2,179 research outputs found

    Moment Boundedness of Linear Stochastic Delay Differential Equation with Distributed Delay

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    This paper studies the moment boundedness of solutions of linear stochastic delay differential equations with distributed delay. For a linear stochastic delay differential equation, the first moment stability is known to be identical to that of the corresponding deterministic delay differential equation. However, boundedness of the second moment is complicated and depends on the stochastic terms. In this paper, the characteristic function of the equation is obtained through techniques of Laplace transform. From the characteristic equation, sufficient conditions for the second moment to be bounded or unbounded are proposed.Comment: 38 pages, 2 figure

    Stability and Boundedness of Stochastic Volterra Integrodifferential Equations with Infinite Delay

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    We make the first attempt to discuss stability and boundedness of solutions to stochastic Volterra integrodifferential equations with infinite delay (IDSVIDEs). By the Lyapunov-Krasovskii functional approach, we get kinds of sufficient criteria for stability and boundedness of solutions to IDSVIDEs. The main innovation here is that stochastic systems with infinite delay can retain stability and boundedness of corresponding deterministic systems under some conditions

    Climate dynamics and fluid mechanics: Natural variability and related uncertainties

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    The purpose of this review-and-research paper is twofold: (i) to review the role played in climate dynamics by fluid-dynamical models; and (ii) to contribute to the understanding and reduction of the uncertainties in future climate-change projections. To illustrate the first point, we focus on the large-scale, wind-driven flow of the mid-latitude oceans which contribute in a crucial way to Earth's climate, and to changes therein. We study the low-frequency variability (LFV) of the wind-driven, double-gyre circulation in mid-latitude ocean basins, via the bifurcation sequence that leads from steady states through periodic solutions and on to the chaotic, irregular flows documented in the observations. This sequence involves local, pitchfork and Hopf bifurcations, as well as global, homoclinic ones. The natural climate variability induced by the LFV of the ocean circulation is but one of the causes of uncertainties in climate projections. Another major cause of such uncertainties could reside in the structural instability in the topological sense, of the equations governing climate dynamics, including but not restricted to those of atmospheric and ocean dynamics. We propose a novel approach to understand, and possibly reduce, these uncertainties, based on the concepts and methods of random dynamical systems theory. As a very first step, we study the effect of noise on the topological classes of the Arnol'd family of circle maps, a paradigmatic model of frequency locking as occurring in the nonlinear interactions between the El Nino-Southern Oscillations (ENSO) and the seasonal cycle. It is shown that the maps' fine-grained resonant landscape is smoothed by the noise, thus permitting their coarse-grained classification. This result is consistent with stabilizing effects of stochastic parametrization obtained in modeling of ENSO phenomenon via some general circulation models.Comment: Invited survey paper for Special Issue on The Euler Equations: 250 Years On, in Physica D: Nonlinear phenomen

    Review on computational methods for Lyapunov functions

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    Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

    Synchronization of globally coupled two-state stochastic oscillators with a state dependent refractory period

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    We present a model of identical coupled two-state stochastic units each of which in isolation is governed by a fixed refractory period. The nonlinear coupling between units directly affects the refractory period, which now depends on the global state of the system and can therefore itself become time dependent. At weak coupling the array settles into a quiescent stationary state. Increasing coupling strength leads to a saddle node bifurcation, beyond which the quiescent state coexists with a stable limit cycle of nonlinear coherent oscillations. We explicitly determine the critical coupling constant for this transition

    Basin stability in delayed dynamics

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    Acknowledgements S.L. was supported by the China Scholarship Council (CSC) scholarship (Grant No. 501100004543). W.L. was supported by the National Natural Science Foundation (NNSF) of China (Grants No. 61273014 and No. 11322111).Peer reviewedPublisher PD

    Attractors for Parametric Delay Differential Equations and Their Continuous Behavior

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    The problem of the continuity of global attractors under minimal assumptions for a general class of parameterized delay differential equations is considered. The theory of equi-attraction developed by Li and Kloeden in [2004a] is adapted to this framework, so it is proved that the continuity of the attractors with respect to the parameter is equivalent to this equi-attraction property

    Exponential stabilization by delay feedback control for highly nonlinear hybrid stochastic functional differential equations with infinite delay

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    Given an unstable hybrid stochastic functional differential equation, how to design a delay feedback controller to make it stable? Some results have been obtained for hybrid systems with finite delay. However, the state of many stochastic differential equations are related to the whole history of the system, so it is necessary to discuss the feedback control of stochastic functional differential equations with infinite delay. On the other hand, in many practical stochastic models, the coefficients of these systems do not satisfy the linear growth condition, but are highly nonlinear. In this paper, the delay feedback controls are designed for a class of infinite delay stochastic systems with highly nonlinear and the influence of switching state
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