On Walsh Noise: Its Properties and Use in Dynamic Stochastic Systems

In order to accelerate computations and improve long time accuracy of numerical simulations, this thesis develops multiscale geometric integrators. For general multiscale stiff ODEs, SDEs, and PDEs, FLow AVeraging integratORs (FLAVORs) have been proposed for the coarse time-stepping without any identification of the slow or the fast variables. In the special case of deterministic and stochastic mechanical systems, symplectic, multisymplectic, and quasi-symplectic multiscale integrators are easily obtained using this strategy. For highly oscillatory mechanical systems (with quasi-quadratic stiff potentials and possibly high-dimensional), a specialized symplectic method has been devised to provide improved efficiency and accuracy. This method is based on the introduction of two highly nontrivial matrix exponentiation algorithms, which are generic, efficient, and symplectic (if the exact exponential is symplectic). For multiscale systems with Dirac-distributed fast processes, a family of symplectic, linearly-implicit and stable integrators has been designed for coarse step simulations. An application is the fast and accurate integration of constrained dynamics. In addition, if one cares about statistical properties of an ensemble of trajectories, but not the numerical accuracy of a single trajectory, we suggest tuning friction and annealing temperature in a Langevin process to accelerate its convergence. Other works include variational integration of circuits, efficient simulation of a nonlinear wave, and finding optimal transition pathways in stochastic dynamical systems (with a demonstration of mass effects in molecular dynamics).

The stability of stochastic dynamic systems have widespread application prospects and greater theoretical significance. Its theory are mainly applied to engineering control, communication equipment, military technology, biology, finance, etc. In this article, we consider the stability of a Class of nonlinear impulsive neutral stochastic differential dynamic systems. A new set of conditions proving the mean square stability of this nonlinear impulsive neutral stochastic dynamic systems are derived by means of the Banach fixed point theorem. Some well-known results are improved and generalized.

Approximation identification for the stochastic time-delayed dynamical system

A full-wave electromagnetic model for analyzing waveguide discontinuities of arbitrarily shaped piecewise planar boundaries is presented. The analysis is facilitated by using the finite plane-wave series expansion of circular cylindrical modal functions. Since electromagnetic fields on each of the planar boundary surfaces of the inhomogeneous region are expressed in terms of plane-wave modal functions, the complete solution is carried out analytically without any numerical integration. To verify the formulation, a number of practical waveguide components are analyzed. The calculated results are compared with other full-wave electromagnetic models. Excellent agreement is obtained for all the cases.

Mean-square practical stability for stochastic large-scale dynamical systems is studied. The vector Lyapunov functions and the basic comparison principle of stochastic systems as well as the decomposition-aggregation method for large-scale systems were used to obtain results for various types of mean-square practical stability of nonlinear stochastic large-scale dynamical systems. The objective is to analyze stochastic large-scale systems in terms of their lower-order subsystems and in terms of their interconnecting structure. These results make it possible to determine the mean-square practical stability of stochastic large-scale dynamical systems by testing the corresponding practical stability of the auxiliary deterministic systems. >

Stochastic dynamical systems always undergo trending mechanisms of transition to criticality

A stochastic dynamical system is a system composed of many interacting components which includes stochastic behavior. Feedback control theory is designed to use information from the current state of the system in order to temper the final state toward the desired outcome. In this dissertation, I perform studies on two choices of classic and quantum dynamical systems to cover vast variety of statistical tools and methods. For classical model, I perform a study on finite-size effects at the phase transition in the Nagel-Schreckenberg traffic model as an example of non-equilibrium many body stochastic dynamical system. For quantum mechanical model, I introduce an optimal measurement-based feedback control protocol for cooling a single qubit as an example of complex system with one cell and many degrees of freedom. I examine the Nagel-Schreckenberg traffic model for a variety of maximum speeds. I show that the low density limit can be described as a dilute gas of vehicles with a repulsive core. At the transition to jamming, I observe finite-size effects in a variety of quantities describing the flow and the density correlations, but only if the maximum speed Vmax is larger than a certain value. A finite-size scaling analysis of several order parameters shows universal behavior, with scaling exponents that depend on Vmax The jamming transition at large Vmax can be viewed as the nucleation of jams in a background of freely flowing vehicles. Feedback control of quantum systems via continuous measurements involves complex nonlinear dynamics. As a result, even for a single qubit the optimal measurement for feedback control is known only in very special cases. I show here that for a broad class of noise processes, a series of compelling arguments can be applied to greatly simplify the problem of steady-state preparation of the ground-state, while loosing little in the way of optimality. Using numerical optimization to solve this simplified control problem, I obtain for the first time a non-trivial feedback protocol valid for all feedback strengths in the regime of good control. The protocol can be described relatively simply, and contains a discontinuity as a function of feedback strength.

Semilinear stochastic dynamic systems in a separable Hilbert space often model some evolution phenomena arising in physics and engineering. In this paper, we study the existence and uniqueness of mild solutions to neutral semilinear stochastic functional dynamic systems under local non-Lipschitz conditions on the coefficients by means of the stopping time technique. We especially generalize and improve the results that appeared in Govinadan (2005), Bao and Hou (2010), and Jiang and Shen (2011).

Stochastic Compound Dynamic Systems (CDS) are complex technical systems that are created through the use of precision mechanics in combination with modern telecommunications and computer technologies. Incertitude in these CDS shows up under the influence of external and internal stochastic perturbations. The constituent elements of CDS are combined into a single system because these elements perform a single complex mission. The information exchange is wireless, there is no mechanical connection between the elements of the CDS. The paper considers groups of unmanned aerial vehicles (UAVs), which are equipped with sensors or multisensors that are able to perform a mobile sensor network. The trajectories of individual elements of the mobile sensor network are trajectories formed under the influence of stochastic perturbations. This fact means that the nature of the mobile sensor network can be classified as a stochastic compound dynamic system and for the mathematical description and optimization of the movement of this system is adequate to use models and methods for optimizing stochastic CDS. The model of CDS motion is considered to be a branching trajectory or, as they say, a branched trajectory. A stochastic mathematical model of the motion of a mobile sensor network, which performs the combined mission of surveying an emergency zone, can be classified as a model of the motion of a stochastic compound dynamic system. This approach is an adequate for mathematical model creation to the mobile sensor network physical condition, for its operation in the zone of natural disaster by natural or anthropogenic origin. This paper is devoted to solving a theoretical problem related to the formulation and proof of the necessary conditions for stochastic CDS moving optimal control along a branched trajectory with an arbitrary branching scheme.

Efficient search and responsiveness trade-offs in a Markov chain model of evolution in dynamic environments

SummaryThe aim of this survey paper is to sketch the problems and results of the qualitative theory of stochastic dynamical systems (SDS) and to give reference to papers where details can be found. Here an SDS is an ordinary differential equation ẋ = f (x,ξ) with a random noise process ξ in the r.h.s. and random initial conditions x(o) = xo. Qualitative theory studies the general nature of a solution in the entire time interval, e.g. recurrence and stability properties. Both the white noise and the real (i.e. non-white) noise case are covered. Emphasis is given to nonlinear systems (including multiplicative noise linear systems) and to stationary and Markovian noise. Standing reference is ARNOLD and KLIEMANN [7].

In the asymptotic limit of weak inhomogeneities and long times or distances, we obtain a system of kinetic equations governing the power transfer among the modes of oscillation of certain stochastic dynamical systems. We include applications to coupled oscillators, waveguides, beams, the quantized motion of a particle in a random potential, and the Klein-Gordon equation with random plasma frequency.

Neutral stochastic dynamic systems have received broad attention, because the problems are not only great theoretical significance, but also have played an important role in many ways such as engineering control, finance, communication equipment, social science, etc. In this paper, we consider stability of neutral stochastic partial differential dynamic systems with Markovian switching. Some new sufficient conditions proving the stability in mean square of this neutral stochastic dynamic systems are derived by employing the fixed point approach. Some well-known results are improved and generalized.

In this paper, a methodology for propagation of uncertainty in stochastic nonlinear dynamical systems is investigated. The process noise is approximated using Karhunen-Loeve (KL) expansion. Perron-Frobenius (PF) operator is used to predict the evolution of uncertainty. A multivariate Kolmogorov-Smirnov test is used to verify the proposed framework. The method is applied to predict uncertainty evolution in a Duffing oscillator and a Vanderpol's oscillator. It is observed that the solution of the approximated stochastic dynamics converges to the true solution in distribution. Finally, the proposed methodology is combined with Bayesian inference to estimate states of a nonlinear dynamical system, and its performance is compared with particle filter. The proposed estimator was found to be computationally superior than the particle filter.

Dealing with a stochastic dynamical system with state vector X(t) satisfying a nonlinear Langevin equation of the type $$dx = f(x)dt + \sigma (x)dW(t),$$ we may only be interested in qualitative changes and their statistics. Typical qualitative changes are the switch from one domain of attraction to a neighbouring one or in the case of population dynamics the extinction or introduction of a species. For that purpose we introduce the Markov chain approximation. For a precise mathematical description of the connection between such a Markov chain and the stochastic dynamical system we refer to Chap. 6 of Freidlin and Wentzell (1984). Here we work out two examples. One modelling the preferent states of the atmospheric circulation and one dealing with the occupation rate of a patch of land by some biological species.