Narrow Results

In this paper, we propose a delayed viral infection model to incorporate a logistic proliferation, a cell pyroptosis effect and the three intracellular time delays. We present the basic reproduction number and investigate the existence and the global stability of equilibria: infection-free equilibrium $p_0$ and infection equilibrium $p^{\ast }$, respectively. By considering different combinations of the time delays, we investigate the existence and the properties of Hopf bifurcation from $p^{\ast }$ when it is unstable. We also numerically explore the viral dynamics beyond stability. Bifurcation diagrams are used to show the stability switches, multiperiod solutions and irregular sustained oscillations with the variation of time delays. The results reveal that both the logistic proliferation term and time delays are responsible for the rich dynamics, but the logistic proliferation term may be the main factor for the occurrence of the Hopf bifurcation. Moreover, we show that ignoring the cell pyroptosis effect may underevaluate the viral infection risk and the sensitivity analysis implies that taking effective strategies for reducing the impact of cell pyroptosis is beneficial for decreasing the viral infection risk.

In this paper, an identification method is proposed for linear continuous time delay processes with unknown non-zero initial condition and disturbance from pulse tests. Multiple integral is used and integral intervals are specifically chosen to enable easy identification of the system parameters in one step. The effectiveness of the identification method is demonstrated through simulations.

We study the effects of time delay on stochastic resonance (SR) in the tumor cell growth model driven by two coupled noises and a weak external periodic signal. Under the condition of small delay time, we obtained the signal-to-noise ratio (SNR) R_SNR from the quasi-steady-state probability distribution function through the adiabatic elimination method and the SR theory about a two-state transition. By the numerical computations, we discussed the effects of the delay time τ on the SNR as a function of the multiplicative noise intensity D, the additive noise intensity α and the cross-correlated strength λ respectively. The appearance of a peak in these curves represents the SR phenomenon. It is found that with the increase of τ, the SR is suppressed in the D–R_SNR plot and weakened in the α–R_SNR plot. However, the SR is strengthened with the increase of τ in the λ–R_SNR plot.

We study dynamical properties of an anti-tumor cell growth system in the presence of time delay and correlations between multiplicative and additive white noise. Using the small time delay approximation, the Novikov theorem and Fox approach, the stationary probability distribution (SPD) is obtained. Based on the SPD, the expressions of the normalized correlation function C(s) and the associated relaxation time T_c are derived by means of Stratonovich decoupling ansatz. Based on numerical computations, we find the following: (i) The SPD exhibits one-peak → two-peaks → one-peak phase transitions as the correlation intensity λ varies. (ii) The relaxation time T_c exhibits a one-peak structure for negatively correlated noise (λ<0), however for positively correlated noise (λ>0), the relaxation time T_c decreases monotonously. (iii) The effects of the delay time τ on T_c and C(s) are entirely the same for λ<0 and for λ>0, i.e. τ enhances the fluctuation decay of the population of tumor cells.

The thermal activation problem of a bistable system driven by correlated noises with time delay is investigated by means of numerical simulations. The simulation results indicate: (1) For the case of the bistable system with linear delay, the phenomenon of noise enhanced stability (NES) is enforced by increasing delay time τ as the multiplicative noise intensity D is smaller, but is weakened as D is larger. (2) For the case of the bistable system with cubic delay, the NES becomes faintness as τ increases. (3) For the case of the bistable system with global delay, the NES is still restrained by increasing τ with smaller D, and in some circumstances, the activation rate as a function of τ exists a peak structure, which demonstrating the emergence of resonant activation.

We investigate the effects of time delay and noise correlation on the stochastic resonance induced by a multiplicative signal in an asymmetric bistable system. By the two-state theory and small delay approximation, the expression of the output signal-to-noise ratio (SNR) is obtained in the adiabatic limit. The results show that SNR as a function of the multiplicative noise intensity D shows a transition from two peaks to one peak with the decreasing of cross-correlation strength λ and the increasing of delay time τ. Moreover, there are the doubly critical phenomena for SNR versus λ and τ, and SNR versus D and α (additive noise intensity).

In this paper, the memristor-based fractional-order neural networks with time delay are analyzed. Based on the theories of set-value maps, differential inclusions and Filippov’s solution, some sufficient conditions for asymptotic stability of this neural network model are obtained when the external inputs are constants. Besides, uniform stability condition is derived when the external inputs are time-varying, and its attractive interval is estimated. Finally, numerical examples are given to verify our results.

In this paper, the issue of synchronization and anti-synchronization for fractional-delayed memristor-based chaotic system is studied by using active control strategy. Firstly, some explicit conditions are proposed to guarantee the synchronization and anti-synchronization of the proposed system. Secondly, the influence of order and time delay on the synchronization (anti-synchronization) is discussed. It reveals that synchronization (anti-synchronization) is faster as the order increases or the time delay decreases. Finally, some numerical simulations are presented to verify the validity of our theoretical analysis.

In traffic systems, cooperative driving has attracted the researchers’ attention. A lot of works attempt to understand the effects of cooperative driving behavior and/or time delays on traffic flow dynamics for specific traffic flow models. This paper is a new attempt to investigate analyses of linear stability and weak nonlinearity for the general car-following model with consideration of cooperation and time delays. We derive linear stability condition and study how the combinations of cooperation and time delays affect the stability of traffic flow. Burgers’ equation and Korteweg de Vries’ (KdV) equation for car-following model considering cooperation and time delays are derived. Their solitary wave solutions and constraint conditions are concluded. We investigate the property of cooperative optimal velocity (OV) model which estimates the combinations of cooperation and time delays about the evolution of traffic waves using both analytic and numerical methods. The results indicate that delays and cooperation are model-dependent, and cooperative behavior could inhibit the stabilization of traffic flow. Moreover, delays of sensing relative motion are easy to trigger the traffic waves; delays of sensing host vehicle are beneficial to relieve the instability effect to a certain extent.

In this paper, the stochastic resonance (SR) phenomenon in a time-delayed tumor cell growth system subjected to a multiplicative periodic signal, the multiplicative and additive noise is investigated. By applying the small time-delay method and two-state theory, the expressions of the mean first-passage time (MFPT) and signal-to-noise ratio (SNR) are obtained, then, the impacts of time delay, noise intensities and system parameters on the MFPT and SNR are discussed. Simulation results show that the multiplicative and additive noise always weaken the SR effect; while time delay plays a key role in motivating the SR phenomenon when noise intensities take a small value, it will restrain SR phenomenon when noise intensities take a large value; the cycle radiation amplitude always plays a positive role in stimulating the SR phenomenon, while, system parameters play different roles in motivating or suppressing SR under the different conditions of noise intensities.

In this paper, the stable state transformation and the effect of the stochastic resonance (SR) for a metapopulation system are investigated, which is disturbed by time delay, the multiplicative non-Gaussian noise, the additive colored Gaussian noise and a multiplicative periodic signal. By use of the fast descent method, the approximation of the unified colored noise and the SR theory, the dynamical behaviors for the steady-state probability function and the SNR are analyzed. It is found that non-Gaussian noise, the colored Gaussian noise and time delay can all reduce the stability of the biological system, and even lead to the population extinction. Inversely, the self-correlation times of two noises can both increase the stability of the population system and be in favor of the population reproduction. As regards the SNR for the metapopulation system induced by the noise terms and time delay, it is discovered that time delay and the correlation time of the multiplicative noise can effectively enhance the SR effect, while the multiplicative noise and the correlation time of the additive noise would all the time suppress the SR phenomena. In addition, the additive noise can effectively motivate the SR effect, but not alter the peak value of the SNR. It is worth noting that the departure parameter from the Gaussian noise plays the diametrical roles in stimulating the SR effect in different cases.

The asymmetric bistable system with time delays in the feedback force and random force under multiplicative and additive Gaussian noise is studied. Using the small time delay approximation approach and time-delayed Fokker–Planck equations (FPE), the signal-to-noise ratio (SNR) of the proposed stochastic system is obtained. The stochastic resonance (SR) phenomena influenced by parameters — including system parameters $a$, $b$, asymmetry parameter $r$, time delay $\tau$, strength $𝜀$ of the time-delayed feedback, noise intensities $D$ and $Q$ of multiplicative and additive noise, and correlation strength $\lambda$ between two noises, are also analyzed by numerical simulations. Results demonstrate that the SR performance of the asymmetric bistable system is superior to one symmetric bistable system. Besides, both time delay and strength of time-delayed feedback could enhance the SR to some extent. Then, the asymmetric time-delayed bistable SR (ATDBSR) method is used to the bearing fault diagnosis. The engineering applications of the ATDBSR method are realized and the value of the method is verified by effective experimental results.

We explore the roles of information time delay and noise correlation on the stability of electricity market by the method of mean first passage time (MFPT) and delayed Heston model. We employ the least square method of probability distribution to estimate the parameters of the proposed model with the closing price data of electricity futures daily of the European Energy Exchange. Then the probability density functions of the price returns are empirically compared between both the simulated data from the delayed Heston model and the electricity futures data, and a good agreement can be found between them. Through the stochastic simulation of the mean first passage time of returns, some results show that: (i) the phenomenon of correlation enhancing stability can be observed in MFPT versus mean reversion of volatility; (ii) we can observe that the delay time and the growth rate can induce the critical phenomenon; (iii) there are optimal values of volatility parameters matching maximum stability of electricity futures price. In addition, the increased growth rate and the delay time enhance the stability of electricity futures price.

In this paper, in order to analyze the coexistent multiple-stability of system, a fractional-order memristive Chua’s circuit with time delay is proposed, which is composed of a passive flux-controlled memristor and a negative conductance as a parallel combination. First, the Chua’s circuit can be considered as a nonlinear feedback system consisting of a nonlinear block and a linear block with low-pass properties. In the complex plane, the nonlinear element of the system can be approximated by a variable gain called a describing function. Second, compared with conventional computation, the describing function can accurately predict the hidden dynamics, fixed points, periodic orbits, unstable behaviors of the system. By using this method, the full mapping of the system dynamics in parameter spaces is presented, and the coexistent multiple-stability of the system is investigated in detail. Third, using bifurcation diagram, phase diagram, time domain diagram and power spectrum diagram, the dynamical behaviors of the system under different system parameters and initial values are discussed. Finally, based on Adams–Bashforth–Moulton (ABM) method, the correctness of theoretical analysis is verified by numerical simulation, which shows that the fractional-order delayed memristive Chua’s system has complex coexistent multiple-stability.

An extended lattice hydrodynamic model with time delay is proposed under non-lane discipline. We try to grasp the impacts of the non-lane discipline of the considered lattice sites. Linear stability analysis of the proposed model is executed and the stability criterion is obtained. Using the reductive perturbation method, we investigate nonlinear analysis of the proposed model and derive the mKdV equation and its solution, which could reveal the propagation of density waves. We analyze the effect of time delay, the ratio of lane deviation and the control coefficient on the stability of traffic flow via numerical experiments. We find that those indices play an important role in the stability of traffic flow. The longer the time delay, the more unstable the system becomes. Also, the ratio of lane deviation and the control coefficient is able to more quickly dissipate the traffic congestions occurring in traffic flow.

This paper investigates the problem of finite-time $H_{\infty }$ synchronization for semi-Markov jump Lur’e systems with time-varying delay and external disturbance. The purpose of this work is to design a mode-dependent state-feedback controller to ensure that the synchronization-error system achieves finite-time synchronization with a prescribed $H_{\infty }$ performance index. A criterion for the finite-time synchronization is proposed by using appropriate Lyapunov functional and two recently developed inequalities. Then, a design method for the required state-feedback controller is presented with the application of several decoupling techniques. Finally, an example is provided to illustrate the applicability of the proposed control method.

In this paper, the motion of Brownian particles driven by a delayed tristable system with multiplicative and additive Gaussian white noise is mainly studied. First, the effective potential function and stable state probability density function (PDF) are derived by using the theory of small-time delay approximation and the approximate Fokker–Planck equation (FPE), and the expression of mean first-passage times (MFPTs) is obtained by using the definition of the MFPTs and the steepest descent method. Then, the effects of the parameters which include noise intensities of multiplicative and additive noise, and correlation strength between two noises, and time delay, and the strength of time-delayed feedback on PDF and MFPTs are analyzed. Results demonstrate that the additive noise intensity has a more profound influence on PDF than the multiplicative noise intensity. The non-equilibrium phase transition of the system can be produced by the correlation strength of noises. In addition, in the behavior of the MFPTs, we can observe the noise-enhanced stability (NES) phenomenon induced by multiplicative noise intensity. Besides, delayed time plays an important role in MFPTs. Moreover, MFPT $T(x_{s1}\to x_{s2})$ (stands for the Brownian particle moving from the left well to the middle well) is greater than $T(x_{s2}\to x_{s1})$ (stands for the Brownian particle moving from the middle well to the left one).

This paper investigates the problem of exponential passive filter design for switched neural networks with time-delay and reaction-diffusion terms. With the aid of a suitable Lyapunov–Krasovskii functional and some inequalities, a linear matrix inequality-based design method is developed that not only makes the filtering error system exponentially stable but also forces it to be passive from external interference to output error. Then, the filter design is extended to the complex-valued case via separating the system into real-valued and complex-valued parts. Finally, a numerical example is utilized to illustrate the effectiveness of the filter design methods for the real-valued and complex-valued cases, respectively.

This paper presents a systematic study on the dynamics of a controlled Duffing oscillator with delayed displacement feedback, especially on the local bifurcations of periodic motions with respect to the time delay. The study begins with the analysis of the stability switches of the trivial equilibrium of the system with various parametric combinations and gives the critical values of time delay, where the trivial equilibrium may change its stability. It shows that as the time delay increases from zero to the positive infinity, the trivial equilibrium undergoes a different number of stability switches for different parametric combinations, and becomes unstable at last for all parametric combinations. Then, the method of multiple scales and the numerical computation method are jointly used to obtain a global diagram of local bifurcations of periodic motions with respect to the time delay for each type of parametric combinations. The diagrams indicate two kinds of local bifurcations. One is the saddle-node bifurcation and the other is the pitchfork bifurcation, of which the former means the sudden emerging of two periodic motions with different stability and the latter implies the Hopf bifurcation in the sense of dynamic bifurcation. A novel feature, referred to as the property of "periodicity in delay", is observed in the global diagrams of local bifurcations and used to justify the validity of infinite number of bifurcating branches in the bifurcation diagrams. The stability of the periodic motions is discussed not only from the high-order approximation of the asymptotic solution, but also from viewpoint of basin of attraction, which gives a good explanation for coexisting periodic motions and quasi-periodic motions, as well as an overall idea of basin of attraction. Afterwards, a conventional Poincaré section technique is used to reveal the abundant dynamic structures of a tori bifurcation sequence, which shows that the system will repeat similar quasi-periodic motions several times, with an increase of time delay, enroute to a chaotic motion. Finally, a new Poincaré section technique is proposed as a comparison with the conventional one, and the results show that the dynamical structures on the two kinds of Poincaré sections are topologically symmetric in rotation.

Some sufficient conditions for the asymptotic stability of cellular neural networks with time delay are derived using the Lyapunov–Krasovskii stability theory for functional differential equations as well as the linear matrix inequality (LMI) approach. The analysis shows how some well-known results can be refined and generalized in a straightforward manner. Moreover, the stability criteria obtained are delay-independent. They are less conservative and restrictive than those reported so far in the literature, and provide a more general set of criteria for determining the stability of delayed cellular neural networks.