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Functional differential equations with state-dependent delay: Theory and applications
... State-dependent delay differential equations (SDDEs) play an important role in the nonlinear dynamics of biological and engineering systems. This class of equations occurs naturally when the time consumed in a given compartment is not constant, but instead is determined by the need that a constant threshold quantity is accumulated during a period of time in that compartment [Hartung et al., 2006]. ...
... Due to the nonsmoothness of solutions of SDDEs in terms of initial conditions or parameters [Walther, 2004], SDDEs can be difficult to analyze and the theory of constant delay differential equations cannot be entirely extended to cover this class of differential equations. For example, applying the Lyapunov-Perron method and the variation of constants formula in [Diekmann et al., 1995], Hartung et al. [2006] proved the existence of local center manifolds which is shown by to be continuously differentiable. Using invariant manifolds of time-t-maps and their invariance properties with respect to the semiflow, Qesmi and Walther [Qesmi & Walther, 2009] proved the existence of continuously differentiable local center-stable manifolds. ...
... As a consequence, local bifurcations for SDDEs, related to singularities of codimension-two or higher, are still hard to be analytically located since they require C k -smooth local center manifolds with k > 1. More considerable efforts have been done recently to overcome many difficulties in the theory of SDDEs [Krisztin, 2003;Walther, 2004;Hartung et al., 2006;Qesmi & Walther, 2009;Hu & Wu, 2010;Mallet-Paret & Nussbaum, 2011;Stumpf, 2012;Sieber, 2012;Getto et al., 2019;Calleja et al., 2017;Eichmann, 2006]. Moreover, the established theoretical results have been used with some success in many fields of application. ...
Dynamic behavior investigations of infectious disease models are central to improve our understanding of emerging characteristics of model states interaction. Here, we consider a Susceptible-Infected (SI) model with a general state-dependent delay, which covers an immuno-epidemiological model of pathogen transmission, developed in our early study, using a threshold delay to examine the effects of multiple exposures to a pathogen. The analysis in the previous work showed the appearance of forward as well as backward bifurcations of endemic equilibria when the basic reproductive ratio R0 is less than unity. The analysis, in the present work, of the endemically infected equilibrium behavior, through the study of a second order exponential polynomial characteristic equation, concludes the existence of a Hopf bifurcation on the upper branch of the backward bifurcation diagram and gives the criteria for stability switches. Furthermore, the inclusion of state-dependent delays is shown to entirely change the dynamics of the SI model and give rise to rich behaviors including periodic, torus and chaotic dynamics.
... Although to the above, most of the theory on SDD differential equations is regarding non-neutral differential equations. Concerning non-neutral equations in finite dimensional spaces, we mention the survey by Hartung, Krisztin, Walther and Wu [21], the recent papers by Walther the [52,53,54] and the references therein. Concerning non-neutral PDEs with SDD and non-neutral ODE's in infinite dimensional space, we cite the interesting papers by Kosovalic et al. [36,37], Rezounenko et al. [38], Yunfei et al. [47] and our recent contributions [27,28,29,30,31]. ...
... The literature on explicit neutral differential equations with state-dependent delay (next ENDE-SDD), with the exception of the recent pioneer Hernandez paper [28], is formed by works on ODE's in finite dimensional spaces. About this matter, we cite the pioneer Driver's papers [11,12,13], the interesting works [14,15,18,19,20,21,35,44] and the papers by Hartung [23] and Walther [55] concerning Well-Posedness. The study of abstract ENDE-SDD in infinite dimensional spaces with applications to PDE's is restrict to the recent work Hernandez [28], where the local and global existence and uniqueness of solution for the problem u (t) = Au(t) + F (t, u(t), t 0 K(t, τ )u (σ(τ, u(τ )))dτ ), t ∈ [0, a], (1.3) is studied assuming A, F (·), K(·) and σ(·) as above. ...
... As pointed out in [28], our interest in the study of abstract ENDE-SDD is motivated by the possibility of create a new and non-trivial research field; to extend and generalize the models and results in the early papers on ENDE-SDD in finite dimensional spaces [11,12,13,14,15,18,19,20,21,35,44] and by some recent works about: explicit neutral ODE's, see [3,4,5], explicit neutral PDE's, see [8,25,26], ordinary neutral equations in population dynamic, see [16,17,39,40,41,43,44,51], neutral equations in economy, see [6,7], and numerical approximation of solution of neutral equations, see [14,15,20,49,50]. In particular, we note that the explicit neutral models u (t) = ∆u(t) + f (t, u t , u (σ(t, u(t)))), u (t) = ∆u(t) + G(t, u t ) + t t−τ (t) K(t, s, u(s), u (σ(s, u(s)))ds, u (t) = Au(t) + u(t) 1 − t 0 β(t, s)u (σ(s, u(s)))ds , are natural generalizations of the neutral problems studied in [6,7,16,17,25,26,39,40,43], [49,50] and [40,41], respectively. ...
... We refer to the works of Hernández in [1][2][3] for g = 0 and [4] for g ≠ 0. Morales et al. [5] proved the existence of mild solution for (1) with infinite delay and 1 (t, u t ) = t with A generating a compact analytic semigroup. SDD differential equations have applications in population growth models, two-body problem in electrodynamics, delay adoption in neutral networks, hematological disorders, mathematical epidemiology, and queuing processes; see [6][7][8][9][10] and the references therein for a brief of natural processes, which are formulated via SDD equations. For SDD differential equations in finite dimension, we refer to [9,[11][12][13][14]. ...
... SDD differential equations have applications in population growth models, two-body problem in electrodynamics, delay adoption in neutral networks, hematological disorders, mathematical epidemiology, and queuing processes; see [6][7][8][9][10] and the references therein for a brief of natural processes, which are formulated via SDD equations. For SDD differential equations in finite dimension, we refer to [9,[11][12][13][14]. For abstract evolution equations, we refer to [1, 3-5, 15, 16]. ...
... Periago et al. [36] proved the following result for the existence of classical solution to (9). ...
We establish the existence and uniqueness of mild solutions of a first‐order neutral differential equation with state‐dependent delay and almost sectorial operator by using the theory of almost sectorial operators and fixed point theorems and show that under certain conditions mild solution becomes the classical solution. Later, we also discuss the local well‐posedness of the associated Cauchy problem.
... Since the sixties [31], it is well known that many concrete systems can be appropriately described by retarded functional differential equations with infinite state-dependent delay. We refer the reader to the chapter of Hartung et al [20] for many examples and models which can be better described by state-dependent delay equations. Due to this reason, the theory of functional differential equations with state-dependent delays has attracted the attention of several authors from the past decades. ...
... A relation between state-dependent retarded delay equations and retarded equations for the case r (t) = t was already established previously in the literature (see [20] and the references therein). However, the relation presented here allows us to consider more general conditions concerning the regularity of the solutions and the involved spaces. ...
... It follows from this expression that (T (t)) t≥0 is a compact semigroup with T (t) ≤ 1 for all t ≥ 0. We study the problem (1.3)-(1.5) in the interval [0, a]. In [20,31] there are many examples of functions ρ that arise in concrete state-dependent delay problems. We consider here the function ρ(t, ψ) = t − p(t) − q(ψ), where p : [0, a] → R + is a regulated function and q : B → R + is a bounded continuous function. ...
This paper is devoted to the investigation of the existence of mild solutions of abstract retarded functional differential equations with infinite state-dependent delay. We obtain the result concerning the existence of mild solutions for the equations with state-dependent delays as a fixed point of the solution operator of an associated abstract retarded functional differential equation with time-varying delays. We are concerned with equations that present a phenomenon of lacunary memory. We apply our results to study the existence of solutions of a state-dependent partial differential equation with infinite state-dependent delay.
... As a result, even if we assume that u stays bounded away from −1 and +∞ such that the delay 1 + u(t − b) of system (3) stays in a finite interval [τ min , τ max ] with 0 < τ min < τ max < ∞, the flow does not possess the regularity properties one would expect given that all terms of the right-hand side of (4) are nested linear functions. Rather, functionals f of state-dependent DDEs satisfy a weaker condition [11], sometimes called mild differentiability [11,42,43]. With the help of this concept, DDEs of type (4) were proven in [51] to be C 1 -regular (that is, once differentiable) dynamical systems on the phase space (manifold) of compatible initial conditions ...
... As a result, even if we assume that u stays bounded away from −1 and +∞ such that the delay 1 + u(t − b) of system (3) stays in a finite interval [τ min , τ max ] with 0 < τ min < τ max < ∞, the flow does not possess the regularity properties one would expect given that all terms of the right-hand side of (4) are nested linear functions. Rather, functionals f of state-dependent DDEs satisfy a weaker condition [11], sometimes called mild differentiability [11,42,43]. With the help of this concept, DDEs of type (4) were proven in [51] to be C 1 -regular (that is, once differentiable) dynamical systems on the phase space (manifold) of compatible initial conditions ...
... with the arbitrarily often mildly differentiable right-hand side given by f ; see [11,42,43] for the definition of mild differentiability. We use the convention that C k = C k ([−τ max , 0; R) is the space of k times continuously differentiable functions on [−τ max , 0]. ...
We study a scalar, first-order delay differential equation (DDE) with instantaneous and state-dependent delayed feedback, which itself may be delayed. The state dependence introduces nonlinearity into an otherwise linear system. We investigate the ensuing nonlinear dynamics with the case of instantaneous state dependence as our starting point. We present the bifurcation diagram in the parameter plane of the two feedback strengths showing how periodic orbits bifurcate from a curve of Hopf bifurcations and disappear along a curve where both period and amplitude grow beyond bound as the orbits become saw-tooth shaped. We then `switch on' the delay within the state-dependent feedback term, reflected by a parameter . Our main conclusion is that the new parameter b has an immediate effect: as soon as the bifurcation diagram for b=0 changes qualitatively and, specifically, the nature of the limiting saw-tooth shaped periodic orbits changes. Moreover, we show numerically and with a normal form analysis that a degeneracy at b=1/3 of an equilibrium with a double real eigenvalue zero leads to a further qualitative change and acts as an organizing center for the bifurcation diagram. Our results demonstrate that state dependence in delayed feedback terms may give rise to new dynamics and, moreover, that the observed dynamics may change significantly when the state-dependent feedback depends on past states of the system. This is expected to have implications for models arising in different application contexts, such as models of human balancing and conceptual climate models of delayed action oscillator type.
... is only continuous, not locally Lipschitz. A result on existence of solutions, uniqueness, and differentiability with respect to initial data which applies to equations with state-dependent delay is the following [19,4]. ...
... (8)- (11) with E + in place of E hold on (γ − , β), and E + (u) = φ 1 on (γ − , β 1 ). 4. Define E by E(u) = E − (u) on (α, γ] and E(u) = E + (u) on (γ, β). ...
... Recall Proposition 4.3. By the results from [19,4] quoted in Section 1 it remains to show X f = ∅. We have U = ∅. ...
Differential equations with state-dependent delays which generalize the scalar example(0)x′(t)=g(x(t),x(t−d(xt))) where g:R2→R and d:C([−r,0],R)→[0,r] are continuously differentiable, and with xt:[−r,0]→R given by xt(s)=x(t+s), define semiflows of differentiable solution operators on an associated submanifold of the state space C1=C1([−r,0],Rn). When written in the general formx′(t)=f(xt) with a map f:C1⊃U→Rn then the associated manifold isXf={ϕ∈U:ϕ′(0)=f(ϕ)}. We obtain results on the nature of Xf.
•If all delays in the system are bounded away from zero then a projection C1→C1 onto the subspaceH={ϕ∈C1:ϕ′(0)=0} defines a diffeomorphism of Xf onto an open subset of H. In other words, Xf is a graph over H.
•There exist g and d with d(ϕ)>0 everywhere and infd=0 so that the manifold Xf associated with Eq. (0) does not admit any graph representation.
•If all delays in the system are strictly positive (but not necessarily bounded away from zero) then Xf has an “almost graph representation” which implies that it is diffeomorphic to an open subset of H.
... Over the past several years, dynamical systems with state-dependent delays have been applied in many fields, such as electrodynamics [Driver, 1984], population growth [Alello et al., 1992], economics [Bélair & Mackey, 1989], engineering [Hartung et al., 2006], and so on. The analysis of SD-DDEs is associated with a number of challenging issues because of the weak smoothness of the righthand side of systems. ...
... After the analysis of its existence in system (2), the stability of the nontrivial equilibrium (x 0 , y 0 ) is explored as follows. It is obvious that the linearized equation at the nontrivial equilibrium (x 0 , y 0 ) can be obtained using formal linearization [Cooke & Huang, 1996;Hartung et al., 2006] as follows, ...
... The linearized equation at the nontrivial equilibrium (x 0 , y 0 ) is obtained by formal linearization [Cooke & Huang, 1996;Hartung et al., 2006] as follows, ...
The reaction time delay in the transcription process depends on the concentration of the protein because the transportation of mRNA from the nucleus to the cytoplasm becomes saturated. Thus the gene regulatory network is a state-dependent delayed model. This study aims to provide some mathematical explanations for the dynamics of the system, such as the linear stability and periodic oscillation, using mathematical techniques, such as formal linearization, linear stability analysis, the method of multiple scale (MMS), and the normal form. First, Hopf bifurcation of the state-dependent delayed gene regulatory networks model in the gene expression is analyzed by the method of multiple scales (MMS). Mechanism of periodic oscillations is obtained by Hopf bifurcation. The findings show that when degradation effects of the mRNA and protein are very strong, the oscillatory gene expression disappears. Then, a more realistic version of the aforementioned model with both constant and state-dependent time delays is established due to the existence of the constant time delay in the protein degradation process. Its nonresonant double Hopf bifurcation is found and analyzed using MMS. Interesting complex dynamic phenomena, such as periodic, quasi-periodic, and global period-2 solutions, are also discovered. These observations indicate that both state-dependent delay and constant delay could induce richer dynamics of the system, and the modified model may potentially describe the real dynamical mechanism (both the transcription process and the degradation process) more accurately in the gene expression. The findings may provide important guidance or hints to understand the real dynamic mechanism of the gene expression process.
... We performed the continuation of three different objects: equilibria, periodic solutions, and Hopf bifurcation curves. We considered as test example the stem cell model with maturation rate (27) and parameters as in Fig. 6 (right). We refer to [4,24] for the details of the pseudospectral approach applied to the delay formulation of the model. ...
... For the maturation rate (27), which is positive and independent of x, the equilibrium distribution m is a linear function of x. Numerical simulations (not included here) show that, for fixed parameter values, the equilibrium values m, w and v are approximated to the tolerance imposed to the software already for M = 1. ...
... In this spirit, B. de Wolff, S.M. Verduyn Lunel, FS and OD investigate in work in progress the convergence of the normal form coefficient of a Hopf bifurcation for delay differential equations. Wishful thinking suggests to go one step beyond and derive properties of the infinite dimensional system by passing to the limit in results for the approximation (note that both models considered here display a form of state-dependent delay and that the theory of state-dependent delay equations is technically demanding because of smoothness problems [27]; since polynomials are C ∞ , the approximation eliminates the smoothness problem; the technically demanding step is now to pass to the limit). ...
Physiologically structured population models are typically formulated as a partial differential equation of transport type for the density, with a boundary condition describing the birth of new individuals. Here we develop numerical bifurcation methods by combining pseudospectral approximate reduction to a finite dimensional system with the use of established tools for ODE. A key preparatory step is to view the density as the derivative of the cumulative distribution. To demonstrate the potential of the approach, we consider two classes of models: a size-structured model for waterfleas (Daphnia) and a maturity-structured model for cell proliferation. Using the package MatCont, we compute numerical bifurcation diagrams, like steady-state stability regions in a two-parameter plane and parametrized branches of equilibria and periodic solutions. Our rather positive conclusion is that a rather low dimension may yield a rather accurate diagram! In addition we show numerically that, for the two models considered here, equilibria of the approximating system converge to the true equilibrium as the dimension of the approximating system increases; this last result is also proved theoretically under some regularity conditions on the model ingredients.
... In the case that the delay is not constant and depends on the state (the unknown function), they are commonly called the state-dependent delay differential equations (SDDEs), which appear in many applications including electrodynamics, control theory, biology, neuroscience, and economics. See Hartung et al. [17] and the references therein for a comprehensive survey on the theory and application. It is also shown in Calleja et al. [4] that even a simple SDDE would exhibit very complicated dynamics, while the dynamics are trivial if the delays therein are made constant. ...
... For instance, to study the local behavior near the equilibrium, one usually resorts to the center manifold theorem. The trouble for SDDEs is that the high regularity of the center manifold is still unknown [17]. ...
We consider the state-dependent delay differential equation (SDDE) obtained by adding delayed perturbation to a one-dimensional ODE with a degenerate equilibrium. We prove the existence of the response solution of the equation, i.e., the quasi-periodic solution with the same frequency as the forcing. The novelty of our paper is to provide a concrete example to discuss the smoothness issues of SDDE, especially showing the analyticity of quasi-periodic solutions in some probability sense.
... The theory of abstract differential equations with state-dependent delay is a field of intensive research because of many applications (see, [11]) and by the fact that the qualitative theory is different to the usual ones on functional differential equations with delay. The main difference arises from the fact that functions of the form u → u(· − σ(·, u (·) )) and u → u µ(·,u (·) ) are not Lipschitz on spaces of continuous functions, which have obvious implications concerning the problem of the existence and "uniqueness" of solution and the well-possedness of ODE's, PDE's, abstract differential equations and abstract fractional differential equations with state ...
... The literature on state-dependent delay problems is extensive. For the case of ODEs on finite dimensional spaces, we note the early papers by Driver [8,9], the excellent survey by Hartung, Krisztin, Walther & Wu [11] and the references therein. For ODEs on infinite dimensional spaces and PDEs, we cite our recent contributions Hernandez et. ...
We study the local and global existence and uniqueness of mild solution for a general class of abstract differential equations with state-dependent argument. In the last section, some examples on partial differential equations with state-dependent argument are presented.
... Differential equations with state-dependent delay (SDD) are well known in modeling many practical problems, and for this reason, the study of this type equations have gain more attention in recent years [16][17][18][19][20]. It should be pointed out that Muthukumar and Rajivganthi [17] investigated the following impulsive neutral stochastic functional differential system with SDD: ...
... This paper is organized as follows. In the next section, we present some notation and preliminaries adopted from [2,8,9,12,16,18,29,32,[39][40][41][42][43][44][45][46]. Section 3 is devoted to the approximate controllability of Eq. (1.6). ...
We consider the approximate controllability for a class of second-order impulsive neutral stochastic differential equations with state-dependent delay and Poisson jumps in a real separable Hilbert space. Under the sufficient conditions, we obtain approximate controllability results by virtue of the theory of a strongly continuous cosine family of bounded linear operators combined with stochastic inequality technique and the Sadovskii fixed point theorem. Finally, we illustrate the main results by an example.
... Differential equations with state-dependent delay are notorious for their lack of smoothness properties and the wealth of associated open problems pertaining to the semiflow and their invariant manifolds. See [19,25] for background. As for the associated Cauchy problem, continuity of the initial condition is typically not enough to ensure uniqueness of solutions, as was demonstrated by the classical counterexample of Winston [42]. ...
... by (19). Since ||ϕ||e rJ(1+ξ −1 ) K 0 < min{ζ, δ, ν} by (20), it follows that ||x t || < min{ζ, δ, ν}. ...
We prove that under fairly natural conditions on the state space and nonlinearities, it is typical for an impulsive differential equation with state-dependent delay to exhibit non-uniqueness of solutions. On a constructive note, we show that uniqueness of solutions can be recovered using a Winston-type condition on the state-dependent delay. Irrespective of uniqueness of solutions, we prove a result on linearized stability. As a specific application, we consider a scalar equation on the positive half-line with continuous-time negative feedback, non-negative state-dependent delayed nonlinearity and impulse effect functional satisfying affine bounds.
... The theory of state-dependent delay differential equations (next, SDD-DE) is a field of intense research motivated by many applications in different E. Hernandez et al. MJOM fields (see [14] and the references therein) and by the remarkable fact that the qualitative theory is different and more complex than the other theories on functional differential equations with delay. The main difference with other theories is the non-Lipschizianity of functions of the form u → u(·−σ(·, u(·))), u → u(· − σ(·, u (·) )) and u → u σ(·,u (·) ) in spaces of continuous functions, which has important implications concerning the study of the existence and "uniqueness" of solutions, the well-posedness and the asymptotical behavior of solutions. ...
... Concerning the associated literature on state-dependent delay differential equations, for the case of ODE's on finite-dimensional spaces, we cite the pioneering work by Driver [9], the interesting papers by Büger and Martin [5], the papers [10,12,13], the excellent work by Aiello, Freedman and Wu [1], the survey by Hartung, Krisztin, Walther and Wu [14] and the references therein. In addition, for abstract differential equations on infinite-dimensional space and PDE's with state-dependent delay, we cite the papers by Kosovalic et al. [22,23], our recent contributions Hernandez et al. [15][16][17][18][19] and Yunfei et al. [24,25]. ...
We study the existence and uniqueness of solutions for a general class of abstract ordinary integro-differential equation with state-dependent delay. In the last section, some examples arising in the population dynamics and in the Solow’s theory of economic growth are presented.
... Differential equations with state-dependent delay are notorious for their lack of smoothness properties and the wealth of associated open problems pertaining to the semiflow and their invariant manifolds. See [19,25] for background. As for the associated Cauchy problem, in order to guarantee uniqueness of solutions it is necessary to impose at the very least absolute continuity [15] or, barring that, Lipschitz [2,33,42] continuity of the initial data. ...
... by (19). Since ||φ||e rJ(1+ξ −1 ) K 0 < min{ζ, δ, ν} by (20), it follows that ||x t || < min{ζ, δ, ν}. ...
We prove that under fairly natural conditions on the state space and nonlinearities, it is typical for an impulsive differential equation with state-dependent delay to exhibit non-uniqueness of solutions. On a constructive note, we show that uniqueness of solutions can be recovered using a Winston-type condition on the state-dependent delay. Irrespective of uniqueness of solutions, we prove a result on linearized stability. As a specific application, we consider a scalar equation on the positive half-line with continuous-time negative feedback, non-negative state-dependent delayed nonlinearity and impulse effect functional satisfying affine bounds
... This reduction eliminates a priori the infinitely many exponentially decaying parts of the solutions, leaving behind the slowly evolving parts on the center manifold. However, the calculation of the center manifold reduction is very complicated, and especially it may not be valid for the systems with the state-dependent time delays [Hartung et al., 2006]. So in this paper, we investigate the Hopf bifurcation of the wireless network congestion model with state-dependent round trip delay by the numerical method, i.e. the bifurcation analysis software, DDE-BIFTOOL, but not the analytical method [Engelborghs et al., 2002]. ...
... It is a very difficult problem to determine if the Hopf bifurcation is supercritical or subcritical for the state-dependent delayed differential equation. Though the local center manifold theorem and local Hopf bifurcation theorem have been obtained [Hartung et al., 2006], there was very little work on the determination of the supercriticality or subcriticality of its Hopf bifurcation. The perturbation method can determine the bifurcation direction successfully [Hou & Guo, 2015]. ...
In this paper, the bifurcation analysis software, DDE-BIFTOOL, is employed to analyze the Hopf bifurcation of the wireless network congestion model with state-dependent round trip delay. Hopf bifurcations are investigated for the four typical work conditions. The corresponding stable and unstable bifurcating periodic solutions are quantitatively and qualitatively verified by nonlinear simulation software, WinPP, respectively, which agree with those of DDE-BIFTOOL very well. The results imply that the channel loss probabilities Pul and Pdl can play a more important role than the speed of the network, i.e. the related link bandwidth C. For larger Pul and Pdl in Cases 3 and 4, the smaller Tp and K can induce Hopf bifurcation. This will result in the loss of stability and performance degradation. So Pul and Pdl should be set smaller to avoid congestion, providing a sound theoretical basis and instructions for the congestion control of the wireless network.
... The study of state-dependent delay equations is motivated by applications and because the qualitative theory is different from the theory of equations with constant and time-dependent delay. For ODEs, we cite the early work by Aiello et al. [2], the survey by Hartung et al. [17], the papers by Hartung et al. [15], Walther [51] and the references therein. Also, for ODEs we cite the recent papers by Li and Wu [39,40]. ...
... (see [17,32,34]) where C i are fixed positive real numbers. From [32] we know that σ(·) is a C 1 function, and then, locally Lipschitz. ...
In this paper, we study the existence and uniqueness of mild and strict solutions for abstract neutral differential equations with state-dependent delay. In the last section, some examples are presented.
... Functional differential equations with state-dependent delay (SDD, in short) have been frequently used as practical models in many applied sub-jects, such as electrodynamics, automatic and remote control, machine cutting, neural networks, population biology, mathematical epidemiology and economics, and for this reason such equations have been investigated widely in the past years [7,13,16,[21][22][23]. On the other hand, controllability has played an important role in both deterministic and stochastic control systems throughout the history of modern control theory. ...
... This paper is organized as follows. In the next section, we present some notation and preliminaries adopted from [1,6,9,10,21,24,[28][29][30][31][32][33][34][35]. Section 3 is devoted to the existence of mild solutions to Eq. (1.7). ...
This paper is considered with the existence and controllability of mild solutions for a class of second-order neutral impulsive stochastic evolution integro-differential equations with state-dependent delay in a real separable Hilbert space.Under the sufficient conditions, the existence and controllability of mild solutions are obtained by means of the fixed point techniques. Finally, an example is given to illustrate the main results. © 2018, Springer International Publishing AG, part of Springer Nature.
... The differences arises from two simples facts, the lack of Lipschitz continuity of functions of the form ↦ (⋅, (⋅) ) , ↦ ( (⋅, (⋅))) and the nonlinear behavior of functions as ↦ ( , ) and ↦ ( ( , ( ))). Concerning ODE's with SDD we site the pioneer papers [7,8], the early papers [9,13], the interesting survey [15], and references therein. For the case of PDEs and abstract ODEs in infinite-dimensional space, we mention the interesting works [25,27,30] and our recent contributions [17-20, 22, 23]. ...
In this work, we study a new class of integro‐differential equations with delay, where the informations from the past are represented as an average of the state over state‐dependent integration intervals. We establish results on the local and global existence and qualitative properties of solutions. The models presented and the ideas developed will allow the generalization of an extensive literature on different classes of functional differential equations. The last section presents some examples motivated by integro‐differential equations arising in the theory of population dynamics.
... In 2006, Hartung et.al. [7] and Krisztin [11] obtained the existence result for continuously differentiable local stable and center manifolds at stationary points. Based on the above results, Qesmi et al. [24] confirmed that the local center-stable manifold for differential equations is continuously differentiable. ...
In this paper, we study the impulsive evolution equation with state-dependent delay by the theory of operator semigroup in Banach spaces. Under conditions that both nonlinearity and impulsive functions are Lipschitz continuous, we obtain the existence and uniqueness results of mild solution. Furthermore, we prove the differentiability of a semi-flow defined by a continuously differentiable solution operator under the appropriate condition.
... Delay-differential equations (DDEs) are important mathematical models in many application areas, including optics [1][2][3][4][5], physiology and infectious disease modeling [6][7][8][9][10][11], mechanics [12][13][14][15][16], neuroscience [17][18][19], and others. More recently, DDEs have become a focus in some areas of machine learning, such as reservoir computing [20][21][22][23][24][25] or deep neural networks [26,27]. ...
We show that delay-differential equations (DDE) exhibit universal bifurcation scenarios, which are observed in large classes of DDEs with a single delay. Each such universality class has the same sequence of stabilizing or destabilizing Hopf bifurcations. These bifurcation sequences and universality classes can be explicitly described by using the asymptotic continuous spectrum for DDEs with large delays. Here, we mainly study linear DDEs, provide a general transversality result for the delay-induced bifurcations, and consider three most common universality classes. For each of them, we explicitly describe the sequence of stabilizing and destabilizing bifurcations. We also illustrate the implications for a nonlinear Stuart-Landau oscillator with time-delayed feedback.
... Due to the influence of circumstances, however, it may be more appropriate who depend on the solution of the governing differential equation than the constant time one. We call such a delay a state-dependent delay [9,8,11,17]. Functional differential equations with state-dependent delays (sd-FDEs) arise in various applications [1,14,12,13]. ...
In this paper, we consider a Van der Pol oscillator with state-dependent delay. Firstly, we seek for a suitable set E on which we construct a Poincar\'e operator F. Then, we prove that the operator F is continuous and compact on . After that, we obtain the ejectivity of the trivial fixed point and the existence of slowly oscillating periodic solution. Finally, we choose an example of state-dependent delay to test our analytical result by comparing it to numerical continuation.This theoretical result generalizes and improves some existing results.
... The theory of differential equations with state-dependent delay has received great attention because they appear frequently in applications as model of equations and the fact that the qualitative theory is quite different from the theories of equations with constant and time-dependent delay. Related to the problems of the first-order abstract differential equations, we refer to [14][15][16][17], the survey by Hartung, Krisztin, Walther and Wu [18]. Recently,in [19], the authors studied the existence of strict solutions to a class of differential equations with state dependent nonlocal conditions. ...
In this paper, we focus on the existence of strict solutions to a class of differential equations with state-dependent nonlocal conditions. Using fixed point theorems and general Gronwall inequality, we give some new conclusions. Moreover, an example is given to illustrate our conclusions valuable.
... Hartung in [7] obtained the differentiability of neutral differential equations (NDEs) with respect to initial data and parameters. The differentiability with respect to initial data is needed in order to guarantee the existence of a C 1 smooth solution semiflow, see [8]. Furthermore, problems of differentiability of stochastic differential equations (SDEs) are also studied, see [2,10]. ...
... For example, Stein's model is a well-known space dependent model which represents a commonly-used description of spontaneous neuronal activity [1]. Many other examples can be found in [2,3]. Extensive numerical methods have been developed for singularly perturbed delay differential equations, such as [4][5][6][7] and references therein. ...
A singularly perturbed delay parabolic problem of convection-diffusion type with a discontinuous convection coefficient and source term is examined. In the problem, strong interior layers and weak boundary layers are exhibited due to a large delay in the spatial variable and discontinuity of convection coefficient and source. The problem is discretized by a nonstandard finite difference scheme in the spatial variable and for the time derivative, we used the Crank–Nicolson scheme. To enhance the order of convergence of the spatial variable, the Richardson extrapolation technique is applied. The error analysis of the proposed scheme was carried out and proved that the scheme is uniformly convergent of second order in both spatial and temporal variables. Numerical experiments are performed to verify the theoretical estimates.
... The theory of state-dependent delay (SDD) differential equations is a field of intense research because they arise naturally in applications and the qualitative theory is different from the theory of differential equations with constant and time-dependent delay. Concerning first order ODEs on finite dimensional spaces, we mention the survey [5]. For first order ordinary abstract differential equations and first order on time partial differential equations, we cite the early paper by Hernández et al. [11] and the recent interesting works [12][13][14][15][17][18][19][20]. ...
We study the global existence and uniqueness of solutions and wellposedness of a general class of abstract second order differential equations with state dependent delay. Some examples are presented.
... This type of time delay is downstream-order controlled state-dependent delay. See [26,27] for recent reviews on differential equations with state-dependent delays. ...
We develop a model of differential equations for a supply network with two suppliers in each level and with state-dependent delivery time delays between every two adjacent levels. With the supply network model, we investigate the bullwhip phenomenon and the stabilities of the equilibrium state of order placements, and further delineate factors which can cause onset of periodic oscillations of the order placement decisions. In particular, the impacts of the disruptions are analyzed when a supplier is dysfunctional. We show that if the supply-demand matrix between every two adjacent levels is a Markov matrix, the supply network is most robust with respect to disruptions. Numerical simulations are given to illustrate Hopf bifurcation and the impacts of disruptions on periodic oscillations in the supply network.
... We note some papers (see [Aiello et al., 1992;Hartung et al., 2006;Hernandez et al., 2016;Hernandez et al., 2020;Kosovalic et al., 2013;Liu, 1999;Walther, 2008]) in which the problem of the behavior of solutions of equations with delay that depend on an unknown function was investigated. In this article, it is assumed that ϕ(v) and f (v) are sufficiently smooth functions. ...
The article considers the question of local dynamics of the logistic equation with delays and the delay coefficients being the nonlinear functions. It is supposed that one of the parameters, characterizing the delay value, is sufficiently large. The stability criterion is formulated and critical cases in the problem of balance state stability are defined. The main content of the paper is focused on studying the local dynamics in cases close to critical. In nature, these critical cases are infinite dimensional. Special nonlinear boundary-value problems of the parabolic type are constructed, which play the role of normal forms. Their nonlocal dynamics defines the behavior of the solutions from the small neighborhood of the balance state of the initial equation. The conclusions on the role of nonlinearity in the delay coefficients, in the behavior of the dynamic properties of the solutions are summarized.
... For a smooth enough function n 0 , N I (0) = Q 0 n 0 (q)dq and N M (0) = +∞ Q n 0 (q)dq, the results in [Hartung et al., 2006] yield that (7) has a unique continuously differentiable solution ϕ I and ϕ M defined on [0, h] when Γ(t) < Q. Note that N I (t) can be determined if N M (t) is known. ...
Considering the mature condition of any individual to have eaten a specific amount of food during the entire period that it can spend at its immature stage, we propose a size-structured model by a first-order quasi-linear partial differential equation. The model can be firstly reduced to a single state-dependent delay differential equation and then to a constant delay differential equation. The state-dependent delay represents intra-specific competition among individuals for limited food resources. A complete analysis of the global dynamics on the positivity and boundedness of solutions, global stability for each equilibrium and Hopf bifurcation is carried out. Our results imply that the delay leads to instability that is shown by a simple example of a certain structured population model.
... For DDEs with state-dependent delays the claim that their local bifurcation theory is identical to the theory of ODEs is not fully resolved. A review of well established results and an exposition of the obstacles that one initially faces are described in the review by Hartung et al. (2006). Even assuming that the state-dependent delays are always bounded within an interval [0, max ], the space of continuous 0 is not a suitable phase space, since no local uniqueness of solutions to initial-value problems can be guaranteed. ...
This chapter presents a dynamical systems point of view of the study of systems with delays. The focus is on how advanced tools from bifurcation theory, as implemented for example in the package DDE-BIFTOOL, can be applied to the study of delay differential equations (DDEs) arising in applications, including those that feature state-dependent delays. We discuss the present capabilities of the most recent release of DDE-BIFTOOL. They include the numerical continuation of steady states, periodic orbits and their bifurcations of codimension one, as well as the detection of certain bifurcations of codimension two and the calculation of their normal forms. Two longer case studies, of a conceptual DDE model for the El Ni{\~n}o phenomenon and of a prototypical scalar DDE with two state-dependent feedback terms, demonstrate what kind of insights can be obtained in this way.
... Reference [15] illustrates the recent work on equations with state dependent delays, with emphasis on particular models and on the emerging theory from the dynamical systems point of view. Several new results are presented. ...
... The delay differential equations can describe many complex processes in nature and technology [14,15]. These equations are the special sort of functional differential equations involving past value of the state variable [16,17]. Recently, study on the solutions of FDEs with delay becomes very attractive [3,8,12,[18][19][20][21][22][23]. ...
In this paper, the system of fractional differential equations with delay is studied. The computational method based on the hybrid functions of the general block-pulse functions and Legendre polynomials is presented to obtain the numerical results for the solution of our problem. In fact, the system under study is converted to the corresponding nonlinear algebraic system of equations, which is resolved by using the collocation technique. Also, the error estimate for the suggested computational method that verifies the convergence is presented. Finally, we examine two models arising in engineering and applied science, including the Chen and human immunodeficiency viruses systems, to illustrate the efficiency of the method.
... Besides that, the qualitative theory is different from those for equations with time-dependent and discrete delay. We refer the reader to the survey paper of Hartung et al. [1] for many examples, and mention the works [2][3][4][5][6][7][8] and the references therein for information on recent results in this area. ...
In this paper, we are concerned with systems determined by partial differential equations with infinite and state-dependent delay. To study the system, we model the equation as an abstract retarded functional differential equations with infinite and state-dependent delay. We establish the existence of mild solutions, and the existence of S-asymptotically periodic solutions and asymptotically periodic solutions using local Lipschitz conditions on the involved functions. We apply our results to study the existence of asymptotically periodic solutions of a partial differential equation with infinite and state-dependent delay.
... Linearization of state dependent delay differential equation is done heuristically by freezing the value of the state dependent delay at its equilibrium value. Justification of this approach is shown in [3]. On the other hand, in [4], range of the solution of this type equation is determined by the means of coefficients and stability of the solution is investigated. ...
The research subject of many studies is the asymptotical behavior of dynamic systems with both continuous and discrete time scales for economical approaches. The qualitative theory of differential or difference equations is the main basis of these studies. When modeling using ordinary differential equations, the delays in the system are always ignored. However, even a small amount of delay in the system can cause significant changes in the system. For this reason, it is more realistic to use delayed differential equations while modeling the majority of the problems encountered. Moreover, some research showed us that some delays in the systems vary with respect to the internal effects of the system. In this case, using of state-dependent delay differential equations is much more suitable.
... This has motivated that these equations are being studied extensively during the last years. The interested reader can see [1,4,7,8,15,16,20,21,29,31,[36][37][38] as well as the references in these papers. ...
In this paper we are concerned with a class of abstract fractional integro-differential inclusions with infinite state-dependent delay. Our approach is based on the existence of a resolvent operator for the homogeneous equation.We establish the existence of mild solutions using both contractive maps and condensing maps. Finally, an application to the theory of heat conduction in materials with memory is given.
This book delves into semilinear evolution equations, impulsive differential equations, and integro-differential equations with different types of delay. The main objective is to investigate the existence of solutions and explore their approximate controllability, complete controllability, and attractivity. The study involves boundary conditions, nonlocal conditions, and impulsive conditions.
The analysis presented in this book goes beyond traditional solutions and encompasses the study of solutions that are asymptotically almost automorphic and integro-differential equations with impulsive effects in both bounded and unbounded domains. The book also contains applications to nuclear physics, elementary particle physics, chemical engineering, and economics.
This book is intended for researchers and professionals in the field of mathematics, physics and industrial engineering, as well as advanced graduate students.
The book has novel results for integro-differential equations.
Studies impulsive equations, equations with delay and coupled systems. Involves approximate and complete controllability, stability, continuous dependence, and attractivity.
We study the local and global existence and uniqueness of a strict solution for a general class of abstract explicit neutral equations with state-dependent delay. Some examples on explicit partial neutral differential equations with state dependent delay are presented.
System simulation is the study of system performance based on the mathematical models and systems behavior of real systems. In particular, it refers to use computers to investigate system performances. The fundamental contents of computer simulation include systems, modeling, algorithms, programming and simulation result display, analysis and validations. In this chapter, the commonly used blocks in Simulink block library are first explained, and the setting and simple use of Simulink are demonstrated, and the modeling strategies for various ODEs are explored.
In this work, we establish sufficient conditions for the existence of solutions for some functional integrodifferential equations with state-dependent delay in Banach spaces. We use -semigroup theory and a fixed point approach based on Banach and Sadovskii’s fixed point theorems, nonlinear alternative for condensing maps, Bihari’s inequality and the technique of equivalent norms. Applications ares provided to a reaction-diffusion equation with state-dependent delay.
Previous chapters have analyzed numerical methods for ODEs according to multistep or multistage strategies, by linear multistep and Runge-Kutta discretizations. In the recent history of numerical methods for ODEs, a third type of strategy for the improvement of the drawbacks of multistep and multistage techniques has been provided, giving rise to the so-called family of multivalue numerical methods. The basic principles of a multivalue discretizations are briefly described, together with the generalization of convergence results for these methods. Example of linear multistep and Runge-Kutta methods recast as multivalue methods are reported, together with examples of genuine multivalue methods that do not fall in these two classes. The generalization of dense output schemes in the case of multivalue numerical methods is also presented.
The study of models which deal with oscillating functions is of great importance in Physics and other sciences. In this work, we are interested in investigating the oscillatory behavior of solutions of delay differential equations subject to impulsive effects and having righthand sides which admit not only many discontinuities but also oscillations. We use the framework of Kurzweil–Henstock and Kurzweil–Henstock–Stieltjes integrals to deal with the integral form of our equations. We establish conditions for all solutions of our linear delay differential equation to be oscillatory. Not only that, but we also deal with generalized ODEs as introduced by Jaroslav Kurzweil, since the latter encompass delay differential equations. We prove an important result which relates oscillatory solutions around zero of retarded Volterra–Stieltjes-type integral equations to oscillatory solutions around zero of the corresponding generalized ODEs. Moreover, an oscillatory criterion not necessarily around zero for linear homogeneous generalized ODEs is obtained. Some examples are brought about in order to illustrate the results.
This chapter is devoted to stability investigation of systems with state-dependent delays under stochastic perturbations. Sufficient conditions of asymptotic mean square stability for the zero solution of a linear stochastic differential equation with distributed delays are obtained via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). Besides delay-independent and delay-dependent conditions of stability in probability are obtained for two equilibria of a nonlinear stochastic differential equation with delay and exponential nonlinearity. The negative definiteness of matrices in the obtained LMIs is checked using the special MATLAB program. It is noted that the proposed research method can be used for the study of other types of linear and nonlinear systems with state-dependent delays. Numerical simulation of solutions of the considered stochastic differential equations with state-dependent delays illustrate the presented here theoretical results and open to readers attention a new unsolved problem of the obtained stability conditions improving.
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.
This paper continues the authors's study of robust numerical scheme for generalized Stein's model of neuronal variability, which is a singularly perturbed parabolic partial differential–difference equation with general values of shift arguments. The work on this class of problems is initiated in the papers [K. Bansal and K.K. Sharma, Parameter-robust numerical scheme for time dependent singularly perturbed reaction- diffusion problem with large delay, Numer. Funct. Anal. Optim. 39(2) (2018), pp. 127–154] for unit shift argument and in [K. Bansal and K.K. Sharma, Parameter uniform numerical scheme for time dependent singularly perturbed convection- diffusion-reaction problems with general shift arguments, Numer. Algorithms 75(1) (2017), pp. 113–145] for general shift arguments. In the present paper, this work is further extended to the development of numerical scheme based on Mickens techniques, interpolation and θ scheme. The advantage of this work over the previous research is that it deals with the problem having general values of the shift arguments with higher order of convergence and without any constraints on the number of intervals.
In this paper, we investigate a class of impulsive differential equations with state-dependent delay under analytic semigroup in Banach spaces. The existence and uniqueness of classical solution is obtained under the assumptions that nonlinear function and impulsive map are Lipschitz continuous. On these assumptions, we further study the existence and smoothness of solution manifold of impulsive differential equations with state-dependent delay.
In this paper, the harmonic balance method with the alternating frequency/time (HB-AFT) domain technique is extended to the dynamical systems with state-dependent delays and non-smooth right-hand side for the first time. Two types of network congestion control models [the modified transmission control protocol-random early detection (TCP-RED) model and the fluid-flow TCP-additive increase multiplicative decease (AIMD)/RED model] with state-dependent round-trip time delays and non-smooth right-hand side are considered in detail. First, their dynamics and bifurcation are analyzed by the numerical analysis method. Then, the analytical approximate expressions of the periodic solutions are obtained by employing the semi-analytical method named as HB-AFT. The results of the numerical simulation and HB-AFT agree with each other very well. It indicates that the HB-AFT technique is simple, valid, effective, and accurate for the non-smooth dynamical systems with state-dependent time delays. Besides, more complicated and rich dynamical phenomena, such as period-2 solution, period-3 solution, chaos and the period-doubling bifurcation, and the window of period-3 solution, which lead to chaos, are explored in the TCP-AIMD/RED model. Moreover, two different kinds of multi-stabilities, such as the coexistence of a stable period-1 solution, a stable period-3 solution, and an asymptotically stable equilibrium, and that of a stable period-1 solution, a stable period-2 solution, chaos, and an asymptotically stable equilibrium, are also discovered in the TCP-AIMD/RED congestion control model. It discloses that the interaction of the state-dependent delay and non-smooth function can easily induce the richer and more complicated dynamics. It can explain the complicated dynamical phenomena in the Internet network. Thus, the control and parameters tuning based on the above results could be used to avoid the congestion and optimize the performance. It can give the theoretical guidance to the application and research in network congestion control, which is very important in practical application.
In this paper, we propose and analyze a cooperation model with harvesting and state-dependent delay, which is assumed to be an increasing function of the population density with lower and upper bound. The main purpose of this article is to obtain the dynamics of our model analytically by controlling the harvesting. We present results on positivity and boundedness of all populations. Criteria for the existence of all equilibria and uniqueness of a positive equilibrium are given by controlling the harvesting. Finally, the global exponentially asymptotical stability criteria of model is obtained by the improved Hanalay inequality.
In this paper, the stability properties of a parabolic partial differential equation with state-dependent delay are investigated by the heuristic approach. The previous works [1,2] obtained a continuously differentiable semiflow with continuously differentiable solution operators defined by the classical solutions, and resolved the problem of linearization for this equation. Here, we clarify the relation between the spectral properties of the linearization of the semiflow at a stationary solution and the strong continuous semigroup defined by the solutions of the linearization of this equation, and consider the local stable and unstable invariant manifolds of the semiflow at a stationary solution. By a biological application, we finally verify all hypotheses for an age structured diffusive model with state-dependent delay and consider its stability behavior.
We construct a delay functional on an infinite-dimensional subset , , so that the delay differential equation has a solution whose short segments are dense in . This implies complicated behaviour of the trajectory in .
We study the existence and uniqueness of mild and strict solutions for abstract neutral differential equations with state-dependent delay. Some examples related to partial neutral differential equations are presented.
Sufficient and realistic conditions are obtained for the existence of positive periodic solutions in periodic equations with state-dependent delay. The method involves the application of the coincidence degree theorem and estimations of uniform upper bounds on solutions. Applications of these results to some population models are presented. These application results indicate that seasonal effects on population models often lead to synchronous solutions. In addition, we may conclude that when both seasonality and time delay are present and deserve consideration, the seasonality is often the generating force for the often observed oscillatory behavior in population densities.
1. A novel approach for analyzing transients in passive structures called "the method of moments" is introduced. It provides, as a special case, an analytic method for calculating the time delay and speed of propagation of electrical signals in any passive dendritic tree without the need for numerical simulations. 2. Total dendritic delay (TD) between two points (y, x) is defined as the difference between the centroid (the center of gravity) of the transient current input, I, at point y[tI(y)] and the centroid of the transient voltage response, V, at point x [tV(x)]. The TD measured at the input points is nonzero and is called the local delay (LD). Propagation delay, PD(y, x), is then defined as TD(y, x)--LD(y) whereas the net dendritic delay, NDD(y, 0), of an input point, y, is defined as TD(y, 0) - LD(0), where 0 is the target point, typically the soma. The signal velocity at a point x0 in the tree, theta(x0), is defined as [1/(dtv(x)/dx)[x = x0. 3. With the use of these definitions, several properties of dendritic delay exist. First, the delay between any two points in a given tree is independent of the properties (shape and duration) of the transient current input. Second, the velocity of the signal at any given point (y) in a given direction from (y) does not depend on the morphology of the tree "behind" the signal, and of the input location. Third, TD(y, x) = TD(x, y), for any two points, x, y. 4. Two additional properties are useful for efficiently calculating delays in arbitrary passive trees. 1) The subtrees connected at the ends of any dendritic segment can each be functionally lumped into an equivalent isopotential R-C compartment. 2) The local delay at any given point (y) in a tree is the mean of the local delays of the separate structures (subtrees) connected at y, weighted by the relative input conductance of the corresponding subtrees. 5. Because the definitions for delays utilize difference between centroids, the local delay and the total delay can be interpreted as measures for the time window in which synaptic inputs affect the voltage response at a target/decision point. Large LD or TD is closely associated with a relatively wide time window, whereas small LD or TD imply that inputs have to be well synchronized to affect the decision point. The net dendritic delay may be interpreted as the cost (in terms of delay) of moving a synapse away from the target point.(ABSTRACT TRUNCATED AT 400 WORDS)
This paper presents existence, uniqueness, and continuous dependence theorems for solutions of initial-value problems for neutral-differential equations of the form where f, g , and h are continuous functions with g ( 0 , x 0 ) = h ( 0 , x 0 ) = 0 g(0,{x_0}) = h(0,{x_0}) = 0 . The existence of a continuous solution of the functional equation z ( t ) = f ( t , z ( h ( t ) ) ) z(t) = f(t,z(h(t))) is proved as a corollary.
This chapter discusses the periodic solutions of some integral equations from the theory of epidemics. It summarizes results on an integral equation. A key role in the abstract theory underlying these results is played by the fixed point index. The chapter presents the proof of some new results about the fixed point index of cone maps. It presents the proof of a nonlinear version of the Krein–Rutman theorem. This theorem has been used to obtain results above the fixed point index of positive linear operators. In the results obtained, the elementary arguments yield information about the fixed point index that in turn yields the Krein–Rutman theorem.
This paper extends back-propagation to continuous-time feed-forward networks with internal, adaptable time delays. The new technique is suitable for parallel hardware implementation, with continuous multidimensional training signals. The resulting networks can be used for signal prediction, signal production, and spatiotemporal pattern recognition tasks. Unlike conventional back-propagation networks, they can easily adapt while performing true signal prediction. We present simulation results for networks trained to predict future values of the Mackey-Glass chaotic signal, using its present value as an input. For this application, networks with adaptable delays had less than half the prediction error of networks with fixed delays, and about one-quarter the error of conventional networks. After training, the network can be operated in a signal production configuration, where it autonomously generates a close approximation to the Mackey-Glass signal. 1 This work was supported by the Natu...
1. The collocation method for ODEs: an introduction 2. Volterra integral equations with smooth kernels 3. Volterra integro-differential equations with smooth kernels 4. Initial-value problems with non-vanishing delays 5. Initial-value problems with proportional (vanishing) delays 6. Volterra integral equations with weakly singular kernels 7. VIDEs with weakly singular kernels 8. Outlook: integral-algebraic equations and beyond 9. Epilogue.
The state dependent delayed differential equation w ' (t)=-g(w(t),w(τ(t))) with ∫ τ(t) t f(t,s,w(s)) ds=m if ∫ 0 t f(t,s,w(s))ds>m, τ(t)=0 otherwise, is studied from the point of view of existence and uniqueness of solutions (w,τ) defined on [0,∞]. It is shown that if (0,0) is a sufficiently strong attractor for the autonomous equation x ' =-g(x,x) and if f forces the delay to grow without bound, then all nontrivial solutions of the problem are oscillatory.
The equations of motion for a one-dimensional two-body problem of electrodynamics are a system of delay differential equations whose delays depend on the unknown trajectories of the charged particles. Some earlier results are extended to include the more general case where the particles are not necessarily in each other's force field at the instant t//0. For this more general case the equations of motion lead to an ordinary vector differential equation x prime equals f(t,x), x(t//0) equals x//0, with the initial point (t//0,x//0) lying on the boundary of the domain D on which the function f is well-defined. Existence and uniqueness for the two-body problem are obtained without requiring the particles to be in each others force field at any specific time.
In this paper we prove that a constant steady-state of an autonomous state-dependent delay equation is exponentially stable if a zero solution of a corresponding linear autonomous equation is exponentially stable.
A class of neutral functional-differential equations with state-dependent delay is considered, and the stability properties are studied in two different ways. The author associates a family of retarded functional-differential equations with the neutral equation and applies a stability result of T. Krisztin [Funkc. Ekvacioj, Ser. Int. 34, No. 2, 241-256 (1991; Zbl 0746.34045)]. The author extends results of J. Wu and J. S. Yu [Dyn. Syst. Appl. 4, No. 2, 279-290 (1995; Zbl 0830.34066)] given to neutral equations with constant delay to neutral equations with state-dependent delay.
A first order method is presented for solution of the initial-value problem for a differential equation of neutral type with implicit delay in the critical case where the time-lag is zero and the method of stepwise integration does not apply. A convergence theorem is proved, and numerical examples are given.
Let a C 1 -function f:ℝ→ℝ be given which satisfies f(0)=0, f ' (ξ)<0 for all ξ∈ℝ, and supf<∞ or -∞<inff. Let C=C([-1,0],ℝ). For an open-dense set of initial data the phase curves [0,∞)→C given by the solutions [-1,∞)→ℝ to the negative feedback equation x ' (t)=-μx(t)+f(x(t-1)),withμ>0, are absorbed into the positively invariant set S⊂C of data ϕ≠0 with at most one sign change. The global attractor A of the semiflow restricted to S ¯ is either the singleton {0} or it is given by a Lipschitz continuous map a with domain pA in a two-dimensional subspace L⊂C and range in a complementary subspace Q; pA is homeomorphic to the closed unit disk in ℝ 2 . We show that a is in fact C 1 -smooth.
We consider a population divided into two groups: the matures and the immatures. A system of nonlinear functional-differential equations is derived which describes the dynamics of this population. The system involves a time-delay which represents the time needed for an immature individual to reach maturity. We take this time-delay to be a function of the total population density. Results dealing with existence and stability of equilibria, positivity of solutions, and boundedness of solutions are proven.