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There are few adapted SIR models in the literature that combine vaccination and logistic growth. In this article, we study bifurcations of a SIR model where the class of Susceptible individuals grows logistically and has been subject to constant vaccination. We explicitly prove that the endemic equilibrium is a codimension two singularity in the parameter space (R 0 , p) , where R 0 is the basic reproduction number and p is the proportion of Susceptible individuals successfully vaccinated at birth. We exhibit explicitly the Hopf, transcritical, Belyakov, heteroclinic and saddle-node bifurcation curves unfolding the singularity. The two parameters (R 0 , p) are written in a useful way to evaluate the proportion of vaccinated individuals necessary to eliminate the disease and to conclude how the vaccination may affect the outcome of the epidemic. We also exhibit the region in the parameter space where the disease persists and we illustrate our main result with numerical simulations, emphasizing the role of the parameters. [ABSTRACT FROM AUTHOR]

The main attention of this work focuses on studying the bifurcations and dispersive optical solitons for the coupled nonlinear Schrödinger equation in optical fiber Bragg gratings. First of all, with the help of bifurcation method of planar dynamical systems, the system's Hamiltonian and the orbits phase portraits have been found. As you can see, we construct the dark and singular solitons to this system. In addition to, some other optical solitons are derived by using the polynomial complete discriminant method and computer algebra with symbolic computation. In particular, it is worth to noting that some of the solutions which have been obtained in this work are new and not available in the known references. As a consequence, the obtained optical soliton solutions substantially improve or complement the corresponding conditions in the known references. Finally, in order to understand mechanisms of complex physical phenomena and dynamical processes for this system, we draw the two-dimensional and three-dimensional graphs. [ABSTRACT FROM AUTHOR]

A new four-dimensional chaotic system with four cross-terms is investigated in this study. The dynamical behaviors of the system and codimension-one bifurcations are analyzed both theoretically and numerically. The influences of the system’s parameters on its nonlinear dynamics are explored. Then, a proposed four-dimensional blinking system is introduced based on the new chaotic system. More interestingly, the possible occurrence of ghost attractors in the phase space of the blinking system is verified and has been illustrated through several examples. The encryption keystream was generated dynamically using pre-treated chaotic mixed sequences and was used to mask the voice bitstream to protect voice data integrity. Finally, a voice encryption example employing the 4-D blinking chaotic system of the proposed system is presented.

Background: Optimal duration of dual antiplatelet therapy (DAPT) in patients undergoing complex percutaneous coronary intervention (PCI) remains under investigation. Our aim is to compare shortened (≤3 months) DAPT with longer DAPT in patients undergoing complex PCIs. Methods: Three major databases (MEDLINE, Cochrane Central Register of Controlled Trials, and Scopus) were screened. The primary endpoint was major bleedings as they are defined by the Bleeding Academic Research Consortium (BARC) 3–5. The secondary endpoints were major adverse cardiovascular events, all-cause and cardiovascular mortality, myocardial infarction, stroke, and stent thrombosis. Results: Five studies were included in our analysis, with a total of 9,115 patients. Our meta-analysis met its primary endpoint, as abbreviated DAPT significantly reduced major bleedings by 43% (95% confidence intervals: 0.35–0.93). Ischemic events and mortality were not affected by the shortening of DAPT. Conclusion: Shortened DAPT significantly reduced the odds of major bleedings in patients undergoing complex PCI without increasing the ischemic events or mortality. Thus, it could be considered a safe and feasible option in such patients.

In this paper, some complicated dynamic characteristics are formulated for a discrete predator-prey model with a prey refuge. After studying the local dynamical properties about fixed points, our main purpose is to investigate condition(s) for the occurrence of flip and hopf bifurcations, respectively. Further, by the bifurcation theory, we have studied flip bifurcation at boundary fixed point, and flip and hopf bifurcations at interior fixed point of the discrete model. We have also studied chaos by state feedback control strategy. Furthermore, theoretical results are numerically verified. Finally, we have also discussed the influence of prey refuge in the discrete model.

This paper proposes a non-smooth human influenza model with logistic source to describe the impact on media coverage and quarantine of susceptible populations of the human influenza transmission process. First, we choose two thresholds $ I_{T} $ and $ S_{T} $ as a broken line control strategy: Once the number of infected people exceeds $ I_{T} $, the media influence comes into play, and when the number of susceptible individuals is greater than $ S_{T} $, the control by quarantine of susceptible individuals is open. Furthermore, by choosing different thresholds $ I_{T} $ and $ S_{T} $ and using Filippov theory, we study the dynamic behavior of the Filippov model with respect to all possible equilibria. It is shown that the Filippov system tends to the pseudo-equilibrium on sliding mode domain or one endemic equilibrium or bistability endemic equilibria under some conditions. The regular/virtulal equilibrium bifurcations are also given. Lastly, numerical simulation results show that choosing appropriate threshold values can prevent the outbreak of influenza, which implies media coverage and quarantine of susceptible individuals can effectively restrain the transmission of influenza. The non-smooth system with logistic source can provide some new insights for the prevention and control of human influenza.

Oscillatory circuits with real memristors have attracted a lot of interest in recent years. The vast majority of circuits involve volatile memristors, while less explored is the use of non-volatile ones. This paper considers a circuit composed by the interconnection of a two-terminal (one port) element, based on the linear part of Chua’s circuit, and a non-volatile memristor obeying the Stanford model. A peculiar feature of such a memristor is that its state displays negligible time-variations under some voltage threshold. Exploiting this feature, the memristor is modeled below threshold as a programmable nonlinear resistor whose resistance depends on the gap distance. Then, the first-order Harmonic Balance (HB) method is employed to derive a procedure to select the parameters of the two-terminal element in order to generate programmable subthreshold oscillatory behaviors, within a given range of the gap, via a supercritical Hopf bifurcation. Finally, the dynamic behaviors of the designed circuits as well as the sensitivity of the procedure with respect to the location of the bifurcating equilibrium point and the range of the gap are discussed and illustrated via some application examples.

The classic Lotka–Volterra model is a two-dimensional system of differential equations used to model population dynamics among two-species: a predator and its prey. In this article, we consider a modified three-dimensional fractional-order Lotka–Volterra system that models population dynamics among three-species: a predator, an omnivore and their mutual prey. Biologically speaking, population models with a discrete and continuous structure often provide richer dynamics than either discrete or continuous models, so we first discretize the model while keeping one time-continuous dependent variable in each equation. Then, we analyze the stability and bifurcation near the equilibria. The results demonstrated that the dynamic behaviors of the discretized model are sensitive to the fractional-order parameter and discretization parameter. Finally, numerical simulations are performed to explain and validate the findings, and the maximum Lyapunov exponents is computed to confirm the presence of chaotic behavior in the studied model.

It is widely known that an appropriately built unpowered bipedal robot can walk down an inclined surface with a passive steady gait. The features of such gait are determined by the robot's geometry and inertial properties, as well as the slope angle. The energy needed to keep the biped moving steadily comes from the gravitational potential energy as it descends the inclined surface. The study of such passive natural motions could lead to ideas for managing active walking devices and a better understanding of the human locomotion. The major goal of this study is to further investigate order, chaos and bifurcations and then to demonstrate the complexity of the passive bipedal walk of the compass-gait biped robot by examining different bifurcation diagrams and also by studying the variation of the eigenvalues of the Poincaré map's Jacobian matrix and the variation of the Lyapunov exponents. We reveal also the exhibition of some additional results by changing the inertial and geometrical parameters of the bipedal robot model.

We report a new batch of wave solutions for the coupled Drinfel’d–Sokolov–Wilson equation which represents a coupled system of nonlinear partial differential equations (NLPDEs). Firstly by making a travelling wave ansatz, we decouple the system and obtain a second-order ordinary differential equation (ODE). Thereafter we perform phase space and bifurcation analysis of that second-order ODE and proceed to construct the general solution for the envelope of the wave packet. The solutions are expressed in terms of the Jacobi elliptic sine function from which one can obtain solitary wave (particular) solutions by imposing appropriate conditions on the roots of certain quartic polynomials as discussed thereafter.

This paper presents a rigorous exploration of the extended Vakhnenko–Parkes-Equation (eVPE), unveiling a rich tapestry of multi solitons, bifurcations, and high-order breathers. Notably, it achieves an impressive feat by deriving solitons up to sixth order and breathers up to third order, pushing the boundaries of the equation’s solutions. These intricate solutions are not only derived but also vividly brought to life through numerical visualizations, underscoring the profound implications of the our work. Through insightful discussions, the manuscript effectively underscores the significance of these findings, opening new avenues for understanding nonlinear dynamics and their applications across various scientific domains.

At the end of the 20th century, analog systems in computer science have been widely replaced by digital systems due to their higher computing power. Nevertheless, the question keeps being intriguing until now: is the brain analog or digital? Initially, the latter has been favored, considering it as a Turing machine that works like a digital computer. However, more recently, digital and analog processes have been combined to implant human behavior in robots, endowing them with artificial intelligence (AI). Therefore, we think it is timely to compare mathematical models with the biology of computation in the brain. To this end, digital and analog processes clearly identified in cellular and molecular interactions in the Central Nervous System are highlighted. But above that, we try to pinpoint reasons distinguishing in silico computation from salient features of biological computation. First, genuinely analog information processing has been observed in electrical synapses and through gap junctions, the latter both in neurons and astrocytes. Apparently opposed to that, neuronal action potentials (APs) or spikes represent clearly digital events, like the yes/no or 1/0 of a Turing machine. However, spikes are rarely uniform, but can vary in amplitude and widths, which has significant, differential effects on transmitter release at the presynaptic terminal, where notwithstanding the quantal (vesicular) release itself is digital. Conversely, at the dendritic site of the postsynaptic neuron, there are numerous analog events of computation. Moreover, synaptic transmission of information is not only neuronal, but heavily influenced by astrocytes tightly ensheathing the majority of synapses in brain (tripartite synapse). At least at this point, LTP and LTD modifying synaptic plasticity and believed to induce short and long-term memory processes including consolidation (equivalent to RAM and ROM in electronic devices) have to be discussed. The present knowledge of how the brain stores and retrieves memories includes a variety of options (e.g., neuronal network oscillations, engram cells, astrocytic syncytium). Also epigenetic features play crucial roles in memory formation and its consolidation, which necessarily guides to molecular events like gene transcription and translation. In conclusion, brain computation is not only digital or analog, or a combination of both, but encompasses features in parallel, and of higher orders of complexity. [ABSTRACT FROM AUTHOR]

This paper considers the production of biomass of two interconnected chemostats in series with biomass mortality and a growth kinetic of the biomass described by an increasing function. A comparison is made with the productivity of a single chemostat with the same mortality rate and with volume equal to the sum of the volumes of the two chemostats. We determine the operating conditions under which the productivity of the serial configuration is greater than the productivity of the single chemostat. Moreover, the differences and similarities in the results corresponding to the case with mortality and the one without mortality, are highlighted. The mortality leads to surprising results where the productivity of a steady state where the bacteria are washed out in the first chemostat is greater than the one where the bacteria are present in both chemostats. [ABSTRACT FROM AUTHOR]

When a laser cavity supports the propagation of several ultrashort pulses, these pulses can form compact bound states called soliton molecules. Soliton molecules are fascinating objects of nonlinear science, presenting striking analogies with their matter molecule counterparts. The relative timing and phase between the copropagating pulses are the most salient internal degrees of freedom of the soliton molecule. The soliton pair, composed of two identical pulses, constitutes the chief soliton molecule of fundamental interest. Its two major internal degrees of freedom allow self‐oscillating soliton molecules, which have been recurrently observed. However, despite theoretical predictions, the low‐dimensional chaotic dynamics of a soliton‐pair molecule remain elusive. This article reports the observation of chaotic soliton‐pair molecules within an ultrafast fiber laser by means of a direct measurement of the relative optical pulse separation with sub‐femtosecond precision in real time. Moreover, it demonstrates an all‐optical control of the chaotic dynamics followed by the soliton molecule by injection of a modulated optical signal that resynchronizes the internal periodic vibration of the soliton molecule. The fast error‐free switching between ordered and chaotic soliton molecules enabled by pump current sweeping and external injection highlights the potential prospects of all‐optical logic gates and chaotic communication using soliton molecules. [ABSTRACT FROM AUTHOR]

This paper examines the relationship between herbivores and plants with a strong Allee effect. When the plant reaches a particular size, the herbivore attacks it. We use the logistic equation to model plant growth and analyze its behavior without herbivores before investigating their interactions. Our study investigates the equilibrium points and their stability, discovering that different fixed points can become unstable due to various bifurcations such as transcritical, saddle-node, period-doubling, and Neimark–Sacker bifurcations. We have identified the Allee threshold, which, if exceeded, can cause both populations to become extinct below that level. However, we have discovered a coexistence equilibrium that is locally asymptotically stable for a range of parameter values above that threshold. Our additional numerical simulations suggest that this area of stability can be expanded. Our results indicate that this system is highly responsive to its parameters. We compare our findings to those of a system without strong Allee effects and conduct numerical simulations to verify our results. By including the Allee effect in the plant population, we enrich the local and global dynamics of the system. [ABSTRACT FROM AUTHOR]

In this paper, we consider a restricted (N + 1) -body problem on surfaces M κ 2 , where the constant κ ≠ 0 is the Gaussian curvature, which by means of a rescaling can be reduced to κ = ± 1. This problem consists in the study of the dynamics of an infinitesimal mass particle attracted by N primaries of identical masses describing elliptic relative equilibria of the N -body problem on M κ 2 , i.e., a solution where the primaries are rotating uniformly with constant angular velocity ω on a fixed parallel of S 2 or H 2 and placed at the vertices of a regular polygon. In a rotating frame, this problem yields a two degrees of freedom Hamiltonian system. The goal of this paper is to describe analytically some dynamics features for κ = ± 1. Precisely, we study the relative location of equilibria, obtaining, in particular, that the poles of S 2 and vertex of H 2 represent equilibrium points for any value of the parameters. Thus, analysis of the nonlinear stability of these equilibria is carried out, as well as two types of bifurcations are detected: Hamiltonian-Hopf and N -bifurcation. Additionally, we prove the existence of a family of Hill's orbits and a family of periodic orbits when the primaries are located near the poles of S 2 or the vertex of H 2. Finally we prove the existence of KAM 2-tori related to these periodic orbits. [ABSTRACT FROM AUTHOR]

In this paper, we compare three energy harvesting systems in which we introduce additional bumpers whose mathematical model is mapped with a non-linear characteristic based on the hyperbolic sine Fibonacci function. For the analysis, we construct non-linear two-well, three-well and four-well systems with a cantilever beam and permanent magnets. In order to compare the effectiveness of the systems, we assume comparable distances between local minima of external wells and the maximum heights of potential barriers. Based on the derived dimensionless models of the systems, we perform simulations of non-linear dynamics in a wide spectrum of frequencies to search for chaotic and periodic motion zones of the systems. We present the issue of the occurrence of transient chaos in the analyzed systems. In the second part of this work, we determine and compare the effectiveness of the tested structures depending on the characteristics of the bumpers and an external excitation whose dynamics are described by the harmonic function, and find the best solutions from the point view of energy harvesting. The most effective impact of the use of bumpers can be observed when dealing with systems described by potential with deep external wells. In addition, the use of the Fibonacci hyperbolic sine is a simple and effective numerical tool for mapping non-linear properties of such motion limiters in energy harvesting systems. [ABSTRACT FROM AUTHOR]

An analytical classification in a three-dimensional parameter space is presented to describe the dynamics for an extended Lorenz system of the Li-Ou type, conditions are given to find supercritical and degenerate Hopf bifurcations and a pitchfork bifurcation. Finally, the theoretical results are compared with numerical simulations and bifurcation diagrams. [ABSTRACT FROM AUTHOR]

Numerical findings of natural convection flows in a trapezoidal cavity are reported in this study. This study focuses on the shift from symmetric steady to chaotic flow within the cavity. This cavity has a heated bottom wall, a cooled top wall, and adiabatic inclined sidewalls. The unsteady natural convection flows occurring within the cavity are numerically simulated using the finite volume (FV) method. The fluid used in the study is air, and the calculations are performed for different dimensionless parameters, including the Prandtl number (Pr), which is kept constant at 0.71, while varying the Rayleigh numbers (Ra) from 100 to 108 and using a fixed aspect ratio (AR) of 0.5. This study focuses on the effect of the Rayleigh numbers on the transition to chaos. In the transition to chaos, a number of bifurcations occur. The first primary transition is found from the steady symmetric to the steady asymmetric stage, known as a pitchfork bifurcation. The second leading transition is found from a steady asymmetric to an unsteady periodic stage, known as Hopf bifurcation. Another prominent bifurcation happens on the changeover of the unsteady flow from the periodic to the chaotic stage. The attractor bifurcates from a stable fixed point to a limit cycle for the Rayleigh numbers between 4 × 106 and 5 × 106. A spectral analysis and the largest Lyapunov exponents are analyzed to investigate the natural convection flows during the shift from periodic to chaos. Moreover, the cavity's heat transfers are computed for various regimes. The cavity's flow phenomena are measured and verified. [ABSTRACT FROM AUTHOR]

Whirl flutter is an aeroelastic instability that affects aircraft with propellers/rotors. With their long and flexible rotor blades, tiltrotor aircraft are particularly susceptible. Whirl flutter is known to have destroyed aircraft and in the best case it constitutes a fatigue hazard. The complexity of whirl flutter analysis increases significantly with the addition of nonlinearities, due to the more complex dynamical behaviours that emerge as a result. Most whirl flutter stability analyses in current literature are grounded in linear theory, preventing the full discovery of the nonlinearities' effects. Continuation and bifurcation methods (CBM) may instead be used to fully appreciate and analyse the effects of the presence of nonlinearities. Previous CBM-based work on nonlinear gimballed hub rotor-nacelle models, representing those found on tiltrotor aircraft, are capable of whirl flutter in parametric regions declared safe by linear analysis. Furthermore, it was found that they are capable of complex behaviours including limit cycle oscillations, quasi-periodic behaviour and even chaos, though the whirl flutter implications of such behaviours has not been explored. This paper investigates the impact of a smooth structural nonlinearity on the whirl flutter stability of a basic gimballed rotor-nacelle model, compared to its baseline linear stiffness version. A 9-DoF model with quasi-steady aerodynamics, a flexible wing and blades that can move both cyclically and collectively in both flapping and lead-lag motions, producing gimbal flap-like behaviour, was adopted from existing literature. A smooth stiffness nonlinearity was introduced in the blade flapping stiffness and CBM was used to find the new whirl flutter behaviours created by the presence of the nonlinearity. Time simulations, Poincaré sections and spectral analysis were then used to investigate the various behaviours found. This in turn allowed recommendations to be made concerning preferable and/or hazardous parameter combinations of use to the tiltrotor designer. [ABSTRACT FROM AUTHOR]

The paper is dedicated to the numerical and experimental study of nonlinear oscillations exhibited by the Vilnius chaotic generator. The motivation for the work is defined by the need for a comprehensive analysis of the dynamics of the oscillators being embedded into chaotic communication systems. These generators should provide low-power operation while ensuring the robustness of the chaotic oscillations, insusceptible to parameter variations and noise. The work focuses on the investigation of the dependence of nonlinear dynamics of the Vilnius oscillator on the operating voltage and component parameter changes. The paper shows that the application of the Method of Complete Bifurcation Groups reveals the complex smooth and non-smooth bifurcation structures, forming regions of robust chaotic oscillations. The novel tool—mode transition graph—is presented, allowing the comparison of experimental and numerical results. The paper demonstrates the applicability of the Vilnius oscillator for the generation of robust chaos, and highlights the need for further investigation of the inherent trade-off between energy efficiency and robustness of the obtained oscillations. [ABSTRACT FROM AUTHOR]

This paper introduces a new chaotic jerk system with three cubic nonlinear terms. The stability properties of the three equilibrium points of the proposed jerk system are analyzed in detail. We show that the three equilibrium points of the new chaotic jerk system are unstable and deduce that the jerk system exhibits self-excited chaotic attractors. The bifurcation structures of the proposed jerk system are investigated numerically, showing period-doubling, periodic windows and coexisting bifurcations. An electronic circuit design of the proposed jerk system is designed using PSPICE. As an engineering application, a new image-encryption approach based on the new chaotic jerk system is presented in this research work. Experimental results demonstrate that the suggested encryption mechanism is effective with high plain-image sensitivity and the reliability of the proposed chaotic jerk system for various cryptographic purposes. [ABSTRACT FROM AUTHOR]

In this paper, we show that oscillations associated with instability in haptic feedback systems can be identified using methods developed to detect impending critical transitions in dynamical systems employing the concept of critical slowing down. We use a state-space stability analysis of a model of a haptic device interfacing with a virtual inertia-spring-damper to show that haptic systems can be subject to secondary Hopf or Niemark-Sacker bifurcations, which often exhibit critical slowing down. We propose the use of indicators of critical slowing down, namely the lag-1 autocorrelation and variance of the haptic device state, to detect the onset of instability in real haptic systems. We present results from experiments with the haptic device interfacing with the virtual inertia-spring-damper, based on which we provide recommendations for the length of rolling window to use when calculating the autocorrelation and variance. We also compare the proposed critical slowing down methods to two common haptic instability detection methods from the literature using a haptic device interfacing with a virtual vehicle model. The critical slowing down method detects instability earlier than the other methods while using about half of the computation time. [ABSTRACT FROM AUTHOR]

In this paper, some complicated dynamic characteristics are formulated for a discrete predator-prey model with a prey refuge. After studying the local dynamical properties about fixed points, our main purpose is to investigate condition(s) for the occurrence of flip and hopf bifurcations, respectively. Further, by the bifurcation theory, we have studied flip bifurcation at boundary fixed point, and flip and hopf bifurcations at interior fixed point of the discrete model. We have also studied chaos by state feedback control strategy. Furthermore, theoretical results are numerically verified. Finally, we have also discussed the influence of prey refuge in the discrete model. [ABSTRACT FROM AUTHOR]

This study examines discrete-time T system. We begin by listing the topological divisions of the system's fixed points. Then, we analytically demonstrate that a discrete T system sits at the foundation of a Neimark Sacker(NS) bifurcation under specific parametric circumstances. With the use of the explicit Flip-NS bifurcation criterion, we establish the flip-NS bifurcation's reality. Center manifold theory is then used to establish the direction of both bifurcations. We do numerical simulations to validate our theoretical findings. Additionally, we employ the 0-1 test for chaos to demonstrate whether or not chaos exists in the system. In order to stop the system's chaotic trajectory, we ultimately employ a hybrid control method. [ABSTRACT FROM AUTHOR]

This work is devoted to present temporal-only and spatio-temporal COVID-19 epidemic models when virus mutations and vaccination influences are considered. Firstly, the proposed non-diffusive COVID-19 model is introduced. The nonlinear incidence rate is employed to better model the strict measures forced by governmental authorities to control pandemic spread. The immunity acquired by vaccinations are assumed to be incomplete for realistic considerations. The existence, uniqueness and continuous dependence on initial conditions are studied for the solution. The study of stability along with bifurcation analysis are carried out to investigate the influences of variations in model's parameters. Moreover, the basic reproduction number is obtained for the proposed model. The stability regions for equilibrium points are depicted in space of parameters to explore their effects. Secondly, the diffusive version of the model is considered where possible occurrence of Turing instability is investigated. Finally, numerical simulations are employed to verify theoretical results of the work. [ABSTRACT FROM AUTHOR]

Caputo fractional (with power-law kernels) and fractional (delta) difference maps belong to a more widely defined class of generalized fractional maps, which are discrete convolutions with some power-law-like functions. The conditions of the asymptotic stability of the fixed points for maps of the orders 0 < α < 1 that are derived in this paper are narrower than the conditions of stability for the discrete convolution equations in general and wider than the well-known conditions of stability for the fractional difference maps. The derived stability conditions for the fractional standard and logistic maps coincide with the results previously observed in numerical simulations. In nonlinear maps, one of the derived limits of the fixed-point stability coincides with the fixed-point—asymptotically period-two bifurcation point. [ABSTRACT FROM AUTHOR]

A powerful method to qualitatively analyze a 2D system is the use of nullclines, curves which separate regions of the plane where the sign of the time derivatives is constant, with their intersections corresponding to steady states. As a quick way to sketch the phase portrait of the system, they can be sufficient to understand the qualitative dynamics at play without integrating the differential equations. While it cannot be extended straightforwardly for dimensions higher than 2, sometimes the phase portrait can still be projected onto a 2-dimensional subspace, with some curves becoming pseudo-nullclines. In this work, we study cell signaling models of dimension higher than 2 with behaviors such as oscillations and bistability. Pseudo-nullclines are defined and used to qualitatively analyze the dynamics involved. Our method applies when a system can be decomposed into 2 modules, mutually coupled through 2 scalar variables. At the same time, it helps track bifurcations in a quick and efficient manner, key for understanding the different behaviors. Our results are both consistent with the expected dynamics, and also lead to new responses like excitability. Further work could test the method for other regions of parameter space and determine how to extend it to three-module systems.

In this paper, we explore two heterogeneous triopoly games where the market demand function is isoelastic. The local stability and the bifurcations of these games are systematically analyzed using a symbolic approach, proposed by the author, of counting real solutions of a parametric system. The novelty of our study is twofold. On one hand, we introduce into the study of oligopoly games several methods of symbolic computation, which can establish analytical results and are different from the existing methods in the literature based on numerical simulations. In particular, we obtain the analytical conditions of the local stability and prove the existence of double routes to chaos through the period-doubling bifurcation and the Neimark-Sacker bifurcation. On the other hand, in the special case of the involved firms having identical marginal costs, we acquire the analytical conditions of the local stability for the two models. By further analyzing these conditions, it seems that the presence of the local monopolistic approximation (LMA) mechanism has a stabilizing effect for heterogeneous triopoly games with the isoelastic demand.

We present direct numerical simulations of the Taylor–Couette flow of a dilute polymer solution when only the inner cylinder rotates and the curvature of the system is moderate (η=0.77). The finitely extensible nonlinear elastic-Peterlin closure is used to model the polymer dynamics. The simulations have revealed the existence of a novel elasto-inertial rotating wave characterized by arrow-shaped structures of the polymer stretch field aligned with the streamwise direction. This rotating wave pattern is comprehensively characterized, including an analysis of its dependence on the dimensionless Reynolds and Weissenberg numbers. Other flow states having arrow-shaped structures coexisting with other types of structures have also been identified for the first time in this study and are briefly discussed. This article is part of the theme issue 'Taylor–Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (part 2)'. [ABSTRACT FROM AUTHOR]

Stably stratified Taylor–Couette flow has attracted much attention due to its relevance as a canonical example of the interplay among rotation, stable stratification, shear and container boundaries, as well as its potential applications in geophysics and astrophysics. In this article, we review the current knowledge on this topic, highlight unanswered questions and propose directions for future research. This article is part of the theme issue 'Taylor–Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (part 2)'. [ABSTRACT FROM AUTHOR]

In this paper we present a mathematical model for studying the interactions between human immune systemand a pathogenic virus, such as Covid-19. Amathematical analysis based on dynamical systems theory is performed. More exactly, we model the interactions between the immune system and the virus by a modified predator-prey method. Several conclusions emerge from this study, and the main two of them are the followings: 1) a deficiency in the concentration of a single type of white blood cells in the early stages of virus proliferation may lead to the virus victory, and 2) if the number of at least one type of white blood cells can be increased beyond the normal threshold by medical interventions in the early stages of virus infection, then the immune system has a better chance to win against the virus. [ABSTRACT FROM AUTHOR]