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In this study, we principally investigate a fractional‐order stage‐structured predator–prey system including distributed time delays and discrete time delays. Taking advantage of transformation of the variable, we obtain an isovalent version of the considered fractional‐order stage‐structured predator–prey system including distributed time delays and discrete time delays. The isovalent version includes fractional‐order and integer‐order equations. Utilizing the stability criterion and bifurcation theory of fractional‐order differential equation, a novel delay‐independent bifurcation condition to ensure the appearance of Hopf bifurcation for the fractional‐order stage‐structured predator–prey system is set up. The impact of time delay on the stability and bifurcation is clearly revealed. Numerical simulation figures are presented to sustain the rationality of the derived key conclusions. [ABSTRACT FROM AUTHOR]

Since the stochastic bifurcation theory is still in its infancy, we try to analyze some stochastic bifurcation phenomenon from a simple mathematical model. Thus, this paper mainly focuses on studying the stochastic bifurcation of a stochastic logistic model with distributed delay in the strong kernel case, which is affected by noise. Therefore, we use the intrinsic growth rate as a bifurcation parameter. First, we study the stochastic D-bifurcation and stochastic P-bifurcation for stochastic logistic model. Furthermore, by deriving the corresponding Fokker–Planck equation, we obtain the expression of the joint density function of the stochastic logistic system near the positive equilibrium point. Finally, some conclusions are given. [ABSTRACT FROM AUTHOR]

We consider a scalar delay differential equation x ˙ (t) = - d x (t) + f ((1 - α) ρ x (t - τ) + α ρ x (t - 2 τ)) with an instant mortality rate d > 0 , the nonlinear Rick reproductive function f, a survival rate during all development stages ρ , and a proportion constant α ∈ [ 0 , 1 ] with which population undergoes a diapause development. We consider global continuation of a branch of periodic solutions locally generated through the Hopf bifurcation mechanism, and we establish the existence of periodic solutions with periods within (3 τ , 6 τ) for a wide range of parameter values. We show this existence of periodic solutions not only for the delay τ near the first critical value τ ∗ when a local Hopf bifurcation takes place near the positive equilibrium, but for all τ > τ ∗ . We obtain this (global) existence of periodic solutions by using the equivalent-degree based global Hopf bifurcation theory, coupled with an application of the Li–Muldowney technique to rule out periodic solutions with period 3 τ . We conduct some numerical simulations to illustrate that this global continuation is completely due to the diapause-delay since solutions of the delay differential equation with only normal development delay in the given biologically realistic range all converge to the positive equilibrium. [ABSTRACT FROM AUTHOR]

This paper concerns a model of hierarchical neural network with D n × D n -symmetry and time delays. The linear stability of the steady state is obtained via an artful factorization technique of the characteristic equation corresponding to the linearized system. And we also perform a delay-induced Hopf bifurcation analysis involving spatio-temporal patterns of bifurcated periodic solutions relying on the equivariant Hopf bifurcation theory for retarded functional differential equations and the representation method of Lie groups. Some numerical simulations for a simple example are carried out to demonstrate the application of our theoretical conclusion that multiple periodic orbits could occur simultaneously from Hopf bifurcation in the model. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove that Wright's equation y ′ (t) = − α y (t − 1) { 1 + y (t) } has a unique slowly oscillating periodic solution for parameter values α ∈ (π 2 , 1.9 ] , up to time translation. This result proves Jones' Conjecture formulated in 1962, that there is a unique slowly oscillating periodic orbit for all α > π 2. Furthermore, there are no isolas of periodic solutions to Wright's equation; all periodic orbits arise from Hopf bifurcations. [ABSTRACT FROM AUTHOR]

We develop a global Hopf bifurcation theory for differential equations with a state-dependent delay governed by an algebraic equation, using the S 1 -equivariant degree. We apply the global Hopf bifurcation theory to a model of genetic regulatory dynamics with threshold type state-dependent delay vanishing at the stationary state, for a description of the global continuation of the periodic oscillations. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a nonlinear delay differential equation applied in a nonstandard finite difference method. By analyzing the numerical discrete system, we show that a sequence of Neimark–Sacker bifurcations occur at the equilibrium as the delay increases. Moreover, the existence of local Neimark–Sacker bifurcations is considered, and the direction and stability of periodic solutions bifurcating from the Neimark–Sacker bifurcation of the discrete model are determined by the Neimark–Sacker bifurcation theory of discrete system. Finally, some numerical simulations are adopted to illustrate the corresponding theoretical results. [ABSTRACT FROM AUTHOR]

We address the problem of the numerical bifurcation analysis of general nonlinear delay equations, including integral and integro-differential equations, for which no software is currently available. Pseudospectral discretization is applied to the abstract reformulation of equations with infinite delay to obtain a finite dimensional system of ordinary differential equations, whose properties can be numerically studied with well-developed software. We explore the applicability of the method on some test problems and provide some numerical evidence of the convergence of the approximations. [ABSTRACT FROM AUTHOR]

In this paper, a time-delayed model of microorganism flocculation with saturated functional responses is presented. We first analyse the local dynamics of this model with bifurcations in parameter fields, and then prove the collection of microorganisms is sustainable as well as obtain an explicit eventual lower bound of microorganism concentration when threshold parameter R0>1. This model has a backward bifurcation if w under an additional condition, which implies that the microorganism-free equilibrium coexists with a microorganism equilibrium. In these cases, we establish some sufficient conditions for the global stability by using a variant of the Lyapunov-LaSalle theorem. [ABSTRACT FROM AUTHOR]

We develop a mathematical model of platelet, megakaryocyte, and thrombopoietin dynamics in humans. We show that there is a single stationary solution that can undergo a Hopf bifurcation, and use this information to investigate both normal and pathological platelet production, specifically cyclic thrombocytopenia. Carefully estimating model parameters from laboratory and clinical data, we then argue that a subset of parameters are involved in the genesis of cyclic thrombocytopenia based on clinical information. We provide model fits to the existing data for both platelet counts and thrombopoietin levels by changing four parameters that have physiological correlates. Our results indicate that the primary change in cyclic thrombocytopenia is an interference with, or destruction of, the thrombopoietin receptor with secondary changes in other processes, including immune-mediated destruction of platelets and megakaryocyte deficiency and failure in platelet production. This study contributes to the understanding of the origin of cyclic thrombocytopenia as well as extending the modeling of thrombopoiesis. [ABSTRACT FROM AUTHOR]

In this article, we present a systematic approach to bifurcation analysis of impulsive systems with autonomous or periodic right-hand sides that may exhibit delayed impulse terms. Methods include Lyapunov-Schmidt reduction and center manifold reduction. Both methods are presented abstractly in the context of the stroboscopic map associated to a given impulsive system, and are illustrated by way of two in-depth examples: the analysis of a SIR model of disease transmission with seasonality and unevenly distributed moments of treatment, and a scalar logistic differential equation with a delayed census impulsive harvesting effort. It is proven that in some special cases, the logistic equation can exhibit a codimension two bifurcation at a 1:1 resonance point. [ABSTRACT FROM AUTHOR]

A common task when analysing dynamical systems is the determination of normal forms near local bifurcations of equilibria. As most of these normal forms have been classified and analysed, finding which particular class of normal form one encounters in a numerical bifurcation study guides follow-up computations. This paper builds on normal form algorithms for equilibria of delay differential equations with constant delay that were developed and implemented in DDE-Biftool recently. We show how one can extend these methods to delay-differential equations with state-dependent delay (sd-DDEs). Since higher degrees of regularity of local center manifolds are still open for sd-DDEs, we give an independent (still only partial) argument which phenomena from the truncated normal must persist in the full sd-DDE. In particular, we show that all invariant manifolds with a sufficient degree of normal hyperbolicity predicted by the normal form exist also in the full sd-DDE. [ABSTRACT FROM AUTHOR]

Passively mode-locked semiconductor lasers are compact, inexpensive sources of short light pulses of high repetition rates. In this work, we investigate the dynamics and bifurcations arising in such a device under the influence of time delayed optical feedback. This laser system is modelled by a system of delay differential equations, which includes delay terms associated with the laser cavity and feedback loop. We make use of specialised path continuation software for delay differential equations to analyse the regime of short feedback delays. Specifically, we consider how the dynamics and bifurcations depend on the pump current of the laser, the feedback strength, and the feedback delay time. We show that an important role is played by resonances between the mode-locking frequencies and the feedback delay time. We find feedback-induced harmonic mode locking and show that a mismatch between the fundamental frequency of the laser and that of the feedback cavity can lead to multi-pulse or quasiperiodic dynamics. The quasiperiodic dynamics exhibit a slow modulation, on the time scale of the gain recovery rate, which results from a beating with the frequency introduced in the associated torus bifurcations and leads to gain competition between multiple pulse trains within the laser cavity. Our results also have implications for the case of large feedback delay times, where a complete bifurcation analysis is not practical. Namely, for increasing delay, there is an everincreasing degree of multistability between mode-locked solutions due to the frequency pulling effect. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of a delay limit cycle oscillator under periodic external forcing. The system exhibits quasiperiodic motion outside of a resonance region where it has periodic motion at the frequency of the forcer for strong enough forcing. By perturbation methods and bifurcation theory, we show that this resonance region is asymmetric in the frequency detuning, and that there are regions where stable periodic and quasiperiodic motions coexist. [ABSTRACT FROM AUTHOR]

We study stability and Hopf bifurcation analysis of a model that refers to the competition between the immune system and an aggressive host such as a tumor. The model which describes this competition is governed by a reaction-diffusion system including time delay under the Neumann boundary conditions, and is based on Kuznetsov-Taylor's model. Choosing the delay parameter as a bifurcation parameter, we first show that Hopf bifurcation occurs. Second, we determine two properties of the periodic solution, namely its direction and stability, by applying the normal form theory and the center manifold reduction for partial functional differential equations. Furthermore, we discuss the effects of diffusion on the dynamics by analyzing a model with constant coefficients and perform some numerical simulations to support the analytical results. The results show that diffusion has an important effects on the dynamics of a mathematical model. [ABSTRACT FROM AUTHOR]

In this paper, the Bogdanov-Takens (B-T) and triple zero bifurcations are investigated for coupled van der Pol-Duffing oscillators with symmetry, in the presence of time delays due to the intrinsic response and coupling. Different from previous works, third order unfolding normal forms associated with B-T and triple zero bifurcations are needed, which are obtained by using the normal form theory of delay differential equations. Numerical simulations are also presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We study a scalar delay differential equation (DDE) with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state-dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three contain only constant delays. We implemented this expansion and the computation of the normal form coeffcients in Matlab using symbolic differentiation and the resulting code HHnfDDEsd is supplied as a supplement to this article. Numerical continuation of the torus bifurcation curves confirms the correctness of our normal form calculations. Moreover, it enables us to compute the curves of torus bifurcations more globally and to find associated curves of saddle-node bifurcations of periodic orbits that bound the resonance tongues. The tori themselves are computed and visualized in a three-dimensional projection, as well as the planar trace of a suitable Poincaré section. In particular, we compute periodic orbits on locked tori and their associated unstable manifolds (when there is a single unstable Floquet multiplier). This allows us to study transitions through resonance tongues and the breakup of a 1 : 4 locked torus. The work presented here demonstrates that state dependence alone is capable of generating a wealth of dynamical phenomena. [ABSTRACT FROM AUTHOR]

Semiconductor lasers exhibit a wealth of dynamics, from emission of a constant beam of light to periodic oscillations and excitability. Self-pulsing regimes, where the laser periodically releases a short pulse of light, are particularly interesting for many applications, from material science to telecommunications. Self-pulsing regimes need to produce pulses very regularly and, as such, they are also known to be particularly sensitive to perturbations, such as noise or light injection. We investigate the effect of delayed optical feedback on the dynamics of a self-pulsing semiconductor laser with saturable absorber (SLSA). More precisely, we consider the Yamada model with delay--a system of three delay--differential equations (DDEs) for two slow and one fast variables--which has been shown to reproduce accurately self-pulsing features as observed in SLSA experimentally. This model is also of broader interest because it is quite closely related to mathematical models of other self-pulsing systems, such as excitable spiking neurons. We perform a numerical bifurcation analysis of the Yamada model with delay, where we consider the feedback delay, the feedback strength, and the strength of pumping as bifurcation parameters. We and a rapidly increasing complexity of the system dynamics when the feedback delay is increased from zero. In particular, there are new feedback-induced dynamics: stable quasi-periodic oscillations on tori, as well as a large degree of multistability, with up to ve five pulse-like stable periodic solutions with different amplitudes and repetition rates. An attractor map in the plane of perturbations on the gain and intensity reveals a Cantor-set-like, intermingled structure of the different basins of attraction. This suggests that, in practice, the multistable laser is extremely sensitive to small perturbations. [ABSTRACT FROM AUTHOR]

Exploring some results of Wang et al. (Adv. Differ. Equ. 2016:33, 2016) from another point of view, we first investigate the stability and direction for a class of Schrödingerean difference equations with Schrödingerean Hopf bifurcation. Next we obtain the stable conditions for these equations and prove that Schrödingerean Hopf bifurcation shall occur when the delay passes through the critical value. [ABSTRACT FROM AUTHOR]

In this work we study local oscillations in delay differential equations with a frequency domain methodology. The main result is a bifurcation equation from which the existence and expressions of local periodic solutions can be determined. We present an iterative method to obtain the bifurcation equation up to a fixed arbitrary order. It is shown how this method can be implemented in symbolic math programs. [ABSTRACT FROM AUTHOR]

We embark on a case study for the scalar delay equation with odd nonlinearity f, real nonzero parameters $$\lambda , \, b$$ , and three positive time delays $$1,\, \vartheta ,\, p/2$$ . We assume supercritical Hopf bifurcation from $$x \equiv 0$$ in the well-understood single-delay case $$b = \infty $$ . Normalizing $$f' (0)=1$$ , branches of constant minimal period $$p_k = 2\pi /\omega _k$$ are known to bifurcate from eigenvalues $$i\omega _k = i(k+\tfrac{1}{2})\pi $$ at $$\lambda _k = (-1)^{k+1}\omega _k$$ , for any nonnegative integer k. The unstable dimension is k, at the local branch k. We obtain stabilization of such branches, for arbitrarily large unstable dimension k. For $$p:= p_k$$ the branch k of constant period $$p_k$$ persists as a solution, for any $$b\ne 0$$ and $$\vartheta \ge 0$$ . Indeed the delayed feedback term controlled by b vanishes on branch k: the feedback control is noninvasive there. Following an idea of Pyragas, we seek parameter regions $$\mathcal {P}$$ of controls $$b \ne 0$$ and delays $$\vartheta \ge 0$$ such that the branch k becomes stable, locally at Hopf bifurcation. We determine rigorous expansions for $$\mathcal {P}$$ in the limit of large k. The only two regions $$\mathcal {P} = \mathcal {P}^\pm $$ which we were able to detect, in this setting, required delays $$\vartheta $$ near 1, controls b near $$(-1)^k \cdot 2/\omega _k$$ , and were of very small area of order $$k^{-4}$$ . Our analysis is based on a 2-scale covering lift for the frequencies involved. [ABSTRACT FROM AUTHOR]

Periodic orbits and associated bifurcations of singularly perturbed state-dependent delay differential equations (DDEs) are studied when the profiles of the periodic orbits contain jump discontinuities in the singular limit. A definition of singular solution is introduced which is based on a continuous parametrisation of the possibly discontinuous limiting solution. This reduces the construction of the limiting profiles to an algebraic problem. A model two state-dependent DDE is studied in detail and periodic singular solutions are constructed with one and two local maxima per period. A complete characterisation of the conditions on the parameters for these singular solutions to exist facilitates an investigation of bifurcation structures in the singular case revealing folds and possible cusp bifurcations. Sophisticated boundary value techniques are used to numerically compute the bifurcation diagram of the state-dependent DDE when the perturbation parameter is close to zero. This confirms that the solutions and bifurcations constructed in the singular case persist when the perturbation parameter is nonzero, and hence demonstrates that the solutions constructed using our singular solution definition are useful and relevant to the singularly perturbed problem. Fold and cusp bifurcations are found very close to the parameter values predicted by the singular solution theory, and we also find period-doubling bifurcations as well as periodic orbits with more than two local maxima per period, and explain the alignment between the folds on different bifurcation branches. [ABSTRACT FROM AUTHOR]

This paper deals with the stability and bifurcation analysis of a general form of equation Dαx(t)=g(x(t),x(t-τ)) involving the derivative of order α ∈ (0, 1] and a constant delay τ ≥ 0. The stability of equilibrium points is presented in terms of the stability regions and critical surfaces. We provide a necessary condition to exist chaos in the system also. A wide range of delay differential equations involving a constant delay can be analyzed using the results proposed in this paper. The illustrative examples are provided to explain the theory. [ABSTRACT FROM AUTHOR]

We analyze canard explosions in delay differential equations with a one-dimensional slow manifold. This study is applied to explore the dynamics of the van der Pol slow-fast system with delayed self-coupling. In the absence of delays, this system provides a canonical example of a canard explosion. We show that as the delay is increased a family of 'classical' canard explosions ends as a Bogdanov-Takens bifurcation occurs at the folds points of the S-shaped critical manifold. [ABSTRACT FROM AUTHOR]

In this work, we develop a mathematical model to study the effect of drug on the development of cancer including the quiescent compartment. The model is governed by a system of delay differential equations where the delay represents the time that the cancer cell take to proliferate. Our analytical study of the stability shows that by considering the time delay as a parameter of bifurcation, it is possible to have stability switch and oscillations through a Hopf bifurcation. Moreover, by introducing the drug intervention term, the critical delay value increases. This indicates that the system can tolerate a longer delay before oscillations start. In the end, we present some numerical simulations illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we give a detailed Hopf bifurcation analysis of a recurrent neural network system involving both discrete and distributed delays. Choosing the sum of the discrete delay terms as a bifurcation parameter the existence of Hopf bifurcation is demonstrated. In particular, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are studied by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included. [ABSTRACT FROM AUTHOR]

We apply the pseudospectral discretization approach to nonlinear delay models described by delay differential equations, renewal equations, or systems of coupled renewal equations and delay differential equations. The aim is to derive ordinary differential equations and to investigate the stability and bifurcation of equilibria of the original model by available software packages for continuation and bifurcation for ordinary differential equations. Theoretical and numerical results confirm the effectiveness and the versatility of the approach, opening a new perspective for the bifurcation analysis of delay equations, in particular coupled renewal and delay differential equations. [ABSTRACT FROM AUTHOR]

In this paper the Bogdanov-Takens (BT) bifurcation of an 2 m coupled neurons network model with multiple delays is studied, where one neuron is excitatory and the next is inhibitory. When the origin of the model has a double zero eigenvalue, by using center manifold reduction of delay differential equations (DDEs), the second-order and third-order universal unfoldings of the normal forms are deduced, respectively. Some bifurcation diagrams and numerical simulations are presented to verify our main results. [ABSTRACT FROM AUTHOR]

In this article, for delayed Nicholson's blowflies equation, we propose a hybrid control nonstandard finite-difference (NSFD) scheme in which state feedback and parameter perturbation are used to control the Neimark-Sacker bifurcation. Firstly, the local stability of the positive equilibria for hybrid control delay differential equation is discussed according to Hopf bifurcation theory. Then, for any step-size, a hybrid control numerical algorithm is introduced to generate the Neimark-Sacker bifurcation at a desired point. Finally, numerical simulation results confirm that the control strategy is efficient in controlling the Neimark-Sacker bifurcation. At the same time, the results show that the NSFD control scheme is better than the Euler control method. [ABSTRACT FROM AUTHOR]

The model analyzed in this paper is based on the model set forth by Aziz Alaoui et al. [Aziz Alaoui & Daher Okiye, 2003; Nindjin et al., 2006] with time delay, which describes the competition between the predator and prey. This model incorporates a modified version of the Leslie-Gower functional response as well as that of Beddington-DeAngelis. In this paper, we consider the model with one delay consisting of a unique nontrivial equilibrium E* and three others which are trivial. Their dynamics are studied in terms of local and global stabilities and of the description of Hopf bifurcation at E*. At the third trivial equilibrium, the existence of the Hopf bifurcation is proven as the delay (taken as a parameter of bifurcation) that crosses some critical values. [ABSTRACT FROM AUTHOR]

This paper studies the existence of bistable states and control strategies to induce state transitions of a delay differential equation model of neutrophil dynamics. We seek the conditions that a stable steady state and an oscillatory state coexist in the neutrophil dynamical system. Physiologically, stable steady state represents the healthy state, while oscillatory state is usually associated with diseases such as cyclical neutropenia. We study the control strategies to induce the transitions from the disease state to the healthy state by introducing temporal perturbations to system parameters. This study is valuable in designing clinical protocols for the treatment of cyclical neutropenia. [ABSTRACT FROM AUTHOR]

Models of global climate phenomena of low to intermediate complexity are very useful for providing an understanding at a conceptual level. An important aspect of such models is the presence of a number of feedback loops that feature considerable delay times, usually due to the time it takes to transport energy (for example, in the form of hot/cold air or water) around the globe. In this paper, we demonstrate how one can perform a bifurcation analysis of the behaviour of a periodically forced system with delay in dependence on key parameters. As an example, we consider the El-Niño Southern Oscillation (ENSO), which is a sea-surface temperature (SST) oscillation on a multi-year scale in the basin of the Pacific Ocean. One can think of ENSO as being generated by an interplay between two feedback effects, one positive and one negative, which act only after some delay that is determined by the speed of transport of SST anomalies across the Pacific. We perform here a case study of a simple delayed-feedback oscillator model for ENSO, which is parametrically forced by annual variation. More specifically, we use numerical bifurcation analysis tools to explore directly regions of delay-induced resonances and other stability boundaries in this delay-differential equation model for ENSO. [ABSTRACT FROM AUTHOR]