Equilibrium and Disequilibrium Dynamics in Cobweb Models with Time Delays

This work provides further contribution to the linear properties in a continuous time linear model. We deal with linear sufficiency and linear completeness properties, together with the linear admissibility property. These concepts were originally introduced and characterized in a discrete time context and subsequently were extended by the authors of the present paper to a continuous time linear model. Our objective is to study in depth these properties showing a general unified context where the classical linear model appears as a particular case.

T he consequences of the decisions and policies taken for the reconstruction of Afghanistan during the past decade, showed that there should be paid more attention to the monetary policies. There is also a question that whether monetary policies have the potential to affect the production and inflation in Afghanistan or not. The aim of this paper is to explore this effect by designing a new Keynesian dynamic stochastic general equilibrium (DSGE) model. The DSGE model used in this paper is taken from the new Keynesian theoretical foundations. It has been addressed to the role of households, firms, the monetary policy of Da Afghanistan Bank, as well as aids of international institutions as shocks that have been able to allocate a particular role in the Afghanistan's economy to itself, with regard to nominal rigidities and monopoly competition in this model. The results show the impulse response functions of production is negative to the external aid shock, financial shock, exchange rate shock, technology shock and external shock and is positive to the monetary shock and cost-shock. An impulse response function of inflation is positive for all shocks except for technology shock.

In this paper, we study a class of markets, among which we can mention agricultural and energy markets, characterized by seasonality, i.e., in which demand and/or supply conditions cyclically alternate with a precise and known periodicity. We propose a new theoretical framework based on a cobweb model with adaptive expectations, accordingly modified to be consistent with market’s seasonality. The model, consisting in a second-order non-autonomous difference equation, is investigated with the aim of understanding how the periodical nature of the market together with the agents’ expectation formation mechanism affects the resulting dynamics. We analytically prove the emergence of dynamical scenarios that are missing in the classic cobweb model for non-seasonal markets, such as quasi-periodic dynamics and an ambiguous role on stability of the expectation weight. Finally, we discuss their economic rationale with the help of numerical simulations. In such a peculiar economic framework, agents’ learning plays a key role to explain the dynamical properties of economic observables.

This paper studied the stability and Hopf bifurcation of a type of protein synthesis system with time delay and negative feedback. Firstly, it is proved theoretically that the time delay, nonlinearity in the protein production and the cooperativity in the negative feedback are key factors to generate circadian oscillation; Taking time delay as a parameter, we obtained the critical value of the time delay that Hopf bifurcation generates. Secondly, based on the center manifold and normal form theorem, we derived the formulas for determining the stability of bifurcating periodic solutions and the supercritical or subcritical Hopf bifurcation. Finally, the matlab program is used to simulate the results.

Circadian rhythm disruptions are linked to increased cancer risk and unfavorable prognosis in patients with cancer, highlighting the critical role of the interplay between the circadian rhythm factor Per2 and the tumor suppressor p53. This brief presents, for the first time, a mathematical model to capture the dynamics of the p53-Per2 network in DNA-damaged cells. The model accurately describes the different stages of the process from unstressed cells to cellular repair and finally to apoptosis as the degree of DNA damage increases. Furthermore, it is found that increasing the inhibition of Per2 by p53 leads to the phase advance of Per2 oscillations, whereas by modulating the inhibition of Mdm2 by Per2, an independent amplitude modulation of active p53 can be achieved, with the range of modulation increasing with the strength of the inhibition. Moreover, the effects of time delays inherent in the transcription, translation, and nuclear translocation of Per2 on the circadian rhythm of DNA-damaged cells are quantitatively investigated by theoretical analyses. It is found that time delays can induce stable oscillations through a supercritical Hopf bifurcation, thereby maintaining the circadian function of DNA-damaged cells and enhancing their DNA-damage repair capacity. This study proposes new insights into cancer prevention and treatment strategies.

Time-delay-induced synchronous behaviors and synchronization transitions have been widely investigated for coupled neurons, and they play important roles for physiological functions. In the present study, time-delay-induced synchronized subthreshold oscillations were simulated, and the bifurcations underlying the synchronized behaviors were identified for a pair of coupled FitzHugh–Nagumo neurons. Multiple transitions between in-phase and anti-phase synchronizations induced by the time delay were simulated for the inhibitory and excitatory couplings. Subcritical or supercritical Hopf bifurcations and the stability of the Hopf-bifurcating periodic subthreshold oscillations were acquired using center manifold reduction and normal form theory. The in-phase or anti-phase synchronizations of the stable periodic subthreshold oscillations, which appear for multiple values of the time delay, were interpreted with the related eigenspace. The distributions of the different dynamical behaviors, including the synchronizations and bifurcations in the two-parameter plane of the time delay and coupling strength, were acquired for both types of synapses, and the different roles of the inhibitory and excitatory couplings on the synchronization transitions were compared.

In this paper, we present an algorithm for deriving the normal forms of Bautin bifurcations in reaction-diffusion systems with time delays and Neumann boundary conditions. On the center manifold near a Bautin bifurcation, the first and second Lyapunov coefficients are calculated explicitly, which completely determine the dynamical behavior near the bifurcation point. As an example, the Segel-Jackson predator-prey model is studied. Near the Bautin bifurcation we find the existence of fold bifurcation of periodic orbits, as well as subcritical and supercritical Hopf bifurcations. Both theoretical and numerical results indicate that solutions with small (large) initial conditions converge to stable periodic orbits (diverge to infinity).

With rapid development of Internet, network security issues become increasingly serious. Temporary patches have been put on the infectious hosts, which may lose efficacy on occasions. This leads to a time delay when vaccinated hosts change to susceptible hosts. On the other hand, the worm infection is usually a nonlinear process. Considering the actual situation, a variable infection rate is introduced to describe the spread process of worms. According to above aspects, we propose a time-delayed worm propagation model with variable infection rate. Then the existence condition and the stability of the positive equilibrium are derived. Due to the existence of time delay, the worm propagation system may be unstable and out of control. Moreover, the threshold τ0 of Hopf bifurcation is obtained. The worm propagation system is stable if time delay is less than τ0. When time delay is over τ0, the system will be unstable. In addition, numerical experiments have been performed, which can match the conclusions we deduce. The numerical experiments also show that there exists a threshold in the parameter a, which implies that we should choose appropriate infection rate β(t) to constrain worm prevalence. Finally, simulation experiments are carried out to prove the validity of our conclusions.

Tracking control for multiple-integrator systems is regarded as a fundamental problem associated with nonlinear dynamic systems in the physical and mathematical sciences, with many applications in engineering fields. In this paper, we adopt the Zhang neural network method to solve this nonlinear dynamic problem. In addition, in order to adapt to the requirements of real-world hardware implementations with higher-order precision for this problem, the multiple-order derivatives in the Zhang neural network method are estimated using backward finite-divided difference formulas with quadratic-order precision, thus producing time delays. As such, we name the proposed method the Zhang neural network method with time delay. Moreover, we present five theorems to describe the convergence property of the Zhang neural network method without time delay and the quadratic-order error pattern of the Zhang neural network method with time delay derived from the backward finite-divided difference formulas with quadratic-order precision, which specifically demonstrate the effect of the time delay. Finally, tracking controllers with quadratic-order precision for multiple-integrator systems are constructed using the Zhang neural network method with time delay, and two numerical experiments are presented to substantiate the theoretical results for the Zhang neural network methods with and without time delay.

In this paper, we introduce and deal with the delayed nutrient-microorganism model with a random network structure. By employing time delay τ as the main critical value of the Hopf bifurcation, we investigate the direction of the Hopf bifurcation of such a random network nutrient-microorganism model. Noticing that the results of the direction of the Hopf bifurcation in a random network model are rare, we thus try to use the method of multiple time scales (MTS) to derive amplitude equation and determine the direction of the Hopf bifurcation. It is showed that the delayed random network nutrient-microorganism model can exhibit a supercritical or subcritical Hopf bifurcation. Numerical experiments are performed to verify the validity of the theoretical analysis.

Application of nonlinear mapping theory to commodity price fluctuatuions

Persistent exchange rate fluctuations and inflationary pressures pose significant economic challenges in Nigeria, affecting businesses, households, and overall economic stability. The depreciation of the naira has consistently driven up consumer prices, eroding purchasing power and increasing production costs, particularly for an import-dependent economy like Nigeria. Despite numerous policy interventions, inflation remains a major concern, exacerbated by external shocks and structural weaknesses. Given these challenges, understanding the precise relationship between exchange rate movements and inflation is critical for designing effective monetary and fiscal policies. This study investigates the impact of exchange rate fluctuations on inflation in Nigeria, utilizing the Consumer Price Index (CPI) and the Implicit Price Deflator (IPD) as key inflation indicators. Using a 24-year time series dataset (2000–2024) and the Autoregressive Distributed Lag (ARDL) model, the study establishes a significant long-run relationship between exchange rate depreciation and inflationary pressures. Findings reveal that naira depreciation leads to rising consumer prices, particularly affecting imported goods and overall cost of living. Additionally, money supply and monetary policy rate significantly influence inflation, even though with a slower adjustment process due to structural challenges in Nigeria’s economy. The study also highlights the impact of major macroeconomic events, including the 2008 global financial crisis, the 2016 exchange rate deregulation, the COVID-19 pandemic, and the 2023 exchange rate unification and fuel subsidy removal, on inflation dynamics. The results underscore the need for effective exchange rate management and monetary policies to stabilize inflation and support economic growth in Nigeria.

Deferred Taxation Under Default Risk

For a wide class of host-parasitoid models, a reduction to Arnold's normal form can be carried out in an explicit way. In the case of Hopf bifurcation, the shape and size of the elliptic limit curve can be derived in terms of the parameters of the model. Some models have a rich bifurcation behaviour with both supercritical and subcritical Hopf bifurcation, and with a transition zone in the parameter plane for which there exists a pair of limit curves, one stable and one unstable. The theory is confirmed and illustrated by numerical experiments.

On the spectral instability and bifurcation of the 2D-quasi-geostrophic potential vorticity equation with a generalized Kolmogorov forcing