28,669 research outputs found
Stability of stochastic impulsive differential equations: integrating the cyber and the physical of stochastic systems
According to Newton's second law of motion, we humans describe a dynamical
system with a differential equation, which is naturally discretized into a
difference equation whenever a computer is used. The differential equation is
the physical model in human brains and the difference equation the cyber model
in computers for the dynamical system. The physical model refers to the
dynamical system itself (particularly, a human-designed system) in the physical
world and the cyber model symbolises it in the cyber counterpart. This paper
formulates a hybrid model with impulsive differential equations for the
dynamical system, which integrates its physical model in real world/human
brains and its cyber counterpart in computers. The presented results establish
a theoretic foundation for the scientific study of control and communication in
the animal/human and the machine (Norbert Wiener) in the era of rise of the
machines as well as a systems science for cyber-physical systems (CPS)
A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods
In this article we compare the mean-square stability properties of the
Theta-Maruyama and Theta-Milstein method that are used to solve stochastic
differential equations. For the linear stability analysis, we propose an
extension of the standard geometric Brownian motion as a test equation and
consider a scalar linear test equation with several multiplicative noise terms.
This test equation allows to begin investigating the influence of
multi-dimensional noise on the stability behaviour of the methods while the
analysis is still tractable. Our findings include: (i) the stability condition
for the Theta-Milstein method and thus, for some choices of Theta, the
conditions on the step-size, are much more restrictive than those for the
Theta-Maruyama method; (ii) the precise stability region of the Theta-Milstein
method explicitly depends on the noise terms. Further, we investigate the
effect of introducing partially implicitness in the diffusion approximation
terms of Milstein-type methods, thus obtaining the possibility to control the
stability properties of these methods with a further method parameter Sigma.
Numerical examples illustrate the results and provide a comparison of the
stability behaviour of the different methods.Comment: 19 pages, 10 figure
Design of quasi-symplectic propagators for Langevin dynamics
A vector field splitting approach is discussed for the systematic derivation
of numerical propagators for deterministic dynamics. Based on the formalism, a
class of numerical integrators for Langevin dynamics are presented for single
and multiple timestep algorithms
Postprocessed integrators for the high order integration of ergodic SDEs
The concept of effective order is a popular methodology in the deterministic
literature for the construction of efficient and accurate integrators for
differential equations over long times. The idea is to enhance the accuracy of
a numerical method by using an appropriate change of variables called the
processor. We show that this technique can be extended to the stochastic
context for the construction of new high order integrators for the sampling of
the invariant measure of ergodic systems. The approach is illustrated with
modifications of the stochastic -method applied to Brownian dynamics,
where postprocessors achieving order two are introduced. Numerical experiments,
including stiff ergodic systems, illustrate the efficiency and versatility of
the approach.Comment: 21 pages, to appear in SIAM J. Sci. Compu
Stochastic analysis of a full system of two competing populations in a chemostat
This paper formulates two 3D stochastic differential equations (SDEs) of two
microbial populations in a chemostat competing over a single substrate. The two
models have two distinct noise sources. One is general noise whereas the other
is dilution rate induced noise. Nonlinear Monod growth rates are assumed and
the paper is mainly focused on the parameter values where coexistence is
present deterministically. Nondimensionalising the equations around the point
of intersection of the two growth rates leads to a large parameter which is the
nondimensional substrate feed. This in turn is used to perform an asymptotic
analysis leading to a reduced 2D system of equations describing the dynamics of
the populations on and close to a line of steady states retrieved from the
deterministic stability analysis. That reduced system allows the formulation of
a spatially 2D Fokker-Planck equation which when solved numerically admits
results similar to those from simulation of the SDEs. Contrary to previous
suggestions, one particular population becomes dominant at large times.
Finally, we brie y explore the case where death rates are added
Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions
The (asymptotic) behaviour of the second moment of solutions to stochastic
differential equations is treated in mean-square stability analysis. This
property is discussed for approximations of infinite-dimensional stochastic
differential equations and necessary and sufficient conditions ensuring
mean-square stability are given. They are applied to typical discretization
schemes such as combinations of spectral Galerkin, finite element,
Euler-Maruyama, Milstein, Crank-Nicolson, and forward and backward Euler
methods. Furthermore, results on the relation to stability properties of
corresponding analytical solutions are provided. Simulations of the stochastic
heat equation illustrate the theory.Comment: 22 pages, 4 figures; deleted a section; shortened the presentation of
results; corrected typo
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