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.

This research focuses on stabilization challenges of a Power System (PS) described by a delayed conformable fractional-order nonlinear model. We adopt the Polynomial Fitting Approximation Algorithm (PFAA) to approximate the cardinal sine function by a Square Of Polynomial (SOP). A polynomial representation is made for PS with different behaviors by dividing the operating range into $r$ regions and then calculating an SOP approximation for each region. Thus, the PS is characterized by $r$ distinct models, each applicable within its respective region. An Observer Based Control (OBC) is designed to stabilize the considered PS. The proposed result ensures the stabilization of the different $r$ models by satisfying sufficient conditions based on the sum-of-squares (SOS) approach.

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.

In this paper, we consider the following nonlinear differential equation

This paper presents a detailed study on the bifurcation of a controlled Duffing oscillator with a time delay involved in the feedback loop. The first objective is to determine the bifurcating periodic motions and to obtain the global diagrams of local bifurcations of periodic motions with respect to time delay. In order to determine the bifurcation point, an analysis on the stability switches of the trivial equilibrium is first performed for all possible parametric combinations. Then, by means of the method of multiple scales, an analysis on the local bifurcation of periodic motions is given. The static bifurcation diagrams on the amplitude-delay plane exhibit two kinds of local bifurcations of periodic motions, namely the saddle-node bifurcation and the pitchfork bifurcation, indicating a sudden emergence of two periodic motions with different stability and a Hopf bifurcation, respectively, in the sense of dynamic bifurcation. The second objective is to develop a shooting technique to locate both stable and unstable periodic motions of autonomous delay differential equations such that the periodic motions and their stability predicted using the method of multiple scales could be verified. The efficacy of the shooting scheme is well illustrated by some examples via phase trajectory and time history. It is shown that periodic motions located by the shooting method agree very well with those predicted on the bifurcation diagrams. Finally, the paper presents some interesting features of this simple, but dynamics-rich system.

It is a rule of thumb that time delay tends to destabilize any dynamical system. This is not true, however, in the case of delayed oscillators, which serve as mechanical models for several surprising physical phenomena. Parametric excitation of oscillatory systems also exhibits stability properties sometimes defying our physical sense. The combination of the two effects leads to challenging tasks when nonlinear dynamic behaviors in these systems are to be predicted or explained as well. This paper gives a brief historical review of the development of stability analysis in these systems, induced by newer and newer models in several fields of engineering. Local and global nonlinear behavior is also discussed in the case of the most typical parametrically excited delayed oscillator, a recent model of cutting applied to the study of high-speed milling processes.

In this paper, a novel method of energy analysis is developed for dynamical systems with time delays that are slightly perturbed from undamped SDOF/MDOF vibration systems. Being served frequently as the mathematical models in many applications, such systems undergo Hopf bifurcation including the classic "Hopf bifurcation" for SDOF systems and "multiple Hopf bifurcation" for MDOF systems, under certain conditions. An interesting observation of this paper is that the local dynamics near a Hopf bifurcation, including the stability of the trivial equilibrium and the bifurcating periodic solutions, of such systems, can be justified simply by the change of the total energy function. The key idea is that for the systems of concern, the total power (the total derivative of the energy function) can be estimated along an approximated solution with harmonic entries, the main part of the solution near the Hopf bifurcation. It shows that the present method works effectively for stability prediction of the trivial equilibrium and the bifurcating periodic solutions, and that it provides a high accurate estimation of the amplitudes of the bifurcating periodic solutions. Compared with the current methods such as the center manifold reduction which involves a great deal of symbolic computation, the energy analysis features a clear physical intuition and easy computation. Two illustrative examples are given to demonstrate the effectiveness of the present method.

This paper presents a detailed analysis on the dynamics of a two-dimensional delayed small-world network under delayed state feedback control. On the basis of stability switch criteria, the equilibrium is studied, and the stability conditions are determined. This study shows that with properly chosen delay and gain in the delayed feedback path, the controlled small-world delayed network may have stable equilibrium, or periodic solutions resulting from the Hopf bifurcation, or the multistability solutions via three types of codimension two bifurcations. Moreover, the direction of Hopf bifurcation and stability of the bifurcation periodic solutions are determined by using the normal form theory and center manifold theorem. In addition, the study shows that the controlled model exhibits period-doubling bifurcations which lead eventually to chaos; and the chaos can also directly occur via the bifurcations from the quasi-periodic solutions. The results show that the delayed feedback is an effective approach in order to generate or annihilate complex behaviors in practical applications.

The stability issues of the equilibrium points of the cellular neural networks (CNN) with single and multiple delays are further investigated. Several novel delay-dependent and delay-independent asymptotical/exponential stability criteria are established by employing parameterized first-order model transformation, Lyapunov–Krasovskii stability theorem and LMI technique in virtue of the linearization of considered model. The stability regions with respect to the delay parameters are formulated by applying the proposed results. To the best of the authors' knowledge, few (if any) reports about such "linearization" approach to stability analysis for delayed neural network models have been presented in the open literatures. Some numerical examples are also given to illustrate the effectiveness of our results and to compare with the recent results.

In this paper, we report a common phenomenon observed in chaotic systems linked by time delay. Recently, the Lorenz chaotic system has been extended to the family of Lorenz systems which includes the Chen and Lü systems. These three chaotic systems, corresponding to different sets of system parameter values, are topologically different. With the aid of numerical simulations, we have surprisingly found that a simple time delay, directly applied to one or more state variables, transforms the Lorenz system to the generalized Chen system or the generalized Lü system without any parameter changes. The existence of this phenomenon has also been found in other known chaotic systems: the Rössler system, the Chua's circuit and the 4-Liu system. This finding has shown a common characteristic of chaotic systems: a new chaotic "branch" can be created from a chaotic attractor by simply adding a time delay.

In this paper, a general two-neuron model with time delay is considered, where the time delay is regarded as a parameter. It is found that Hopf bifurcation occurs when this delay passes through a sequence of critical value. By analyzing the characteristic equation and using the frequency domain approach, the existence of Hopf bifurcation is determined. The stability of bifurcating periodic solutions are determined by the harmonic balance approach, Nyquist criterion and the graphic Hopf bifurcation theorem. Numerical results are given to justify the theoretical analysis.

In this paper, a novel method named pseudo-oscillator analysis is developed for the local dynamics near a Hopf bifurcation of scalar nonlinear dynamical systems with time delays. For this purpose, a pseudo-oscillator that is slightly perturbed from an undamped oscillator is firstly constructed, its fundamental frequency is the same as the frequency at the bifurcation point, and the disturbance is associated with the original system. Next, the pseudo-power function, defined as the power function of the pseudo-oscillator, is estimated along a harmonic function. Then we conclude that the local dynamics near the Hopf bifurcation can be justified from the variation of the averaged pseudo-power function. The new method features a clear physical intuition and easy computation, and it yields very accurate prediction for the periodic solution resulted from the Hopf bifurcation, as shown in three illustrative examples.

We investigate the behavior of a neural network model consisting of three neurons with delayed self and nearest-neighbor connections. We give analytical results on the existence, stability and bifurcation of nontrivial equilibria of the system. We show the existence of codimension two bifurcation points involving both standard and D₃-equivariant, Hopf and pitchfork bifurcation points. We use numerical simulation and numerical bifurcation analysis to investigate the dynamics near the pitchfork–Hopf interaction points. Our numerical investigations reveal that multiple secondary Hopf bifurcations and pitchfork bifurcations of limit cycles may emanate from the pitchfork–Hopf points. Further, these secondary bifurcations give rise to ten different types of periodic solutions. In addition, the secondary bifurcations can lead to multistability between equilibrium points and periodic solutions in some regions of parameter space. We conclude by generalizing our results into conjectures about the secondary bifurcations that emanate from codimension two pitchfork–Hopf bifurcation points in systems with D_n symmetry.

We study a learning and recalling model of phase patterns in a two- or three-coupled BVP oscillators system with a time delay δ. The coupling strengths are modulated by the Hebbian learning rule. Assuming the first-order approximation, we calculate the optimal condition of δ for exact recall by applying the phase dynamics theory. When α = 0, where α represents the coupling from activator to inhibitor, the correlation between the learning and the retrieval phase depends on δ. When α = 1, exact recall is achieved independent of δ. The results can be explained by the phase dynamics theory.

The main purpose of this paper is to investigate the stability and Hopf bifurcation for a delayed two-species cooperative diffusion system with Neumann boundary conditions. By linearizing the system at the positive equilibrium and analyzing the corresponding characteristic equation, the asymptotic stability of positive equilibrium and the existence of Hopf oscillations are demonstrated. It is shown that, under certain conditions, the system undergoes only a spatially homogeneous Hopf bifurcation at the positive equilibrium when the delay crosses through a sequence of critical values; under the other conditions, except for the previous spatially homogeneous Hopf bifurcations, the system also undergoes a spatially inhomogeneous Hopf bifurcation at the positive equilibrium when the delay crosses through another sequence of critical values. In particular, in order to determine the direction and stability of periodic solutions bifurcating from spatially homogeneous Hopf bifurcations, the explicit formulas are given by using the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs). Finally, to verify our theoretical predictions, some numerical simulations are also included.

The paper presents a detailed analysis on the dynamics of a two-neuron network with time-delayed connections between the neurons and time-delayed feedback from each neuron to itself. On the basis of characteristic roots method and Hopf bifurcation theorems for functional differential equations, we investigate the existence of local Hopf bifurcation. In addition, the direction of Hopf bifurcation and stability of the periodic solutions bifurcating from the trivial equilibrium are determined based on the normal form theory and center manifold theorem. Moreover, employing the global Hopf bifurcation theory due to [Wu, 1998], we study the global existence of periodic solutions. It is shown that the local Hopf bifurcation indicates the global Hopf bifurcation after the second group critical value of the delay.

When two phase oscillators interact with a time-delay, new synchronized and desynchronized regimes appear, in this way giving rise to the phenomenon of multistability. The number of coexisting stable states grows with an increase of the delay, and their frequencies quantize. We show that, while the number of synchronized solutions grows linearly with the delay and/or coupling, the set of the desynchronized solutions, i.e. those with different average frequencies of the individual oscillators, raises quadratically with increasing delay. For the synchronized states, we analyze the mutual arrangement of the basins of attraction, and conclude that the structure and size of the basins are apparently the same for each state. We discuss possible implications for desynchronizing brain stimulation techniques.

This paper reveals the dynamics of a delayed neural network of four neurons, with a short-cut connection through a theoretical analysis and some case studies of both numerical simulations and experiments. It presents a detailed analysis of the stability and the stability switches of the network equilibrium, as well as the Hopf bifurcation and the bifurcating periodic responses on the basis of the normal form and the center manifold reduction. Afterwards, the study focuses on the validation of theoretical results through numerical simulations and circuit experiments. The numerical simulations and the circuit experiments not only show good agreement with theoretical results, but also show abundant effects of the short-cut connection on the network dynamics.

In this paper, the classical Liénard equation with a discrete delay is considered. Under the assumption that the classical Liénard equation without delay has a unique stable trivial equilibrium, we consider the effect of the delay on the stability of zero equilibrium. It is found that the increase of delay not only can change the stability of zero equilibrium but can also lead to the occurrence of periodic solutions near the zero equilibrium. Furthermore, the stability of bifurcated periodic solutions is investigated by applying the normal form theory and center manifold reduction for functional differential equations. Finally, in order to verify these theoretical conclusions, some numerical simulations are given.