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In this paper we study the scalar delay differential equation x ′ (t) = α (t) x (t - g 1 (t)) f (a (t) , x (t - g 2 (t))) - β (t) x (t) where f is decreasing in both arguments and the coefficients are positive and bounded. Sufficient conditions for the permanence and global attractivity for a fixed positive solution are derived. We apply our results to nonautonomous variants of Nicholson's blowfly equation and the Beverton–Holt model. [ABSTRACT FROM AUTHOR]

In 1973 Nussbaum proved that certain bounded solutions of autonomous delay-differential equations with analytic nonlinearities are themselves analytic. On the other hand, the two authors of this paper more recently showed that bounded solutions of certain delay-differential equations, again with analytic nonlinearities, can be C ∞ smooth, yet not be analytic for certain ranges of the independent variable t. In this paper we extend the 1973 results to obtain analytic solutions of a broader class of delay-differential equations, including a wide variety of nonautonomous equations. Nevertheless, there are still equations with analytic nonlinearities possessing global bounded C ∞ solutions for which analyticity is unknown. This is the case, for example, for the equation y ′ (t) = g (y (t - 1)) + ε sin (t 2) where g is analytic and where y = 0 is a hyperbolic equilibrium when ε = 0 . [ABSTRACT FROM AUTHOR]

We consider a periodic function p : R → R of minimal period 4 which satisfies a family of delay differential equations 0.1 x ′ (t) = g (x (t - d Δ (x t))) , Δ ∈ R , with a continuously differentiable function g : R → R and delay functionals d Δ : C ([ - 2 , 0 ] , R) → (0 , 2). The solution segment x t in Eq. (0.1) is given by x t (s) = x (t + s) . For every Δ ∈ R the solutions of Eq. (0.1) defines a semiflow of continuously differentiable solution operators S Δ , t : x 0 ↦ x t , t ≥ 0 , on a continuously differentiable submanifold X Δ of the space C 1 ([ - 2 , 0 ] , R) , with codim X Δ = 1 . At Δ = 0 the delay is constant, d 0 (ϕ) = 1 everywhere, and the orbit O = { p t : 0 ≤ t < 4 } ⊂ X 0 of the periodic solution is extremely stable in the sense that the spectrum of the monodromy operator M 0 = D S 0 , 4 (p 0) is σ 0 = { 0 , 1 } , with the eigenvalue 1 being simple. For | Δ | ↗ ∞ there is an increasing contribution of variable, state-dependent delay to the time lag d Δ (x t) = 1 + ⋯ in Eq. (0.1). We study how the spectrum σ Δ of M Δ = D S Δ , 4 (p 0) changes if | Δ | grows from 0 to ∞ . A main result is that at Δ = 0 an eigenvalue Λ (Δ) < 0 of M Δ bifurcates from 0 ∈ σ 0 and decreases to - ∞ as | Δ | ↗ ∞ . Moreover we verify the spectral hypotheses for a period doubling bifurcation from the periodic orbit O at the critical parameter Δ ∗ where Λ (Δ ∗) = - 1 . [ABSTRACT FROM AUTHOR]

Long-time dynamics of the solutions for the suspension bridge equation with constant and time-dependent delays have been investigated, but there are no works on suspension bridge equation with state-dependent delay. Thus, we first consider the existence of pullback attractor for the non-autonomous suspension bridge equation with state-dependent delay by using the contractive function methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the long term dynamics of the stochastic Schrödinger delay lattice systems when the nonlinear drift and diffusion terms are both locally Lipschitz continuous. Based on the well-posedness of the system, we first prove the existence and uniqueness of weak pullback mean random attractors in a product Hilbert space. We then show the tightness of distribution laws of solutions and the existence of invariant measures. We further prove the set of all invariant measures of the delay system is tight and every limit point of invariant measures of the delay system must be an invariant measure of the limiting system as time delay approaches zero. The idea of uniform tail-estimates is employed to to establish the tightness of distributions of solutions as well as the set of invariant measures of the delay system. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of solutions and periodic solutions for a class of semilinear neutral functional differential equations with state-dependent delay, in which the linear part is non-autonomous and generates a linear evolution operator. We first establish the existence and regularity of bounded solutions for the considered equation, and then we show by using Banach fixed point theorem that these solutions have periodicity property or asymptotic periodicity property respectively under some conditions. Finally, an example to illustrate the obtained results is given. [ABSTRACT FROM AUTHOR]

A computational method is developed for a family of functional differential equations in one independent variable with a single deviating argument, assumed to be a delay. These equations are mapped into a finite system of ODEs, a subsystem of which involves only the function controlling the delay. Key features include efficiency during method of steps, freedom from Jacobians and root finding techniques, and computing a continuous approximation. The nonlinear differential equations may be retarded, neutral or advanced. The method is established for state dependent delays, stiff equations, discontinuous initial history, a specific loss of monotonicity in the delay and extended naturally to distributed delays. When restricted to ODEs, the method extends naturally to PDEs; the hope is the delayed version here can eventually be extended to delayed PDEs. Conditions for convergence of the approximation are established, and results of numerical experiments are reported to indicate robustness of the implementation. [ABSTRACT FROM AUTHOR]

In this paper we analyze a coupled system between a transport equation and an ordinary differential equation with time delay (which is a simplified version of a model for kidney blood flow control). Through a careful spectral analysis we characterize the region of stability, namely the set of parameters for which the system is exponentially stable. Also, we perform a bifurcation analysis and determine some properties of the stable steady state set and the limit cycle oscillation region. Some numerical examples illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a neutral functional differential equation with multiple delays is considered. In a first step, we assumed some sufficient hypotheses to guarantee the existence of the Bogdanov–Takens and the triple-zero bifurcations. In a second step, the normal form of the two bifurcations is obtained by using the reduction on the center manifold and the theory of the normal form. Finally, we applied our study to a class of three-neuron bidirectional associative memory networks, its dynamic behaviors are studied and proved by an example and its numerical simulations. [ABSTRACT FROM AUTHOR]

We construct a delay functional d on an open subset of the space C r 1 = C 1 ([ - r , 0 ] , R) and find h ∈ (0 , r) so that the equation x ′ (t) = - x (t - d (x t)) defines a continuous semiflow of continuously differentiable solution operators on the solution manifold X = { ϕ ∈ C r 1 : ϕ ′ (0) = - ϕ (- d (ϕ)) } , and along each solution the delayed argument t - d (x t) is strictly increasing, and there exists a solution whose short segments x t , s h o r t = x (t + ·) ∈ C h 2 , t ≥ 0 , are dense in an infinite-dimensional subset of the space C h 2 . The result supplements earlier work on complicated motion caused by state-dependent delay with oscillatory delayed arguments. [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]

The first part of the paper is devoted to studying the continuous dependence of the solutions of Carathéodory constant delay differential equations where the vector fields satisfy classical cooperative conditions. As a consequence, when the set of considered vector fields is invariant with respect to the time-translation map, the continuity of the respective induced skew-product semiflows is obtained. These results are important for the study of the long-term behavior of the trajectories. In particular, the construction of semicontinuous semiequilibria and equilibria is extended to the context of ordinary and delay Carathéodory differential equations. Under appropriate assumptions of sublinearity, the existence of a unique continuous equilibrium, whose graph coincides with the pullback attractor for the evolution processes, is shown. The conditions under which such a solution is the forward attractor of the considered problem are outlined. Two examples of application of the developed tools are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical behaviors of neutral differential equations with small delays. We first establish the existence and smoothness of the global inertial manifolds for these equations. Then we further prove the smoothness of inertial manifolds with respect to small delays for a certain class of neutral differential equations. The method can be also applied to deal with more general small-delay systems. Finally, we apply our main results to the van der Pol oscillator model with small delay. [ABSTRACT FROM AUTHOR]

We study stability of so-called synchronous slowly oscillating periodic solutions (SOPSs) for a system of identical delay differential equations (DDEs) with linear decay and nonlinear delayed negative feedback that are coupled through their nonlinear term. Under a row sum condition on the coupling matrix, existence of a unique SOPS for the corresponding scalar DDE implies existence of a unique synchronous SOPS for the coupled DDEs. However, stability of the SOPS for the scalar DDE does not generally imply stability of the synchronous SOPS for the coupled DDEs. We obtain an explicit formula, depending only on the spectrum of the coupling matrix, the strength of the linear decay and the values of the nonlinear negative feedback function near plus/minus infinity, that determines the stability of the synchronous SOPS in the asymptotic regime where the nonlinear term is heavily weighted. We also treat the special cases of so-called weakly coupled systems, near uniformly coupled systems, and doubly nonnegative coupled systems, in the aforementioned asymptotic regime. Our approach is to estimate the characteristic (Floquet) multipliers for the synchronous SOPS. We first reduce the analysis of the multidimensional variational equation to the analysis of a family of scalar variational-type equations, and then establish limits for an associated family of monodromy-type operators. We illustrate our results with examples of systems of DDEs with mean-field coupling and systems of DDEs arranged in a ring. [ABSTRACT FROM AUTHOR]

This paper studies the local unstable manifold attached to an equilibrium solution of a system of delay differential equations (DDEs). Two main results are developed. The first is a general method for computing the formal Taylor series coefficients of a function parameterizing the unstable manifold. We derive linear systems of equations whose solutions are the Taylor coefficients, describe explicit formulas for assembling the linear equations for DDEs with polynomial nonlinearities. We also discuss a scheme for transforming non-polynomial DDEs into polynomial ones by appending auxiliary equations. The second main result is an a-posteriori theorem which—when combined with deliberate control of rounding errors—leads to mathematically rigorous computer assisted convergence results and error bounds for the truncated series. Our approach is based on the parameterization method for invariant manifolds and requires some mild non-resonance conditions between the unstable eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper we develop a general computer-assisted proof method for periodic solutions to delay differential equations. The class of problems considered includes systems of delay differential equations with an arbitrary number of (forward and backward) delays. When the nonlinearities include nonpolynomial terms we introduce auxiliary variables to first rewrite the problem into an equivalent polynomial one. We then apply a flexible fixed point technique in a space of geometrically decaying Fourier coefficients. We showcase the efficacy of this method by proving periodic solutions in the well-known Mackey–Glass delay differential equation for the classical parameter values. [ABSTRACT FROM AUTHOR]

We study the asymptotic dynamics of stochastic Young differential delay equations under the regular assumptions on Lipschitz continuity of the coefficient functions. Our main results show that, if there is a linear part in the drift term which has no delay factor and has eigenvalues of negative real parts, then the generated random dynamical system possesses a random pullback attractor provided that the Lipschitz coefficients of the remaining parts are small. [ABSTRACT FROM AUTHOR]

We study the stability of general n-dimensional nonautonomous linear differential equations with infinite delays. Delay independent criteria, as well as criteria depending on the size of some finite delays are established. In the first situation, the effect of the delays is dominated by non-delayed diagonal negative feedback terms, and sufficient conditions for both the asymptotic and the exponential asymptotic stability of the system are given. In the second case, the stability depends on the size of some bounded diagonal delays and coefficients, although terms with unbounded delay may co-exist. Our results encompass DDEs with discrete and distributed delays, and enhance some recent achievements in the literature. [ABSTRACT FROM AUTHOR]

In this paper, the well-posedness of stochastic time fractional 2D-Stokes equations of order α ∈ (0 , 1) containig finite or infinite delay with multiplicative noise is established, respectively, in the spaces C ([ - h , 0 ] ; L 2 (Ω ; L σ 2)) and C ((- ∞ , 0 ] ; L 2 (Ω ; L σ 2)) . The existence and uniqueness of mild solution to such kind of equations are proved by using a fixed-point argument. Also the continuity with respect to initial data is shown. Finally, we conclude with several comments on future research concerning the challenging model: time fractional stochastic delay 2D-Navier–Stokes equations with multiplicative noise. Hence, this paper can be regarded as a first step to study this challenging topic. [ABSTRACT FROM AUTHOR]

We develop validated numerical methods for the computation of Floquet multipliers of equilibria and periodic solutions of delay differential equations, as well as impulsive delay differential equations. Using our methods, one can rigorously count the number of Floquet multipliers outside a closed disc centered at zero or the number of multipliers contained in a compact set bounded away from zero. We consider systems with a single delay where the period is at most equal to the delay, and the latter two are commensurate. We first represent the monodromy operator (period map) as an operator acting on a product of sequence spaces that represent the Chebyshev coefficients of the state-space vectors. Truncation of the number of modes yields the numerical method, and by carefully bounding the truncation error in addition to some other technical operator norms, this leads to the method being suitable to computer-assisted proofs of Floquet multiplier location. We demonstrate the computer-assisted proofs on two example problems. We also test our discretization scheme in floating point arithmetic on a gamut of randomly-generated high-dimensional examples with both periodic and constant coefficients to inspect the precision of the spectral radius estimation of the monodromy operator (i.e. stability/instability check for periodic systems) for increasing numbers of Chebyshev modes. [ABSTRACT FROM AUTHOR]

In this paper, fractionally dissipative 2D quasi-geostrophic equations with an external force containing infinite delay is considered in the space H s with s ≥ 2 - 2 α and α ∈ (1 2 , 1) . First, we investigate the existence and regularity of solutions by Galerkin approximation and the energy method. The continuity of solutions with respect to initial data and the uniqueness of solutions are also established. Then we prove the existence and uniqueness of a stationary solution by the Lax–Milgram theorem and the Schauder fixed point theorem. Using the classical Lyapunov method, the construction method of Lyapunov functionals and the Razumikhin–Lyapunov technique, we analyze the local stability of stationary solutions. Finally, the polynomial stability of stationary solutions is verified in a particular case of unbounded variable delay. [ABSTRACT FROM AUTHOR]

We present a new approach to validated numerical integration for systems of delay differential equations. We focus on the case of a single constant delay though the method generalizes to systems with multiple lags. The method provides mathematically rigorous existence results as well as error bounds for both the solution and the Fréchet derivative of the solution with respect to a given past history segment. We use Chebyshev series to discretize the problem, and solve approximately using a standard numerical scheme corrected via Newton's method. The existence/error analysis exploits a Newton–Kantorovich argument. We present examples of the rigorous time stepping procedure, and illustrate the use of the method in computer-assisted existence proofs for periodic solutions of the Mackey–Glass equation. [ABSTRACT FROM AUTHOR]

The object of study is an autonomous impulsive system proposed as a model of drugs absorption by living organisms consisting of a linear differential delay equation and an impulsive self-support condition. We get a representation of the general solution in terms of the fundamental solution of the differential delay equation. The impulsive self-support generates periodic auto-oscillations given by fixed points of a first return map. [ABSTRACT FROM AUTHOR]

In this paper we prove that the following delay differential equation d dt x (t) = r x (t) 1 - ∫ 0 1 x (t - s) d s , has a periodic solution of period two for r > π 2 2 (when the steady state, x = 1 , is unstable). In order to find the periodic solution, we study an integrable system of ordinary differential equations, following the idea by Kaplan and Yorke (J Math Anal Appl 48:317–324, 1974). The periodic solution is expressed in terms of the Jacobi elliptic functions. [ABSTRACT FROM AUTHOR]

We construct a delay functional d Y on an infinite-dimensional subset Y ⊂ C 1 ([ - r , 0 ] , R) , r > 1 , so that the delay differential equation x ′ (t) = - α x (t - d Y (x t)) , α > 0 , has a solution whose short segments x t | [ - 1 , 0 ] are dense in C 1 ([ - 1 , 0 ] , R) . This implies complicated behaviour of the trajectory t ↦ x t in C 1 ([ - r , 0 ] , R) . [ABSTRACT FROM AUTHOR]

The paper is concerned with a broad family of scalar periodic delay differential equations with linear impulses, for which the existence of a positive periodic solution is established under very general conditions. The proofs rely on fixed point arguments, employing either the Schauder theorem or Krasnoselskii fixed point theorem in cones. The results are illustrated with applications to an impulsive hematopoiesis model or generalized Nicholson's equations, among other selected examples from mathematical biology. The method presented here turns out to be very powerful, in the sense that the derived theorems largely generalize and improve other results in recent literature, even for the situation without impulses. [ABSTRACT FROM AUTHOR]

Consider the delay differential equation x ˙ (t) = a ∫ 0 r x (t - s) d η (s) - g (x (t)) and the neutral type differential equation y ˙ (t) = a ∫ 0 r y ˙ (t - s) d μ (s) - g (y (t)) where a > 0 , g : R → R is smooth, u g (u) > 0 for u ≠ 0 , ∫ 0 s g (u) d u → ∞ as | s | → ∞ , r > 0 , η and μ are nonnegative functions of bounded variation on [ 0 , r ] , η (0) = η (r) = 0 , ∫ 0 r η (s) d s = 1 , μ is nondecreasing, μ does not have a singular part, ∫ 0 r d μ = 1 . Both equations can be interpreted as price models. Global asymptotic stability of y = 0 is obtained, in case a ∈ (0 , 1) , for the neutral equation by using a Lyapunov functional. Then this result is applied to get global asymptotic stability of x = 0 for the (non-neutral) delay differential equation provided a ∈ (0 , 1) . As particular cases, two related global stability conjectures are solved, with an affirmative answer. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in classical delay differential equations by using the Lyapunov–Perron method. The results are based on the rigorous functional analytic perturbation framework for dual semigroups (sun–star calculus). The generality of the dual perturbation framework ensures that the results extend to a much broader class of evolution equations. [ABSTRACT FROM AUTHOR]

Scalar difference equations x k + 1 = f (x k , x k - d) with delay d ∈ N are well-motivated from applications e.g. in the life sciences or discretizations of delay-differential equations. We investigate their global dynamics by providing a (nontrivial) Morse decomposition of the global attractor. Under an appropriate feedback condition on the second variable of f, our basic tool is an integer-valued Lyapunov functional. [ABSTRACT FROM AUTHOR]

This work focuses on the existence of quasi-periodic solutions for ordinary and delay differential equations (ODEs and DDEs for short) with an elliptic-type degenerate equilibrium point under quasi-periodic perturbations. We prove that under appropriate hypotheses there exist quasi-periodic solutions for perturbed ODEs and DDEs near the equilibrium point for most parameter values, then apply these results to the delayed van der Pol's oscillator with zero-Hopf singularity. [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]

We consider the question of when delay systems, which are intrinsically infinite dimensional, can be represented by finite dimensional systems. Specifically, we give conditions for when all the information about the solutions of the delay system can be obtained from the solutions of a finite system of ordinary differential equations. For linear autonomous systems and linear systems with time-dependent input we give necessary and sufficient conditions and in the nonlinear case we give sufficient conditions. Most of our results for linear renewal and delay differential equations are known in different guises. The novelty lies in the approach which is tailored for applications to models of physiologically structured populations. Our results on linear systems with input and nonlinear systems are new. [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]

In this paper, we use the parameterization method to construct quasi-periodic solutions of state-dependent delay differential equations. For example Under the assumption of exponential dichotomies for the $$\epsilon =0$$ case, we use a contraction mapping argument to prove the existence and smoothness of the quasi-periodic solution. Furthermore, the result is given in an a posteriori format. The method is very general and applies also to equations with several delays, distributed delays etc. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a method to prove existence of several rapidly and slowly oscillating periodic solutions of a delayed Van der Pol oscillator. The proof is a combination of pen and paper analytic estimates, the contraction mapping theorem and a computer program using interval arithmetic. Using this approach we extend some existence results obtained by Nussbaum (Ann Mat Pura Appl 4(101):263-306, 1974). [ABSTRACT FROM AUTHOR]

We exhibit a differential delay equation with state-dependent delay for which the familiar non-increasing 'oscillation speed' is defined and for which there exists an asymptotically stable rapidly oscillating periodic solution. [ABSTRACT FROM AUTHOR]

The first part of this paper is a general approach towards chaotic dynamics for a continuous map $$f:X\supset M\rightarrow X$$ which employs the fixed point index and continuation. The second part deals with the differential equation with state-dependent delay. For a suitable parameter $$\alpha $$ close to $$5\pi /2$$ we construct a delay functional $$d_{{\varDelta }}$$ , constant near the origin, so that the previous equation has a homoclinic solution, $$h(t)\rightarrow 0$$ as $$t\rightarrow \pm \infty $$ , with certain regularity properties of the linearization of the semiflow along the flowline $$t\mapsto h_t$$ . The third part applies the method from the beginning to a return map which describes solution behaviour close to the homoclinic loop, and yields the existence of chaotic motion. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of differential equations with state-dependent delays. We show differentiability of the solution with respect to the initial function and the initial time for each fixed time value assuming that the state-dependent time lag function is piecewise monotone increasing. Based on these results, we prove a nonlinear variation of constants formula for differential equations with state-dependent delay. As an application, we discuss asymptotic properties of perturbed nonlinear differential equations with state-dependent delays. [ABSTRACT FROM AUTHOR]

This is the third part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general theory, developed in the authors' paper Mierczyński and Shen (Trans Am Math Soc 365(10):5329-5365, 2013), to positive continuous-time random dynamical systems on infinite dimensional ordered Banach spaces arising from random parabolic equations and random delay systems. It is shown under some quite general assumptions that measurable linear skew-product semidynamical systems generated by random parabolic equations and by cooperative systems of linear delay differential equations admit measurable families of generalized principal Floquet subspaces, and generalized principal Lyapunov exponents. [ABSTRACT FROM AUTHOR]

We study asymptotic stability of a class of non-autonomous stochastic delay lattice systems driven by a multiplicative white noise. Under certain conditions, we prove such systems have a unique tempered complete quasi-solution which exponentially pullback attracts all solutions starting from a tempered random set. When the non-autonomous deterministic forcings are time-periodic, we obtain the existence, uniqueness and exponential stability of pathwise random periodic solutions for the stochastic lattice systems with delay. The convergence of the tempered complete quasi-solution (periodic solution) is also established when time delay approaches zero. [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]

We consider a nonlocal delayed reaction-diffusion equation in a semi-infinite interval that describes mature population of a single species with two age stages (immature and mature) and a fixed maturation period living in a spatially semi-infinite environment. Homogeneous Dirichlet condition is imposed at the finite end, accounting for a scenario that boundary is hostile to the species. Due to the lack of compactness and symmetry of the spatial domain, the global dynamics of the equation turns out to be a very challenging problem. We first establish a priori estimate for nontrivial solutions after exploring the delicate asymptotic properties of the nonlocal delayed effect and the diffusion operator. Using the estimate, we are able to show the repellency of the trivial equilibrium and the existence of a positive heterogeneous steady state under the Dirichlet boundary condition. We then employ the dynamical system arguments to establish the global attractivity of the heterogeneous steady state. As a byproduct, we also obtain the existence and global attractivity of the heterogeneous steady state for the bistable evolution equation in the whole space. [ABSTRACT FROM AUTHOR]

The paper deals with a class of cooperative functional differential equations (FDEs) with infinite delay, for which sufficient conditions for persistence and permanence are established. Here, the persistence refers to all solutions with initial conditions that are positive, continuous and bounded. The present method applies to a very broad class of abstract systems of FDEs with infinite delay, both autonomous and non-autonomous, which include many important models used in mathematical biology. Moreover, the hypotheses imposed are in general very easy to check. The results are illustrated with some selected examples. [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]

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]

The analysis of general linear variable coefficient delay differential-algebraic systems (DDAEs) is presented. The solvability for DDAEs is investigated and a reformulation procedure to regularize a given DDAE is developed. Based on this regularization procedure existence and uniqueness of solutions and consistency of initial functions is analyzed as well as other structural properties of DDAEs like smoothness requirements. We also present some examples to demonstrate that for the numerical solution of a DDAE, a reformulation of the system before applying numerical methods is essential. [ABSTRACT FROM AUTHOR]

In this paper we study the discretization effects on the Takens-Bogdanov bifurcation of delay differential equations by forward Euler scheme. We show that the Takens-Bogdanov point is inherited by the discretization without any shift and turns into a 1:1 resonance point. The normal form on the center manifold near this singular point of the forward Euler method is calculated by applying a new technique, which is developed in this work for a general class of parameterized maps. The local dynamical behaviors are investigated in detail through this normal form. We show that the bifurcated Hopf point branch and the homoclinic branch of the numerical method are $$O(h)$$ close to their continuous counterparts, where $$h$$ is stepsize. A numerical experiment is presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

For a Nicholson's blowflies system with patch structure and multiple discrete delays, we analyze several features of the global asymptotic behavior of its solutions. It is shown that if the spectral bound of the community matrix is non-positive, then the population becomes extinct on each patch, whereas the total population uniformly persists if the spectral bound is positive. Explicit uniform lower and upper bounds for the asymptotic behavior of solutions are also given. When the population uniformly persists, the existence of a unique positive equilibrium is established, as well as a sharp criterion for its absolute global asymptotic stability, improving results in the recent literature. While our system is not cooperative, several sharp threshold-type results about its dynamics are proven, even when the community matrix is reducible, a case usually not treated in the literature. [ABSTRACT FROM AUTHOR]

We discuss ordinary differential equations with delay and memory terms in Hilbert spaces. By introducing a time derivative as a normal operator in an appropriate Hilbert space, we develop a new approach to a solution theory covering integro-differential equations, neutral differential equations and general delay differential equations within a unified framework. We show that reasonable differential equations lead to causal solution operators. [ABSTRACT FROM AUTHOR]

We indicate how a Hopf bifurcation theorem for retarded functional differential equations can be proved correctly, following an approach that is described, but not carried out properly in the literature. Then we state and prove a Hopf bifurcation theorem for general semiflows in Banach spaces, using different methods. The general statement includes the case of retarded functional differential equations. [ABSTRACT FROM AUTHOR]