Critical Value Of Delay Research Articles

Bifurcation and chaos analysis of a fractional-order delay financial risk system using dynamic system approach and persistent homology

A comprehensive theoretical and numerical analysis of the dynamical features of a fractional-order delay financial risk system(FDRS) is presented in this paper. Applying the linearization method and Laplace transform, the critical value of delay when Hopf bifurcation first appears near the equilibrium is firstly derived in an explicit formula. Comparison simulations clarify the reasonableness of fractional-order derivative and delay in describing the financial risk management processes. Then we employ persistent homology and six topological indicators to reveal the geometric and topological structures of FDRS in delay interval. Persistence barcodes, diagrams, and landscapes are utilized for visualizing the simplicial complex’s information. The approximate values of delay when FDRS undergoes different periodic oscillations and even chaos are determined. The existence of periodic windows within the chaotic interval is correctly decided. The results of this paper contribute to capturing intricate information of underlying financial activities and detecting the critical transition of FDRS, which has promising and reliable implications for a deeper comprehension of complex behaviors in financial markets.

Bifurcation Exploration and Controller Design in a Fractional Oxygen–Plankton Model with Delay

Fractional-order differential equations have been proved to have great practical application value in characterizing the dynamical peculiarity in biology. In this article, relying on earlier work, we formulate a new fractional oxygen–plankton model with delay. First of all, the features of the solutions of the fractional delayed oxygen–plankton model are explored. The judgment rules on non-negativeness, existence and uniqueness and the boundedness of the solution are established. Subsequently, the generation of bifurcation and stability of the model are dealt with. Delay-independent parameter criteria on bifurcation and stability are presented. Thirdly, a hybrid controller and an extended hybrid controller are designed to control the time of onset of bifurcation and stability domain of this model. The critical delay value is provided to display the bifurcation point. Last, software experiments are offered to support the acquired key outcomes. The established outcomes of this article are perfectly innovative and provide tremendous theoretical significance in balancing the oxygen density and the phytoplankton density in biology.

Echo threshold measurement based on headphone presentation

Echo threshold(ET), as the upper threshold of the precedence effect, refers to the critical delay value when the listener transitions from perceiving a single fused sound image to two clear and separated sound images in the presence of two relatively delayed sounds. This study investigates the ET measurement using two kinds of headphone-based virtual sound source presentation methods, i.e., interaural time difference (ITD) lateralization and virtual auditory synthesis (VAS). The “lead-lag” paradigm was adopted in the experiment and an adaptive up-down procedure was used to measure ETs of four stimuli, including noise burst, speech, music and pink noise. The results demonstrate that ET is related to the type of signal and the presentation method. Long-duration signals (i.e., speech, music and pink noise) have greater ETs than short-duration signal(noise burst), which is due to the fact that the temporal overlap between lead and lag sound of long-duration signal would suppress the identification of the lag. ETs of ITD-based presentation were significantly greater than that of VAS-based method for long-duration signal, which indicate that different spatial cues produced by different presentation methods would influence the ET of long-duration signal.

Hopf Bifurcation in a Modified Leslie-Gower Two Preys One Predator Model and Holling Type II Functional Response with Harvesting and Time-Delay

In this paper, a modified Leslie-Gower two preys one predator model and Holling type II functional response with harvesting and time-delay were discussed. Model analysis is carried out by determining fixed points, then analyzing the stability of the fixed points and discussing the existence of the Hopf bifurcation. In some conditions that occur in nature indicate the occurrence of hunting of prey and predator species by humans. Therefore, this model is modified by adding the assumption that prey and predators are being harvested. Another modification given to the model is the use of time delays.The delay time term is for taking into account the case that the members of the predator species need time from birth to predation for being active predators. The first case is a model without time delay, it is obtained that 3 fixed points are unstable and 7 fixed points are stable. One of them is the interior fixed point tested with the Routh-Hurwitz criteria. The second case is a model with a delay time, the critical delay value is obained. Hopf bifurcation occurs when the delay time value is equal to the critical delay value and also fulfills the transversality condition. Observations on the model simulation are carried out by varying the value of the delay time. When the Hopf bifurcation occurs, the graph on the solution plane shows a constant oscillatory movement. If the value of the delay time given is less than the critical value of the delay, the controlled system solution goes to a balanced state. Then when the delay time value is greater than the critical delay value, the system solution continues to fluctuate causing an unstable system condition.

An image encryption algorithm based on the double time-delay Lorenz system.

The traditional image encryption technology has the disadvantages of low encryption efficiency and low security. According to the characteristics of image information, an image encryption algorithm based on double time-delay chaos is proposed by combining the delay chaotic system with traditional encryption technology. Because of the infinite dimension and complex dynamic behavior of the delayed chaotic system, it is difficult to be simulated by AI technology. Furthermore time delay and time delay position have also become elements to be considered in the key space. The proposed encryption algorithm has good quality. The stability and the existence condition of Hopf bifurcation of Lorenz system with double delay at the equilibrium point are studied by nonlinear dynamics theory, and the critical delay value of Hopf bifurcation is obtained. The system intercepts the pseudo-random sequence in chaotic state and encrypts the image by means of scrambling operation and diffusion operation. The algorithm is simulated and analyzed from key space size, key sensitivity, plaintext image sensitivity and plaintext histogram. The results show that the algorithm can produce satisfactory scrambling effect and can effectively encrypt and decrypt images without distortion. Moreover, the scheme is not only robust to statistical attacks, selective plaintext attacks and noise, but also has high stability.

Dynamic Analysis of a New Financial System with Diffusion Effect and Two Delays

The dynamic behavior of a new financial system with diffusion effect and two delays is studied. The local stability of equilibrium is studied by analyzing the corresponding characteristic equation and developing the crossing curve method. Hopf bifurcation curves are obtained by three procedures and the critical value of delay of Hopf bifurcation is obtained. By using the central manifold theorem and the normal form theory of partial differential equations, the normal form of Hopf bifurcation with two delays is derived, and the bifurcation direction and the stability of periodic solution are determined. The numerical simulations not only verify these theoretical results, but also show that double Hopf bifurcation may occur and the two delays play a key role in chaos suppression. This study has important theoretical value of innovation and practical value in the macro-economic system.

Bifurkasi Hopf Pada Model Siklus Bisnis Investment Saving-Liquidity Money dengan Waktu Tunda

The economy is not always good, meaning it experiences ups and downs. This situation requires forecasting. One of which is the Investment Saving-Liquidity Money business cycle model, which was first introduced by Kalecki in 1935 with the assumption that profits will be saved to be used as initial investment capital, thus causing a delay in the investment process (delay time). The addition of time delay in the system of differential equations causes a change in stability at the equilibrium point so that bifurcation occurs. Another modification given to the model is the use of time delays. The model analyzed in this study is a non-linear IS-LM business cycle model with time delay. Delay time is required for the gross product, capital, and interest rate to be used as an investment. This model was observed in two cases. First Case is a model without time delay. The second case is a model with a time delay. In this case, the critical value of delay is obtained. Hopf bifurcation occurs when the value of delay time is equal to the critical value of delay and satisfies the transversality condition. Observations on the model simulation are carried out by varying the delay time value. When the Hopf bifurcation occurs, the graph on the solution plane shows a constant oscillatory movement. If the value of delay time given is less than the critical value of delay, the controlled system solution goes to a balanced condition. Then when the delay time value is greater than the critical value of delay, the system solution continues to fluctuate, causing an unstable system condition.

Stability and bifurcation analysis in a diffusive predator–prey model with delay and spatial average

In this paper, we study the effect of spatial average and time delay on the dynamics of a diffusive predator–prey model under the Neumann boundary condition. Compared to the model without spatial average, the delay‐induced Hopf bifurcation at the first critical value of delay is nonhomogeneous due to the joint effects of spatial average and delay, and spatiotemporal patterns arise. Also, we show that the spatially homogeneous and nonhomogeneous periodic patterns may switch for different bifurcation parameter values. Moreover, a double Hopf bifurcation occurs at the intersection point of the homogeneous and nonhomogeneous Hopf bifurcation curves, and spatially nonhomogeneous quasi‐periodic patterns can be observed via numerical simulations near the double Hopf bifurcation point. The normal form algorithm for the spatially nonhomogeneous/homogeneous Hopf bifurcation is derived for a general reaction‐diffusion system with spatial average and delay.

Stability and Hopf bifurcation of a love model with two delays

We analyze the stability and Hopf bifurcation of a proposed nonlinear love model with two delays. The nonlinearity stems from a logistic term injected to meticulously depict a couple’s cautious love affairs. The analysis is based on the linearization theory and starts with a single-delay case assuming that one of the two characters is always impulsive. The critical delay values and existence conditions for the Hopf bifurcation are obtained. Subsequently, we extend to the more practical but also more involved dual-delay case where the delays affect the emotions of both individuals. Accordingly, a non-trivial resultant-based analysis framework is proposed, and the corresponding Hopf bifurcation is detected intuitively and exhaustively by an obtained stability map in the delay domain. The results explain an interesting and well-known feature in our daily life, i.e., proper time delays stabilize the love dynamics. Furthermore, the spectral analysis based on a numerical case reveals another common emotional experience, that proper delays not only lead to a smoother transition process but also yield a shorter settling time for the love convergence. The numerical bifurcation diagram and computational cost are also considered to demonstrate the effectiveness and efficiency of the presented analysis framework. Finally, some additional suggestions are provided for the studied couple to earn a sweet love relationship.

Stability Switches and Chaos in a Diffusive Toxic Phytoplankton–Zooplankton Model with Delay

To explore the effect of toxins produced by phytoplankton in algal blooms, a diffusive toxic phytoplankton–zooplankton system with delay is studied through detailed bifurcation analysis. The local stability of the positive equilibrium is studied by analyzing the characteristic equation. The critical values of delay at which Hopf bifurcations occur are obtained and ordered, and it is proved that the delay can induce stability switches, which means that the delay has a significant impact on both the formation and the termination of algal blooms. During the process of stability switching, double Hopf bifurcation arises. Taking the rate of toxin liberation and time delay as the bifurcation parameters, the normal form for double Hopf bifurcation is derived, from which the classification for dynamics and bifurcation sets near double Hopf bifurcation point is obtained. It is proved that the system has complex dynamics, such as periodic oscillations, quasi-periodic oscillations on two- or three-torus and even chaos. Numerical simulations are presented to support the theoretical results.

Multidelay Differential Equations: A Taylor Expansion Approach

It is already well-understood that many delay differential equations with only a single constant delay exhibit a change in stability according to the value of the delay in relation to a critical delay value. Finding a formula for the critical delay is important to understanding the dynamics of delayed systems and is often simple to obtain when the system only has a single constant delay. However, if we consider a system with multiple constant delays, there is no known way to obtain such a formula that determines for what values of the delays a change in stability occurs. In this paper, we present some single-delay approximations to a multidelay system obtained via a Taylor expansion as well as formulas for their critical delays which are used to approximate where the change in stability occurs in the multidelay system. We determine when our approximations perform well and we give extra analytical and numerical attention to the two-delay and three-delay settings.

Vladimir Kharitonov: Achievements in research

AbstractA survey of the main research results of Professor Kharitonov is provided. The following topics are considered: stability analysis of interval polynomials, quasi‐polynomials, and multivariate polynomials; Kalman's hypothesis on the generalization of the Lyapunov matrix equation; critical delay values, complete type functionals and prediction‐based control for time‐delay systems. The fundamentals of Kharitonov's work and the influence of his results on other researchers are discussed.

Analysis of Mangrove Forest Resource Depletion Models due to The Opening of Fish Pond Land with Time Delay

Mangrove forest is an ecosystem with many resources and high biological diversity that contain many species of animal, such as proboscis monkeys. One of the problems that keep happening on mangrove forest is the opening of fish pond. Opening of fish pond can damage mangrove resource by the degradation of the mangrove habitat. This study aims to analyze the depletion stability model of mangrove forest resources with time delay, and to show effect of time delay and the presence of change in the equilibrium point (Hopf bifurcation) on depletion stability model of mangrove forest resources due to fish pond openings. The depletion stability model of mangrove forest resources can be modeled as a system of nonlinear differential equations. In the numerical simulation results, critical value of delay (τ0) is 7.05 and the transverse conditions are not fulfilled. This mean there is no change in stability and the delay time parameter (τ) does not affect the stability of the system. System will remain stable when the mangrove forests, proboscis monkeys, and fish ponds are in equilibrium. In other words, mangrove forests, proboscis monkeys, and fish ponds can coexist even though time delays, and the analysis using the Hopf bifurcation cannot be carried out.

Bifurcation analysis in a diffusive Logistic population model with two delayed density-dependent feedback terms

The present paper is concerned with a diffusive population model of Logistic type with an instantaneous density-dependent term and two delayed density-dependent terms and subject to the zero-Dirichlet boundary condition. By regarding the delay as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation for the sufficiently small positive steady state solution are shown. It is found that under the suitable condition, the positive steady state solution of the model will become ultimately unstable after a single stability switch (or change) at a certain critical value of delay through a Hopf bifurcation. However, under the other condition, the positive steady state solution of the model will become ultimately unstable after multiple stability switches at some certain critical values of delay through Hopf bifurcations. In addition, the direction of the above Hopf bifurcations and the stability of the bifurcating periodic solutions are analyzed by means of the center manifold theory and normal form method for partial functional differential equations. Finally, in order to illustrate the correction of the obtained theoretical results, some numerical simulations are also carried out.

Machine Learning Assisted Chimera and Solitary States in Networks

Chimera and Solitary states have captivated scientists and engineers due to their peculiar dynamical states corresponding to co-existence of coherent and incoherent dynamical evolution in coupled units in various natural and artificial systems. It has been further demonstrated that such states can be engineered in systems of coupled oscillators by suitable implementation of communication delays. Here, using supervised machine learning, we predict (a) the precise value of delay which is sufficient for engineering chimera and solitary states for a given set of system's parameters, as well as (b) the intensity of incoherence for such engineered states. Ergo, using few initial data points we generate a machine learning model which can then create a more refined phase plot as well as by including new parameter values. We demonstrate our results for two different examples consisting of single layer and multi layer networks. First, the chimera states (solitary states) are engineered by establishing delays in the neighboring links of a node (the interlayer links) in a 2-D lattice (multiplex network) of oscillators. Then, different machine learning classifiers, K-nearest neighbors (KNN), support vector machine (SVM) and multi-layer perceptron neural network (MLP-NN) are employed by feeding the data obtained from the network models. Once a machine learning model is trained using the limited amount of data, it predicts the precise value of critical delay as well as the intensity of incoherence for a given unknown systems parameters values. Testing accuracy, sensitivity, and specificity analysis reveal that MLP-NN classifier is better suited than Knn or SVM classifier for the predictions of parameters values for engineered chimera and solitary states. The technique provides an easy methodology to predict critical delay values as well as intensity of incoherence for that delay value for designing an experimental setup to create solitary and chimera states.

Hopf bifurcation in a delayed predator-prey system with asymmetric functional response and additional food

<abstract><p>In this paper, a delayed predator-prey system with additional food and asymmetric functional response is investigated. We discuss the local stability of equilibria and the existence of local Hopf bifurcation under the influence of the time delay. By using the normal form theory and center manifold theorem, the explicit formulas which determine the properties of bifurcating periodic solutions are obtained. Further, we prove that global periodic solutions exist after the second critical value of delay via Wu's theory. Finally, the correctness of the previous theoretical analysis is demonstrated by some numerical cases.</p></abstract>

On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect

For a stage-structured population model in periodic discrete habitat, with periodic initial values it reduces to a system of two differential equations with time delay. Assuming the birth rate is of unimodal type, we obtain the influence of time delay on the local and global dynamics. In particular, large delay leads to population vanishing. As delay decreases, we found three critical values of delay for the emergence of different dynamics, by appealing to a combination of monotone dynamical system theory, Hopf bifurcation theory and the fluctuation method. Numerical simulations are also performed to illustrate the results.

Multiple stability switches and Hopf bifurcations induced by the delay in a Lengyel-Epstein chemical reaction system

This paper examines the dynamical analysis of the Lengyel-Epstein system with a discrete delay in detail. Under the assumption that the unique positive equilibrium of the model is locally asymptotically stable in the absence of the delay, the effect of the increase of delay on the stability of the unique positive equilibrium is analyzed in detail. It is found that under suitable conditions on the other parameters, the delay doesn’t affect the stability of the equilibrium, namely, the equilibrium is absolutely stable while under the other conditions on the other parameters, the equilibrium will become ultimately unstable after passing through multiple stability switches and Hopf bifurcations at some certain critical values of delay. Particularly, by means of the normal form method and the center manifold reduction for retarded functional differential equations, the explicit formulae determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions are obtained. To verify our theoretical conclusions, some numerical simulations for specific examples are also included at the end of this article.

Rumor model on homogeneous social network incorporating delay in expert intervention and government action

Rumors are spreading all over the network drastically, affecting millions of people in a minuscule instance. This malicious news can be a cause of panic, social unrest, political imbalance, and slow economic growth of the country. Epidemiological modeling is a valuable tool to describe not only the dynamics of rumor on a social network but also analyzing the effect of various control strategies for handling emergencies. The time delay is viewed as a latent period and immune period in epidemics. Similarly, time delay exists in rumor spreading on the social network not only to influence thinkers by rumor adopters but also in expert intervention and government policy. On the basis of these premises, we propose a computational mathematical model of malicious news dynamics on a homogeneous social network and demonstrate the effect of delay in expert intervention and government action. Here, we observe that it is difficult to judge the opinion of the social network user about any particular event, if experts take more time to respond and are not continuously active to make aware of the newcomers. Also, social network data become unpredictable, so this work can be helpful for the information security of social network data.The condition of existence and stability of steady states are studied. We also evaluate the Hopf bifurcation condition and calculate the critical value of delay in expert intervention. Moreover, we recognize the most sensitive parameter to the system and numerically simulate the model with some useful parameter sets to justify the theoretical findings.

Stability and Hopf bifurcation for a delayed diffusive competition model with saturation effect.

This paper presents an investigation on the dynamics of a delayed diffusive competition model with saturation effect. We first perform the stability analysis of the positive equilibrium and the existence of Hopf bifurcations. It is shown that the positive equilibrium is asymptotically stable under some conditions, and that there exists a critical value of delay, when the delay increases across it, the positive equilibrium loses its stability and a spatially homogeneous or inhomogeneous periodic solution emerges from the positive equilibrium. Then, we derive the formulas for the determination of the direction of Hopf bifurcation and the properties of the bifurcating periodic solutions. Finally, some numerical simulations are performed to illustrate the obtained results.