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Discrete & Continuous Dynamical Systems - B

2013 , Volume 18 , Issue 6

Special Issue on on Deterministic and Stochastic Dynamical Systems with delays

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Preface
Tomás Caraballo and Hans-Otto Walther
2013, 18(6): i-ii doi: 10.3934/dcdsb.2013.18.6i +[Abstract](317) +[PDF](77.1KB)
Abstract:
Studying delay differential equations is motivated by the fact that the evolution of systems in physics, chemistry, the life sciences, engineering, and economics, may, and often does, depend not only on the present state of the system but also on earlier states. Examples arise from the 2-body problem of electrodynamics (which is barely understood), in laser physics, materials with thermal memory, biochemical reactions, population growth, physiological regulatory systems, and business cycles, among many others. Delays may also appear when one wants to control a system by applying an external force which takes into account the history of the solution. Also mathematical problems in geometry and probability yield delay differential equations. In modeling real world phenomena another important aspect is uncertainty. It is often useful to take into account some randomness or environmental noise. This Special Issue addresses both aspects of dynamical systems, namely, hereditary characteristics and stochasticity. We have selected a dozen of papers which illustrate some lines of recent research.

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A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations
Evelyn Buckwar and Girolama Notarangelo
2013, 18(6): 1521-1531 doi: 10.3934/dcdsb.2013.18.1521 +[Abstract](772) +[PDF](328.8KB)
Abstract:
The stability of equilibrium solutions of a deterministic linear system of delay differential equations can be investigated by studying the characteristic equation. For stochastic delay differential equations stability analysis is usually based on Lyapunov functional or Razumikhin type results, or Linear Matrix Inequality techniques. In [7] the authors proposed a technique based on the vectorisation of matrices and the Kronecker product to transform the mean-square stability problem of a system of linear stochastic differential equations into a stability problem for a system of deterministic linear differential equations. In this paper we extend this method to the case of stochastic delay differential equations, providing sufficient and necessary conditions for the stability of the equilibrium. We apply our results to a neuron model perturbed by multiplicative noise. We study the stochastic stability properties of the equilibrium of this system and then compare them with the same equilibrium in the deterministic case. Finally the theoretical results are illustrated by numerical simulations.
Invariance and monotonicity for stochastic delay differential equations
Igor Chueshov and Michael Scheutzow
2013, 18(6): 1533-1554 doi: 10.3934/dcdsb.2013.18.1533 +[Abstract](486) +[PDF](464.8KB)
Abstract:
We study invariance and monotonicity properties of Kunita-type sto-chastic differential equations in $\mathbb{R}^d$ with delay. Our first result provides sufficient conditions for the invariance of closed subsets of $\mathbb{R}^d$. Then we present a comparison principle and show that under appropriate conditions the stochastic delay system considered generates a monotone (order-preserving) random dynamical system. Several applications are considered.
Exponential stability for a class of linear hyperbolic equations with hereditary memory
Monica Conti, Elsa M. Marchini and Vittorino Pata
2013, 18(6): 1555-1565 doi: 10.3934/dcdsb.2013.18.1555 +[Abstract](390) +[PDF](411.0KB)
Abstract:
We establish a necessary and sufficient condition of exponential stability for the contraction semigroup generated by an abstract version of the linear differential equation $$∂_t u(t)-\int_0^\infty k(s)\Delta u(t-s)ds = 0 $$ modeling hereditary heat conduction of Gurtin-Pipkin type.
Asymptotic behaviour for a class of delayed cooperative models with patch structure
Teresa Faria
2013, 18(6): 1567-1579 doi: 10.3934/dcdsb.2013.18.1567 +[Abstract](404) +[PDF](334.6KB)
Abstract:
For a class of cooperative population models with patch structure and multiple discrete delays, we give conditions for the absolute global asymptotic stability of both the trivial solution and -- when it exists -- a positive equilibrium. Under a sublinearity condition, sharper results are obtained. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. As a by-product, we obtain a criterion for the existence of positive traveling wave solutions for an associated reaction-diffusion model with patch structure. Our results improve and generalize criteria in the recent literature.
Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions
Ciprian G. Gal and Maurizio Grasselli
2013, 18(6): 1581-1610 doi: 10.3934/dcdsb.2013.18.1581 +[Abstract](552) +[PDF](614.6KB)
Abstract:
We consider a modified Cahn-Hiliard equation where the velocity of the order parameter $u$ depends on the past history of $\Delta \mu $, $\mu $ being the chemical potential with an additional viscous term $ \alpha u_{t},$ $\alpha >0.$ In addition, the usual no-flux boundary condition for $u$ is replaced by a nonlinear dynamic boundary condition which accounts for possible interactions with the boundary. The aim of this work is to analyze the passage to the singular limit when the memory kernel collapses into a Dirac mass. In particular, we discuss the convergence of solutions on finite time-intervals and we also establish stability results for global and exponential attractors.
Parameter estimation by quasilinearization in differential equations with state-dependent delays
Ferenc Hartung
2013, 18(6): 1611-1631 doi: 10.3934/dcdsb.2013.18.1611 +[Abstract](562) +[PDF](322.2KB)
Abstract:
In this paper we study a parameter estimation method in functional differential equations with state-dependent delays using a quasilinearization technique. We define the method, prove its convergence under certain conditions, and test its applicability in numerical examples. We estimate infinite dimensional parameters such as coefficient functions, delay functions and initial functions in state-dependent delay equations. The method uses the derivative of the solution with respect to the parameters. The proof of the convergence is based on the Lipschitz continuity of the derivative with respect to the parameters.
A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions
Benjamin B. Kennedy
2013, 18(6): 1633-1650 doi: 10.3934/dcdsb.2013.18.1633 +[Abstract](610) +[PDF](505.0KB)
Abstract:
We consider state-dependent delay equations of the form \[ x'(t) = f(x(t - d(x(t)))) \] where $d$ is smooth and $f$ is smooth, bounded, nonincreasing, and satisfies the negative feedback condition $xf(x) < 0$ for $x \neq 0$. We identify a special family of such equations each of which has a ``rapidly oscillating" periodic solution $p$. The initial segment $p_0$ of $p$ is the fixed point of a return map $R$ that is differentiable in an appropriate setting.
    We show that, although all the periodic solutions $p$ we consider are unstable, the stability can be made arbitrarily mild in the sense that, given $\epsilon > 0$, we can choose $f$ and $d$ such that the spectral radius of the derivative of $R$ at $p_0$ is less than $1 + \epsilon$. The spectral radii are computed via a semiconjugacy of $R$ with a finite-dimensional map.
Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays
Kai Liu
2013, 18(6): 1651-1661 doi: 10.3934/dcdsb.2013.18.1651 +[Abstract](508) +[PDF](342.9KB)
Abstract:
A class of stochastic optimal control problems of infinite dimensional Ornstein-Uhlenbeck processes of neutral type are considered. One special feature of the system under investigation is that time delays are present in the control. An equivalent formulation between an adjoint stochastic controlled delay differential equation and its lifted control system (without delays) is developed. As a consequence, the finite time quadratic regulator problem governed by this formulation is solved based on a direct solution of some associated Riccati equation.
Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay
Arne Ogrowsky and Björn Schmalfuss
2013, 18(6): 1663-1681 doi: 10.3934/dcdsb.2013.18.1663 +[Abstract](600) +[PDF](370.0KB)
Abstract:
In this paper we deal with a nonautonomous differential equation with a nonautonomous delay. The aim is to establish the existence of an unstable invariant manifold to this differential equation for which we use the Lyapunov-Perron transformation. However, the delay is assumed to be unbounded which makes it necessary to use nonclassical methods.
Exponential growth rate for a singular linear stochastic delay differential equation
Michael Scheutzow
2013, 18(6): 1683-1696 doi: 10.3934/dcdsb.2013.18.1683 +[Abstract](514) +[PDF](371.4KB)
Abstract:
We establish the existence of a deterministic exponential growth rate for the norm (on an appropriate function space) of the solution of the linear scalar stochastic delay equation $d X(t) = X(t-1) d W(t)$ which does not depend on the initial condition as long as it is not identically zero. Due to the singular nature of the equation this property does not follow from available results on stochastic delay differential equations. The key technique is to establish existence and uniqueness of an invariant measure of the projection of the solution onto the unit sphere in the chosen function space via asymptotic coupling and to prove a Furstenberg-Hasminskii-type formula (like in the finite dimensional case).
Khasminskii-type theorems for stochastic functional differential equations
Minghui Song, Liangjian Hu, Xuerong Mao and Liguo Zhang
2013, 18(6): 1697-1714 doi: 10.3934/dcdsb.2013.18.1697 +[Abstract](566) +[PDF](384.1KB)
Abstract:
For a stochastic functional differential equation (SFDE) to have a unique global solution it is in general required that the coefficients of the SFDE obey the local Lipschitz condition and the linear growth condition. However, there are many SFDEs in practice which do not obey the linear growth condition. The main aim of this paper is to establish existence-and-uniqueness theorems for SFDEs where the linear growth condition is replaced by more general Khasminskii-type conditions in terms of a pair of Laypunov-type functions.
Mean-square random attractors of stochastic delay differential equations with random delay
Fuke Wu and Peter E. Kloeden
2013, 18(6): 1715-1734 doi: 10.3934/dcdsb.2013.18.1715 +[Abstract](563) +[PDF](414.7KB)
Abstract:
The existence of a random attactor is established for a mean-square random dynamical system (MS-RDS) generated by a stochastic delay equation (SDDE) with random delay for which the drift term is dominated by a nondelay component satisfying a one-sided dissipative Lipschitz condition. It is shown by Razumikhin-type techniques that the solution of this SDDE is ultimately bounded in the mean-square sense and that solutions for different initial values converge exponentially together as time increases in the mean-square sense. Consequently, similar boundedness and convergence properties hold for the MS-RDS and imply the existence of a mean-square random attractor for the MS-RDS that consists of a single stochastic process.

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