcation of codimension 2, and a Hopf bifurcation. It reveals that there exist some critical harvesting values such that the species are in danger of extinction when the harvesting rate is greater than the critical values, which indicates that the dynamics of the system are sensitive to the constant prey harvesting. Moreover, numerical simulations are presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

An age‐structured pertussis model with covert infection is proposed to understand the effect of covert infection on the recurrence of pertussis. It is found that vaccination only for young children does not have a decisive effect on whooping cough control. It is shown that although the vaccine coverage rate is relatively high, the model has a backward bifurcation for a larger covert infection rate. In addition, sufficient conditions for the disease‐free steady state to be globally asymptotically stable are obtained. [ABSTRACT FROM AUTHOR]

The consumer–resource relationships are among the most fundamental of all ecological relationships and have been the focus of ecology since its beginnings. Usually are described by nonlinear differential equation systems, putting the emphasis in the effect of antipredator behavior (APB) by the prey; nevertheless, a minor quantity of articles has considered the social behavior of predators. In this work, two predator–prey models derived from the Volterra model are analyzed, in which the equation of predators is modified considering cooperation or collaboration among predators. It is well known that competition among predators produces a stabilizing effect on system describing the model, since there exists a wide set in the parameter space where the system has a unique equilibrium point in the phase plane, which is globally asymptotically stable. Meanwhile, the cooperation can originate more complex and unusual dynamics. As we will show, it is possible to prove that for certain subset of parameter values the predator population sizes tend to infinite when the prey population goes to extinct. This apparently contradicts the idea of a realistic model, when it is implicitly assumed that the predators are specialist, ie, the prey is its unique source of food. However, this could be a desirable effect when the prey constitutes a plague. To reinforce the analytical result, numerical simulations are presented. [ABSTRACT FROM AUTHOR]

A proposed discretized form of fractional‐order prey‐predator model is investigated. A sufficient condition for the solution of the discrete system to exist and to be unique is determined. Jury stability test is applied for studying stability of equilibrium points of the discretized system. Then, the effects of varying fractional order and other parameters of the systems on its dynamics are examined. The system undergoes Neimark‐Sacker and flip bifurcation under certain conditions. We observe that the model exhibits chaotic dynamics following stable states as the memory parameter α decreases and step size h increases. Theoretical results illustrate the rich dynamics and complexity of the model. Numerical simulation validates theoretical results and demonstrates the presence of rich dynamical behaviors include S‐asymptotically bounded periodic orbits, quasi‐periodicity, and chaos. The system exhibits a wide range of dynamical behaviors for fractional‐order α key parameter. [ABSTRACT FROM AUTHOR]

We study the maximum mean discrepancy (MMD) in the context of critical transitions modelled by fast‐slow stochastic dynamical systems. We establish a new link between the dynamical theory of critical transitions with the statistical aspects of the MMD. In particular, we show that a formal approximation of the MMD near fast subsystem bifurcation points can be computed to leading order. This leading order approximation shows that the MMD depends intricately on the fast‐slow systems parameters, which can influence the detection of potential early‐warning signs before critical transitions. However, the MMD turns out to be an excellent binary classifier to detect the change‐point location induced by the critical transition. We cross‐validate our results by numerical simulations for a van der Pol‐type model. [ABSTRACT FROM AUTHOR]

Many components of machines and other technological devices (chains and bodies hanging on the ropes) can be modeled as multiple pendulums situated in tubes or holes of limited space. The investigation of motion of such systems represents the substance of many real technological problems; therefore, it stirred up motivation to perform research on vibration of a double pendulum situated between two skew rigid walls. The analyzed system was set into motion by the horizontal movement of its suspending. The results of the simulations show that the system can exhibit both the regular and chaotic movements depending on the excitation frequency. The development of the computational model, which is applicable for more complex investigations, and learning more on the motion character of a double pendulum, the movement of which is limited by skew walls, are the principal contributions of the presented article. [ABSTRACT FROM AUTHOR]

A deterministic model for the transmission dynamics of Zika is designed and rigorously analysed. A model consisting of mutually exclusive compartments representing the human and mosquito dynamics takes into account both direct (human‐human) and indirect modes of transmissions. The basic offspring number of the mosquito population is computed and condition for existence and stability of equilibria is investigated. Using the centre manifold theory, the model (with and without direct transmission) is shown to exhibit the phenomenon of backward bifurcation (where a locally asymptotically stable disease free equilibrium coexists with a locally asymptotically stable endemic equilibrium) whenever the associated reproduction number is less than unity. The study shows that the models with and without direct transmission exhibit the same qualitative dynamics with respect to the local stability of their associated disease‐free equilibrium and backward bifurcation phenomenon. The main cause of the backward bifurcation is identified as Zika induced mortality in humans. Sensitivity (local and global) analysis of the model parameters are conducted to identify crucial parameters that influence the dynamics of the disease. Analysis of the model shows that an increase in the mating rate with sterile mosquito decreases the mosquito population. Numerical simulations, using parameter values relevant to the transmission dynamics of Zika are carried out to support some of the main theoretical findings. [ABSTRACT FROM AUTHOR]

Syphilis, a major sexually transmitted disease, continues to pose major public health burden in both underdeveloped and developed nations of the world. This study presents a new 2‐group sex‐structured model for assessing the community‐level impact of treatment and condom use on the transmission dynamics and control of syphilis. Rigorous analysis of the model shows that it undergoes the phenomenon of backward bifurcation. In the absence of this phenomenon (which is shown to arise because of the reinfection of recovered individuals), the disease‐free equilibrium of the model is shown to be globally asymptotically stable when the associated reproduction number is less than unity. Furthermore, the model can have multiple endemic equilibria when the reproduction threshold exceeds unity. Numerical simulations of the model, using data relevant to the transmission dynamics of the disease in Nigeria, show that, with the assumed 80% condom efficacy, the disease will continue to persist (ie, remain endemic) in the population regardless of the level of compliance in condom usage by men. Furthermore, detailed optimal control analysis (using Pontraygin's maximum principle) reveals that, for situations where the cost of implementing the controls (treatment and condom‐use) considered in this study is low, channelling resources to a treatment‐only strategy is more effective than channelling them to a condom‐use only strategy. Furthermore, as expected, the combined condom‐treatment strategy provides a higher population‐level impact than the treatment‐only strategy or the condom‐use only strategy. When the cost of implementing the controls is high, the 3 strategies are essentially equally as ineffective. [ABSTRACT FROM AUTHOR]

A 4‐dimensional mathematical model of the hypothalamus‐pituitary‐adrenal axis is investigated, incorporating the influence of the glucocorticoid receptors concentration and general feedback functions. The inclusion of distributed time delays provides a more realistic modeling approach, since the whole past history of the variables is taken into account. The positivity of the solutions and the existence of a positively invariant bounded region are proved. It is shown that the considered 4‐dimensional system has at least 1 equilibrium state, and a detailed local stability and Hopf bifurcation analysis is given. Numerical results reveal the fact that an appropriate choice of the system's parameters leads to the coexistence of 2 asymptotically stable equilibria in the nondelayed case. When the total average time delay of the system is large enough, the coexistence of 2 stable limit cycles is revealed, which successfully model the ultradian rhythm of the hypothalamus‐pituitary‐adrenal axis both in a normal disease‐free situation and in a diseased hypocortisolism state, respectively. Numerical simulations reflect the importance of the theoretical results. [ABSTRACT FROM AUTHOR]

For two‐dimensional autonomous linear incommensurate fractional‐order dynamical systems with Caputo derivatives of different orders, necessary and sufficient conditions are obtained for the asymptotic stability and instability of the null solution. These conditions are expressed in terms of the elements of the system's matrix, as well as of the fractional orders of the Caputo derivatives, leading to a generalization of the well known Routh‐Hurwitz conditions. These theoretical results are then used to investigate the stability properties of a two‐dimensional fractional‐order FitzHugh‐Nagumo neuronal model. The occurrence of Hopf bifurcations is also discussed. Numerical simulations are provided with the aim of exemplifying the theoretical results, revealing rich spiking behavior, in comparison with the classical integer‐order FitzHugh‐Nagumo model. [ABSTRACT FROM AUTHOR]

This research was motivated by a real technological problem of vibrations of bodies hanging on chains or ropes in tubes or spaces limited by walls or other bodies. The studied system has two degrees of freedom. It is formed by two pendulums moving between two walls. Its movement is governed by a set of nonlinear ordinary differential equations. The results of the simulations shown that the system exhibits regular and chaotic movement. The simulations were performed for 3 excitation amplitudes and the range of the excitation frequencies between 1 and 30 rad s−1. The subject of the investigations was the determination of the character of the pendulums' motions and identification of their collisions with the sided walls. [ABSTRACT FROM AUTHOR]

In this work, a modified Holling-Tanner predator-prey model is analyzed, considering important aspects describing the interaction such as the following: the predator growth function is of a logistic type, and a weak Allee effect acting in the prey growth function and the functional response is of hyperbolic type. By making a change of variables and a time rescaling, we obtain a polynomial differential equations system topologically equivalent to the original one, in which the nonhyperbolic equilibrium point (0,0) is an attractor for all parameter values. An important consequence of this property is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane, and the system exhibits the bistability phenomenon, because the trajectories can have different ω-limit sets, as an example, the origin (0,0) or a stable limit cycle surrounding an unstable positive equilibrium point. We show that, under certain parameter conditions, a positive equilibrium may undergo saddle-node, Hopf, and Bogdanov-Takens bifurcations; the existence of a homoclinic curve on the phase plane is also proved, which breaks in an unstable limit cycle. Some simulations to reinforce our results are also shown. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A theoretical eco-epidemiological model of a prey-predator interaction system with disease in prey species is studied. Predator consumes both susceptible and infected prey population, but predator also feeds preferentially on many numerous species, which are over represented in the predator's diet. Equilibrium points of the system are determined, and the dynamic behaviour of the system is investigated around equilibrium points. Death rate of predator species is considered as a bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighbourhood of the coexisting equilibria. Numerical simulations are carried out to support the analytical results. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

T. Wanner We investigate global dynamics of the equation [ABSTRACT FROM AUTHOR]

In this work, a modified Holling-Tanner predator-prey model is analyzed, considering important aspects describing the interaction such as the predator growth function is of a logistic type; a weak Allee effect acting in the prey growth function, and the functional response is of hyperbolic type. Making a change of variables and time rescaling, we obtain a polynomial differential equations system topologically equivalent to the original one in which the non-hyperbolic equilibrium point (0,0) is an attractor for all parameter values. An important consequence of this property is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane, and the system exhibits the bistability phenomenon, because the trajectories can have different ? - limit sets; as example, the origin (0,0) or a stable limit cycle surrounding an unstable positive equilibrium point. We show that, under certain parameter conditions, a positive equilibrium may undergo saddle-node, Hopf, and Bogdanov-Takens bifurcations; the existence of a homoclinic curve on the phase plane is also proved, which breaks in an unstable limit cycle. Some simulations to reinforce our results are also shown. [ABSTRACT FROM AUTHOR]

Dramatic increases in jellyfish populations, which lead to the collapse of formally healthy ecosystems, are repeatedly reported from many different sites. Because of their devastating effects on fish populations, understanding of the causes for such bloomings are of major ecological as well as economical importance. Based on a previous work, we set up a two-dimensional predator-prey model for the fish-jellyfish interactions by taking fishery as well as environmental conditions into account. It assumes the fish as the dominant predator species. By totally analytic means, we completely classify all equilibria in terms of existence and non-linear stability and give a description of this system's non-linear global dynamics. We analytically study the non-equilibrium dynamics and detect homoclinic, Andronov-Hopf, and Takens-Bogdanov bifurcations as well as the non-existence of periodic and homoclinic orbits in certain relevant regions of the phase space. A hereby found, and analytically rigorously stated, funnel phenomenon gives rise to a mathematical explanation of jellyfish blooming beyond the typical one in terms of large-amplitude limit cycles based on the well-known Rosenzweig-MacArthur equations. Also, we illustrate the plethora of bifurcation scenarios by numerical results. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Criteria for the bifurcation of small solutions of an equation F( λ, u) = 0 from a line [ABSTRACT FROM AUTHOR]

We consider the existence of spots and stripes for a class of nonlinear Schrödinger-type equations in the presence of nearly one-dimensional localized potentials. Under suitable assumptions on the potential, we construct various types of waves that are localized in the direction of the potential and have single- or multihump, or periodic profile in the perpendicular direction. The analysis relies upon a spatial dynamics formulation of the existence problem, together with a center manifold reduction. This reduction procedure allows these waves to be realized as unipulse or multipulse homoclinic orbits, or periodic orbits in a reduced system of ordinary differential equations. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]