Collective behavior of multi-agent network dynamic systems under internal and external random perturbations

In this work, by formulating stochastic model for multi-agent network dynamic systems, the qualitative and quantitative properties of these dynamic models are investigated. In particular the cohesiveness, convergence, stability and invariant sets are addressed.     

Dynamical systems under the action of fast random perturbations

Dynamical systems, depending on a randomly varying parameter, are considered. Conditions, under which such a dynamical system turns into a diffusion process, are investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 3–21, January, 1991.     

Collapse of attractors for ODEs under small random perturbations

This paper concerns comparisons between attractors for random dynamical systems and their corresponding noiseless systems. It is shown that if a random dynamical system has negative time trajectories that are transient or explode with probability one, then the random attractor cannot contain any open set. The result applies to any Polish space and when applied to autonomous stochastic differential equations with additive noise requires only a mild dissipation of the drift. Additionally, following observations from numerical simulations in a previous paper, analytical results are presented proving that the random global attractors for a class of gradient-like stochastic differential equations consist of a single random point. Comparison with the noiseless system reveals that arbitrarily small non-degenerate additive white noise causes the deterministic global attractor, which may have non-zero dimension, to ‘collapse’. Unlike existing results of this type, no order preserving property is necessary.      

A regularity theorem for quasilinear parabolic systems under random perturbations

Several aspects of regularity theory for parabolic systems are investigated under the effect of random perturbations. The deterministic theory, when strict parabolicity is assumed, presents both classes of systems where all weak solutions are in fact more regular, and examples of systems with weak solutions which develop singularities in finite time. Our main result is the extension of a regularity result due to Kalita to the stochastic case, which concerns local Hölder continuity of weak solutions in the vectorial case. For the proof, we apply stochastic versions of methods, which are classical in the deterministic case (such as difference quotient techniques, higher integrability by embedding theorems, and a version of Moser’s iteration technique). This might be of interest on their own.     

Forward dynamics simulation of human body under tilting perturbations

Human body uses different strategies to maintain its stability and these strategies vary from fixed-foot strategies to strategies which foot is moved in order to increase the support base. Tilting movement of foot is one type of the perturbations usually is exposed to human body. In the presence of such perturbations human body must employ appropriate reactions to prevent threats like falling. But it is not clear that how human body maintains its stability by central nervous system (CNS). At present study it is tried that by presenting a musculoskeletal model of human lower extremity with four links, three degrees of freedom (DOF) and eight skeletal muscles, the level of muscle activations causes the maintenance of stability, be investigated. Using forward dynamics solution, leads to a more general problem, rather than inverse dynamics. Hence, forward dynamics solution by forward optimization has been used for solving this highly nonlinear problem. To this end, first the system’s equations of motion has been derived using lagrangian dynamics. Eight Hill-type muscles as actuators of the system were modeled. Because determination of muscle forces considering their number is an undetermined problem, optimization of an appropriate goal function should be practiced. For optimization problem, the characteristics of genetic algorithms as a method based on direct search, and the direct collocation method, has been profited. Also by considering requirements of problem, some constraints such as conservation of model stability are entered into optimization procedure. Finally to investigate validation of model, the results from optimization and experimental data are compared and good agreements are obtained.     

Stability of spectral types for Jacobi matrices under decaying random perturbations

We study stability of spectral types for semi-infinite self-adjoint tridiagonal matrices under random decaying perturbations. We show that absolutely continuous spectrum associated with bounded eigenfunctions is stable under Hilbert-Schmidt random perturbations. We also obtain some results for singular spectral types.     

Stability of Hausdorff dimension of Axiom A attractors under random perturbations

This paper proves that the Hausdorff dimension of an Axiom A attractor is stable under random perturbations.     

The multifractal analysis of Birkhoff averages for conformal repellers under random perturbations

We study the stability of multifractal structures for dynamical systems under small perturbations. For a repeller associated with an expanding C ^1+β -conformal topological mixing map, we show that the multifractal structure of Birkhoff averages is stable under small random perturbations.     

Singular continuous spectrum under rank one perturbations and localization for random hamiltonians

We consider a selfadjoint operator, A, and a selfadjoint rank-one projection, P, onto a vector, φ, which is cyclic for A. In terms of the spectral measure dμ^A_φ, we give necessary and sufficient conditions for A + λ P to have empty singular continuous spectrum or to have only point spectrum for a.e. λ. We apply these results to questions of localization in the one- and multi-dimensional Anderson models.     

The multifractal analysis of Birkhoff averages for conformal repellers under random perturbations

We study the stability of multifractal structures for dynamical systems under small perturbations. For a repeller associated with an expanding C ^1+β -conformal topological mixing map, we show that the multifractal structure of Birkhoff averages is stable under small random perturbations.     

Modeling reorientation phenomena in nonwoven materials with random fiber network microstructure

In this work, a micromechanically motivated affine full network model (AFNM) is used to simulate the load-displacement response of nonwoven materials. These materials are made from synthetic advanced fibers by bonding or interlocking networks of randomly laid fibers through mechanical, chemical or thermal processes. This results into a random fibrous network microstructure and a highly inhomogeneous response to external loadings with a very high degree of anisotropy. The load-displacement response under large strains is observed to be highly non-linear with a stiffening behavior which is also accompanied by complex dissipative phenomena. The preliminary simulation results with the AFNM, after reducing to two-dimensional setting, highlight its limitations and help identify the areas of improvement for a realistic material modeling. Reorientation of fibers are shown to play a critical role in the overall macroscale response of the materials. A new evolution law for reorientation is presented in the form of a first order ordinary differential equation where an exact analytical solution to the same is also computed. Improvements of the predicted load-displacement response signify the microscopic origins of the non-linear stiffening behavior as reorientation of fibers. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Small random perturbations of one-dimensional diffusion processes

In the present paper we consider the small random perturbations of one-dimensional diffusion processes. By virtue of stochastic analysis methods, we investigate the asymptotics of the mean exit times and probabilistical estimates of the exit times as the perturbations tend to zero. Partially supported by the Young Teachers Foundation of Beijing Institute of Technology     

Near-resonant motions in systems with random perturbations

The effect of random perturbations on near-resonant motions in non-linear oscillatory systems is investigated. It is assumed that the equations of motion of the system can be reduced to standard form with a small parameter ϵ, and that an isolated primary resonance exists in the unperturbed system [1]. The behaviour of the perturbed system in the ϵ-neighbourhood of the resonance surface is considered and an effect analogous to deterministic “capture in resonance” [1] in an asymptotically long time interval is investigated.     

Small random perturbations in second-order oscillatory systems

The limit behavior of the solutions of a nonlinear differential equation that describes an oscillatory system with small random perturbations of the type of multidimensional white and shot noises is studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 1, pp. 11–16, January, 1992.     

Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain

In this paper we obtain the continuity of attractors for semilinear parabolic problems with Neumann boundary conditions relatively to perturbations of the domain. We show that, if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the Neumann Laplacian is granted then, we obtain the upper semicontinuity of the attractors. If, moreover, every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors. We also give necessary and sufficient conditions for the spectral convergence of Neumann problems under perturbations of the domain.