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Building upon our previous work on the Wilson-Cowan equations with distributed delays (Kaslik et al., 2022), we study the dynamic behavior in a system of two coupled Wilson-Cowan pairs. We focus in particular on understanding the mechanisms that govern the transitions in and out of oscillatory regimes associated with pathological behavior. We investigate these mechanisms under multiple coupling scenarios, and we compare the effects of using discrete delays versus a weak Gamma delay distribution. We found that, in order to trigger and stop oscillations, each kernel emphasizes different critical combinations of coupling weights and time delay, with the weak Gamma kernel restricting oscillations to a tighter locus of coupling strengths, and to a limited range of time delays. We finally illustrate the general analytical results with simulations for two particular applications: generation of beta-rhythms in the basal ganglia, and alpha oscillations in the prefrontal-limbic system. • We study long-term dynamics in coupled Wilson–Cowan systems with distributed delays. • Our analysis focuses on understanding the system's transitions in and out of stable oscillations. • We study how these transitions change under different connectivity schemes and strengths. • We explore the effects of different delay distributions and average delay values. • The analysis is illustrated with simulations in two different functional brain networks. [ABSTRACT FROM AUTHOR]

This study aims to present a complete Hopf bifurcation analysis of a model describing the relationship between prey and predator. A ratio-dependent reaction–diffusion system with two discrete time delays operating under Neumann boundary conditions governs the model that represents this competition. The bifurcation parameter for the analysis is a delay parameter that reflects the amount of time needed for the predator to be able to hunt. Bilazeroğlu and Merdan's algorithm (Bilazeroğlu et al., 2021), which is developed by using the center manifold theorem and normal form theory, is used to establish the existence of Hopf bifurcations and also the stability of the bifurcating periodic solutions. The same procedure is used to illustrate some specific bifurcation properties, such as direction, stability, and period. Furthermore, by examining a model with constant coefficients, we also analyze how diffusion and the amount of time needed for prey to mature impact the model's dynamics. To support the obtained analytical results, we also run some numerical simulations. The results indicate that the dynamic of the mathematical model is significantly influenced by diffusion, the amount of time needed for the predator to gain the capacity to hunt, and the amount of time required for prey to reach maturity that the predator can hunt. [ABSTRACT FROM AUTHOR]

Highlights • A network with wireless access links with a delay differential equation (DDE) model and input delay is considered. • Integral sliding mode control is assumed as the nonlinear robust active queue management in the network. • Hopf bifurcation analysis is investigated with the transmission delay considered as the Hopf bifurcation parameter. • The direction and the stability of the bifurcation periodic solutions are determined. • Simulation results are provided to illustrate the system behavior with different parameters. Abstract Transmission control protocol (TCP), which is usually implemented in the transport layer of a communication network, can detect the congestion after its occurrence. But congestion control can be more complicated in networks with wireless access links due to time-varying fading and packet error rate (PER). Thus, it is essential to use robust active queue management (RAQM) to prevent congestion with certain tolerance against serious disturbance of the wireless environment. As the stability of a TCP/RAQM is tremendously related to the transmission delay, this paper focuses on Hopf bifurcation analysis of a network governed by nonlinear delayed differential equations and an integral sliding mode control as a nonlinear RAQM. The transmission delay is assumed as the bifurcation parameter. It is shown that if the delay passes through a critical value, the closed loop system would not remain stable and the Hopf bifurcation occurs. Moreover, based on the normal form theory and the center manifold theorem, the direction and the stability of the bifurcation periodic solutions are determined. Finally, simulation results are given to confirm the analytical achievements and to show the effectiveness of the procedure compared to some recent protocols. [ABSTRACT FROM AUTHOR]

We study the dynamics of a damped harmonic oscillator in the presence of a retarded potential with state-dependent time-delayed feedback. In the limit of small time-delays, we show that the oscillator is equivalent to a Liénard system. This allows us to analytically predict the value of the first Hopf bifurcation, unleashing a self-oscillatory motion. We compute bifurcation diagrams for several model parameter values and analyze multistable domains in detail. Using the Lyapunov energy function, two well-resolved energy levels represented by two coexisting stable limit cycles are discerned. Further exploration of the parameter space reveals the existence of a superposition limit cycle, encompassing two degenerate coexisting limit cycles at the fundamental energy level. When the system is driven very far from equilibrium, a multiscale strange attractor displaying intrinsic and robust intermittency is uncovered. • A retarded harmonic oscillator with state-dependent delay is studied. • The oscillator is related to a Liénard system proving that the zero point fluctuations arise via Hopf bifurcation. • Multistability domains with two well-resolved quantized orbits are described by means of bifurcation diagrams and the Lyapunov energy function. • When the retarded potential is comparable to the external well, a superposition limit cycle encompassing two degenerate limit cycles appears. • For high time-delays we also find a multiscale strange attractor displaying robust intermittency. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a diffusive predator–prey system with Crowley–Martin functional response and delay subject to Neumann boundary condition. More precisely, we study the stability and Turing instability of positive equilibrium for non-delay system, instability and Hopf bifurcation induced by time delay for delay system. In addition, by the theory of normal form and center manifold method, we derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution. [ABSTRACT FROM AUTHOR]

In this study, we are concerned with the dynamics of a new established fractional-order delayed zooplankton–phytoplankton system. The existence and uniqueness of the solution are proved via Banach fixed point theorem. Non-negativeness of the solution is studied by mathematical inequality technique. The boundedness of the solution is analyzed by virtue of constructing an appropriate function. A novel delay-independent sufficient condition ensuring the stability and the onset of Hopf bifurcation for the established fractional-order delayed zooplankton–phytoplankton system is derived by means of Laplace transform, stability criterion and bifurcation knowledge of fractional-order differential equation. The global stability condition for the involved fractional-order delayed zooplankton–phytoplankton system is built by using a suitable positive definite function. Taking advantage of hybrid control tactics, we effectively control the time of occurrence of Hopf bifurcation for the established fractional-order delayed zooplankton–phytoplankton system. The study manifests that delay plays a vital role in controlling the stability and the time of occurrence of Hopf bifurcation for the involved fractional-order delayed zooplankton–phytoplankton system and the fractional-order controlled zooplankton–phytoplankton system involving delays. To verify the correctness of established chief results, computer simulation figures are distinctly displayed. The derived conclusions of this research are entirely new and possess potential theoretical value in preserving the balance of biological population. Up to now, there are few publications on detailed and comprehensive dynamic analysis on fractional-order delayed zooplankton–phytoplankton system via various exploration ways. [ABSTRACT FROM AUTHOR]

In this paper, the stochastic time-delay mathematical model of the feedback system of glucose-insulin endocrine regulation with noise effects is proposed. The hopf bifurcation of the system is obtained with the delay as the parameter. Considering the influence of time delay and noise, the random bifurcation of the system is obtained, and through numerical analysis, the changement figure of the numerical solution of the model before and after the occurrence of random bifurcation is obtained. The results show that a high-noise and high-sugar diet can lead to the occurrence of random bifurcations, thereby destabilizing the system. In addition, we found the possible bursting phenomenon in the model, and through the analysis, it is concluded that bursting phenomenon is more likely to occur in the noise environment than in the no-noise environment. • Noise and erratic diet cause glucose-insulin system to shift from cycle to chaos. • Noise and diet can cause bursts in the system. • The presence or absence of noise has a great influence on the bursting phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, we delve into the dynamics of a tree-insect model that considers size structure in trees, aiming to explore the effect of the maturity size at which the saplings are just coming into adulthood on the insect population in forests. By making full use of mathematical techniques and an ingenious variable transformation, we convert the model of nonlinear ordinary differential equation coupled to a partial differential equation into a model of delay differential equations with fixed delay. Combining the method of fluctuations and bifurcation theory, a relatively complete qualitative analysis of the model is conducted, including uniqueness, nonnegativity and boundedness of the solutions, global and local asymptotic stability of the equilibria, uniform persistence of the model, stability switches occurring at the positive equilibrium. In addition, numerical simulations indicate that the maturity size can lead to richer dynamics, such as regime shifts, bistability and long transients. • A size-structured model describing insect destruction in forests is investigated. • The conditions for the stability, Hopf bifurcation and persistence are obtained. • Rich dynamics such as regime shifts, bistability and long transients are observed. • The maturity size of tree significantly affects the dynamics of insect population. [ABSTRACT FROM AUTHOR]

Among many other factors that affect the preventive interventions to any infectious disease, not reporting timely in a hospital is also one of the catastrophic behavior of human beings in any society. Similarly, masses who do not report make it difficult for healthcare researchers to measure the actual data and develop prevention strategies, accordingly. Therefore, there is a critical need to structure a potential epidemic model with the unreported class of individuals. This novel idea is deliberated in this paper to study the profiles of the epidemic model of virulent diseases due to the individuals that report timely and those who don't report in hospitals for any reason. Mathematically, a system of seven equations is taken into consideration, which describes the susceptible individuals, exposed, people who do not report to the hospital and those who report to the hospital, and individuals who are quarantined, infected, and recovered. So, with the consideration of new compartments, the conventional SIR epidemic model expands to SEU R R P QIR. The innovative design is made more realistic by utilizing proportional fractional-order differential equations with time delay. A special simplified expansion of this derivative reduces its computational cost and produces the results with fractional index, which helps to predict each fractional change. In addition, an optimal control methodology is also carried out to analyze the effectiveness of the awareness campaign in shifting the unreported individuals to the reported class, with optimal cost function for the unreported cases. Discussions are supported through the very recent deadly pandemic as an example to conclude the practical advantage of the model. The sensitivity analysis of basic reproduction numbers based on effective awareness campaigns is also the part of this study, which infers public awareness campaigns may be devised to motivate and guide such individuals to approach any healthcare center or a hospital. • Compartments for unreported and reported cases in the epidemic model. • Analysis through system of fractional order differential equations with time lags. • Realistic profiles of disease dynamics at different states. • Awareness campaign as optimal control approach to restrain unreported class. [ABSTRACT FROM AUTHOR]

• We considered an opinion model of threshold-type delay differential equations. • We proved that the model with either a transcritical forward or a backward bifurcation, can exhibit Hopf bifurcation. • We showed that the obtained Hopf bifurcation appears only on upper bifurcated branches of the equilibria. • We performed numerical simulations to illustrate and support our theoretical results. In a recent paper [R. Qesmi, Dynamics of an opinion model with threshold-type delay, Chaos, Solitons & Fractals 98 (2020). (https://doi.org/10.1016/j.chaos.2020.110379)], we proposed a mathematical model of threshold-type delay differential equations describing the relationship between two subpopulations with opposite opinions and the opinion spread dynamics. The study there showed the possibility of a transcritical forward and backward bifurcations of positive equilibria. In the present paper, we show that the opinion model undergoes a Hopf bifurcation through which one of the bifurcation branches loses the stability and periodic solutions appear. One of the important consequences of the obtained dynamics is that the consensus of the both opinions could be lost by maintaining the balance between the time taken for an individual to become convinced of the outsider opinion, which need be short, and the number of individuals converted to the local opinion which need be low. Finally, we provide numerical simulations to illustrate and support our theoretical results. [ABSTRACT FROM AUTHOR]

• The first contribution of this research is Hopf bifurcation analysis of a class of system of reaction-diffusion equations involving two discrete time delays under Newmann boundary conditions. The research presents the condition on parameters at which a periodic solution arises locally which is important for the control of a system governed by this class of equations. • The second contribution is determining some properties of existing periodic solutions, including its stability and period. These properties are important to understand the behavior of solutions of a system belonging to this class of equations. • The third contribution is the derivation of an algorithm that will be used for the existence and direction analysis of Hopf bifurcation of a system of equations which lie in that class by simply utilizing the cooefficients in the characteristic equation of its linearized form. This algorithm reduces the complex and time consuming computations to several steps through a systematic schema. We analyze Hopf bifurcation and its properties of a class of system of reaction-diffusion equations involving two discrete time delays. First, we discuss the existence of periodic solutions of this class under Neumann boundary conditions, and determine the required conditions on parameters of the system at which Hopf bifurcation arises near equilibrium point. Bifurcation analysis is carried out by choosing one of the delay parameter as a bifurcation parameter and fixing the other in its stability interval. Second, some properties of periodic solutions such as direction of Hopf bifurcation and stability of bifurcating periodic solution are studied through the normal form theory and the center manifold reduction for functional partial differential equations. Moreover, an algorithm is developed in order to determine the existence of Hopf bifurcation (and its properties) of variety of system of reaction-diffusion equations that lie in the same class. The benefit of this algorithm is that it puts a very complex and long computations of existence of Hopf bifurcation for each equation in that class into a systematic schema. In other words, this algorithm consists of the conditions and formulae that are useful for completing the existence analysis of Hopf bifurcation by only using coefficients in the characteristic equation of the linearized system. Similarly, it is also useful for determining the direction analysis of the Hopf bifurcation merely by using the coefficients of the second degree Taylor polynomials of functions in the right hand side of the system. Finally, the existence of Hopf bifurcation for three different problems whose governing equations stay in that class is given by utilizing the algorithm derived, and thus the feasibility of the algorithm is presented. [ABSTRACT FROM AUTHOR]

• We study the impact of two discrete-time delays on the basic Goodwin growth cycle model. • We study a delayed differential equation system with discrete-time delays and delay-dependent coefficients depending only on one of the time delay. • Having chosen the time delays as bifurcation parameters, we study the stability-switching properties of the transcendental characteristic equation. • We analyze the direction and stability of Hopf bifurcation by means of the first Lyapunov coefficient. This paper deals with the impact of two discrete-time delays on the basic Goodwin growth cycle model. The former concerns the existence of a finite time delay for building capital goods as suggested by Kalecki. The latter pertains to the wage lag hypothesis. This is because, taking the current change of the employment rate into account, workers and capitalists bargain new wage periodically. There are no examples in the literature on the Goodwin model of the use of both those lags in order to explore the GDP dynamics. From the analytical point-of-view, what we obtain is a delayed differential equation system with discrete-time delays and delay-dependent coefficients depending only on one of the time delays. Having chosen the time delays as bifurcation parameters, we study the stability-switching properties of the transcendental characteristic equation resulting from the stability analysis and the direction of the Hopf bifurcations. Although the system with no lag displays a stable focus, the introduction of the two lags preserves the stable solution only for particular combinations of parameters and length of the lags. In any other case, instability prevails and regular cycles or chaotic fluctuations emerge. Finally, we provide the analytical results with the necessary economic interpretations. [ABSTRACT FROM AUTHOR]