Stochastic Chaos Research Articles

Topological supersymmetry breaking: The definition and stochastic generalization of chaos and the limit of applicability of statistics

The concept of deterministic dynamical chaos has a long history and is well established by now. Nevertheless, its field theoretic essence and its stochastic generalization have been revealed only very recently. Within the newly found supersymmetric theory of stochastics (STS), all stochastic differential equations (SDEs) possess topological or de Rahm supersymmetry and stochastic chaos is the phenomenon of its spontaneous breakdown. Even though the STS is free of approximations and thus is technically solid, it is still missing a firm interpretational basis in order to be physically sound. Here, we make a few important steps toward the construction of the interpretational foundation for the STS. In particular, we discuss that one way to understand why the ground states of chaotic SDEs are conditional (not total) probability distributions, is that some of the variables have infinite memory of initial conditions and thus are not “thermalized”, i.e., cannot be described by the initial-conditions-independent probability distributions. As a result, the definitive assumption of physical statistics that the ground state is a steady-state total probability distribution is not valid for chaotic SDEs.

Rationalize the irrational and fractional expressions in nonlinear analysis

Chebyshev polynomial approximation is an effective method to study the stochastic bifurcation and chaos. However, due to irrational and fractional expressions existing in the denominator of some mechanical systems, the integral process is very complicated. The Taylor series expansion is proposed to expand the irrational and fractional expressions into a series of polynomials. Smooth and discontinuous oscillator was taken as an example, and the results show that the Taylor series expansion method is acceptable. The rub-impact force was taken as another example. Numerical results indicate that the method is suitable for the rub-impact rotor system.

Fixture Capability Optimisation for Early-stage Design of Assembly System with Compliant Parts Using Nested Polynomial Chaos Expansion

This paper introduces the novel concept of fixture capability measure to determine fixture layout for the best assembly process yield by optimizing position of locators and reference clamps to compensate stochastic product variations and part deformation. This allows reducing the risk of product failures caused by product and process variation. The method is based on three main steps: (i) physics-based modelling of parts and fixtures, (ii) stochastic polynomial chaos expansion to calculate fixture capability, and (iii) fixture capability optimisation using surrogate modelling. The methodology is demonstrated and validated using the results of an aerospace wing sub-assembly joined by riveting technique.

Parallel preconditioners and multigrid solvers for stochastic polynomial chaos discretizations of the diffusion equation at the large scale

SummaryThis paper presents parallel preconditioners and multigrid solvers for solving linear systems of equations arising from stochastic polynomial chaos formulations of the diffusion equation with random coefficients. These preconditioners and solvers are extensions of the preconditioner developed in an earlier paper for strongly coupled systems of elliptic partial differential equations that are norm equivalent to systems that can be factored into an algebraic coupling component and a diagonal differential component. The first preconditioner, which is applied to the norm equivalent system, is obtained by sparsifying the inverse of the algebraic coupling component. This sparsification leads to an efficient method for solving these systems at the large scale, even for problems with large statistical variations in the random coefficients. An extension of this preconditioner leads to stand‐alone multigrid methods that can be applied directly to the actual system rather than to the norm equivalent system. These multigrid methods exploit the algebraic/differential factorization of the norm equivalent systems to produce variable‐decoupled systems on the coarse levels. Moreover, the structure of these methods allows easy software implementation through re‐use of robust high‐performance software such as the Hypre library package. Two‐grid matrix bounds will be established, and numerical results will be given. Copyright © 2015 John Wiley & Sons, Ltd.

Stochastic polynomial chaos based algorithm for solving PDEs with random coefficients

Abstract A generalization of a polynomial chaos-based algorithm for solving PDEs with random input data is suggested. The input random field is assumed to be defined by its mean and correlation function. The method uses the Karhunen–Loève expansion, in its analytical form, for the input random field. Potentially, however, if desired, the Karhunen–Loève expansion can be also constructed by a randomized singular value decomposition of the correlation function recently suggested in our paper [Math. Comput. Simulation 82 (2011), 295–317]. The polynomial chaos expansion is then constructed by resolving a probabilistic collocation-based system of linear equations. The method is compared against a direct Monte Carlo method which solves repeatedly many times the PDE for a set of samples of the input random field. Along with the commonly used statistical characteristics like the mean and variance of the solution, we were able to calculate more sophisticated functionals like the instant velocity samples and the mean for Eulerian and Lagrangian velocity fields.

Vibration of a shape memory alloy beam and detecting the threshold value of chaos induced by noise

The aim of this study is to describe the hysteretic nonlinear characteristic of the strain–stress relationship of a shape memory alloy; a Van-der-Pol hysteretic cycle is applied to simulate the hysteretic loops. Then, the model of a simply supported shape memory alloy beam with nonlinear damping was proposed. The stability of the free vibration is studied. The criterion determining the stochastic chaos is obtained by the random Melnikov approach. To verify the theoretical analysis, numerical experiments are carried out. The results totally agreed with the analytical work. Clear fractal boundaries of the system's safe basin are observed.

Stochastic bifurcation and fractal and chaos control of a giant magnetostrictive film-shape memory alloy composite cantilever plate subjected to in-plane harmonic and stochastic excitation

Stochastic bifurcation and fractal and chaos control of a giant magnetostrictive film–shape memory alloy (GMF–SMA) composite cantilever plate subjected to in-plane harmonic and stochastic excitation were studied. Van der Pol items were improved to interpret the hysteretic phenomena of both GMF and SMA, and the nonlinear dynamic model of a GMF–SMA composite cantilever plate subjected to in-plane harmonic and stochastic excitation was developed. The probability density function of the dynamic response of the system was obtained, and the conditions of stochastic Hopf bifurcation were analyzed. The conditions of noise-induced chaotic response were obtained in the stochastic Melnikov integral method, and the fractal boundary of the safe basin of the system was provided. Finally, the chaos control strategy was proposed in the stochastic dynamic programming method. Numerical simulation shows that stochastic Hopf bifurcation and chaos appear in the parameter variation process. The boundary of the safe basin of the system has fractal characteristics, and its area decreases when the noise intensifies. The system reliability was improved through stochastic optimal control, and the safe basin area of the system increased.

Non-linear dynamic characteristics and optimal control of giant magnetostrictive film subjected to in-plane stochastic excitation

The non-linear dynamic characteristics and optimal control of a giant magnetostrictive film (GMF) subjected to in-plane stochastic excitation were studied. Non-linear differential items were introduced to interpret the hysteretic phenomena of the GMF, and the non-linear dynamic model of the GMF subjected to in-plane stochastic excitation was developed. The stochastic stability was analysed, and the probability density function was obtained. The condition of stochastic Hopf bifurcation and noise-induced chaotic response were determined, and the fractal boundary of the system's safe basin was provided. The reliability function was solved from the backward Kolmogorov equation, and an optimal control strategy was proposed in the stochastic dynamic programming method. Numerical simulation shows that the system stability varies with the parameters, and stochastic Hopf bifurcation and chaos appear in the process; the area of the safe basin decreases when the noise intensifies, and the boundary of the safe basin becomes fractal; the system reliability improved through stochastic optimal control. Finally, the theoretical and numerical results were proved by experiments. The results are helpful in the engineering applications of GMF.

Modeling of a Shape Memory Alloy Beam and its Stochastic Chaos

The van-der-pol hysteretic cycle was applied to describe the hysteretic nonlinear characteristic of the strain-stress relation of a shape memory alloy (SMA). A new model with nonlinear damping of a simply supported SMA beam was proposed. The Criterions determining the stochastic chaos is obtained by the random Melnikov approach. The numerical results show the effectiveness of the theoretical analysis. Clear fractal boundaries of the system's safe basin is observed.

Reduced chaos expansions with random coefficientsin reduced‐dimensional stochastic modeling of coupled problems

SUMMARYWe address the curse of dimensionality in methods for solving stochastic coupled problems with an emphasis on stochastic expansion methods such as those involving polynomial chaos expansions. The proposed method entails a partitioned iterative solution algorithm that relies on a reduced‐dimensional representation of information exchanged between subproblems to allow each subproblem to be solved within its own stochastic dimension while interacting with a reduced projection of the other subproblems. The proposed method extends previous work by the authors by introducing a reduced chaos expansion with random coefficients. The representation of the exchanged information by using this reduced chaos expansion with random coefficients enables an expeditious construction of doubly stochastic polynomial chaos expansions that separate the effect of uncertainty local to a subproblem from the effect of statistically independent uncertainty coming from other subproblems through the coupling. After laying out the theoretical framework, we apply the proposed method to a multiphysics problem from nuclear engineering. Copyright © 2013 John Wiley & Sons, Ltd.

Restoration of Coherent Population Movement from Noise-Induced Chaos in the Chemotaxis of E. Coli

Bacteria navigating in a chemically guided manner are under the impact of noise from at least three sources – inside the cells, at the binding sites between chemoattractants in the environment and corresponding receptors of the cells, and in the environment itself. For Escherichia coli as model system, compounded effects of these sources of noise were investigated recently by using the fractal dimensions of the trajectories of the cells as an index of the nature of population motility. It was observed that environmental noise can drive synchronized movement of a population toward a chemoattractant into stochastic chaos. Those results have been used here to explore the effectiveness of different kinds of noise filters in restoring coherent motion of the cells. An auto-associative neural filter was the best, followed by the extended Kalman filter. The performance of either filter depended on the relative rates of motion of the bacteria and the chemoattractant, and on whether the responses of the cells to fluctuations in the external chemoattractant was non-adaptive or adaptive. The results establish: (a) the validity and usefulness of fractal indexes to characterize noise-affected chemotaxis, (b) the significance of the effect of environmental noise on chemotactic motility, and (c) the effectiveness of a neural filter in rescuing coherent population movement from noise-induced chaos.

Statistics of Poincaré recurrences in local and global approaches

The basic statistical characteristics of the Poincaré recurrence sequence are obtained numerically for the logistic map in the chaotic regime. The mean values, variance and recurrence distribution density are calculated and their dependence on the return region size is analyzed. It is verified that the Afraimovich–Pesin dimension may be evaluated by the Kolmogorov–Sinai entropy. The peculiarities of the influence of noise on the recurrence statistics are studied in local and global approaches. It is shown that the obtained numerical data are in complete agreement with the theoretical results. It is demonstrated that the Poincaré recurrence theory can be applied to diagnose effects of stochastic resonance and chaos synchronization and to calculate the fractal dimension of a chaotic attractor.

FRACTAL ANALYSIS OF LIPASE–CATALYSED SYNTHESIS OF BUTYL BUTYRATE IN A MICROBIOREACTOR UNDER THE INFLUENCE OF NOISE

Microbioreactors operated in real environments are often subject to noise from the environment. This is commonly manifested as fluctuations in the flow rates of the feed streams. Previous studies with larger bioreactors have shown that noise can seriously impair the performance. Given this possibility, the effects of noise on the performance of a microbioreactor have been analyzed for the trans-esterification of vinyl butyrate by 1-butanol by immobilized lipase B to produce butyl butyrate. As in previous work for macrobioreactors, the analysis was done with (i) no noise, (ii) unfiltered noise, and (iii) noise filtered by four different methods, and the fractal dimension of the product was used as an index of the performance. All fractal dimensions decreased with increasing dilution rates, and significant stochastic chaos was likely at low dilution rates. Of the four types of filters, the auto-associative neural filter (ANF) was the most effective in reducing chaos and restoring of smooth, nearly noise-free performance. The ANF also does not require a process model, which is a significant advantage for real systems. Simulations also revealed that even in the absence of noise, deterministic chaos is possible at low dilution rates; this underscores the importance of efficient filtering under such conditions when external noise too is present. The results thus establish the importance of noise in microbioreactor behavior and the usefulness of the fractal dimension in characterizing the effects.

Stochastic and deterministic multicloud parameterizations for tropical convection

Despite recent advances in supercomputing, current general circulation models poorly represent the variability associated with organized tropical convection. In a recent study, the authors have shown, in the context of a paradigm two baroclinic mode system, that a stochastic multicloud convective parameterization based on three cloud types (congestus, deep and stratiform) can be used to improve the variability and the dynamical structure of tropical convection. Here, the stochastic multicloud model is modified with a lag type stratiform closure and augmented with an explicit mechanism for congestus detrainment moistening. These modifications improve the representation of intermittent coherent structures such as synoptic and mesoscale convective systems. Moreover, the new stratiform-lag closure allows for increased robustness of the coherent features of the model with respect to the amount of stochastic noise and leading to a multi-scale organization of slowly moving waves envelopes in which short-lived and chaotic convective events persist. Congestus cloud decks dominate the suppressed-dry phase of the wave envelopes. The simulations with the new closure have a higher amount of stochastic noise and result in a Walker type circulation with realistic mean and coherent variability which surpasses results of previous deterministic and stochastic multicloud models in the same parameter regime. Further, deterministic mean field limit equations (DMFLE) for the stochastic multicloud model are considered. Aside from providing a link to the deterministic multicloud parameterization, the DMFLE allow a judicious way of determining the amount of deterministic and stochastic “chaos” in the system. It is shown that with the old stratiform heating closure, the stochastic process accounts for most of the chaotic behavior. The simulations with the new stratiform heating closure exhibit a mixture of stochastic and deterministic chaos. The highly chaotic dynamics in the simulations with congestus detrainment mechanism is due to the strongly nonlinear and numerically stiff deterministic dynamics. In the latter two cases, the DMFLE can be viewed as a “standalone” parameterization, which is capable of capturing some dynamical features of the stochastic parameterization. Furthermore, it is shown that, in spatially extended simulations, the stochastic multicloud model can capture qualitatively two local statistical features of the observations: long and short auto-correlation times of moisture and precipitation, respectively and the approximate power-law in the probability density of precipitation event size for large precipitation events. The latter feature is not reproduced in the column simulations. This fact underscores the importance of gravity waves and large scale moisture convergence.

Adaptive Synchronization for Two Different Stochastic Chaotic Systems with Unknown Parameters via a Sliding Mode Controller

This paper investigates the problem of synchronization for two different stochastic chaotic systems with unknown parameters and uncertain terms. The main work of this paper consists of the following aspects. Firstly, based on the Lyapunov theory in stochastic differential equations and the theory of sliding mode control, we propose a simple sliding surface and discuss the occurrence of the sliding motion. Secondly, we design an adaptive sliding mode controller to realize the asymptotical synchronization in mean squares. Thirdly, we design an adaptive sliding mode controller to realize the almost surely synchronization. Finally, the designed adaptive sliding mode controllers are used to achieve synchronization between two pairs of different stochastic chaos systems (Lorenz-Chen and Chen-Lu) in the presence of the uncertainties and unknown parameters. Numerical simulations are given to demonstrate the robustness and efficiency of the proposed robust adaptive sliding mode controller.

Simulation-based time-dependent reliability analysis for composite hydrokinetic turbine blades

The reliability of blades is vital to the system reliability of a hydrokinetic turbine. A time-dependent reliability analysis methodology is developed for river-based composite hydrokinetic turbine blades. Coupled with the blade element momentum theory, finite element analysis is used to establish the responses (limit-state functions) for the failure indicator of the Tsai---Hill failure criterion and blade deflections. The stochastic polynomial chaos expansion method is adopted to approximate the limit-state functions. The uncertainties considered include those in river flow velocity and composite material properties. The probabilities of failure for the two failure modes are calculated by means of time-dependent reliability analysis with joint upcrossing rates. A design example for the Missouri river is studied, and the probabilities of failure are obtained for a given period of operation time.

Stochastic equilibria control and chaos suppression for 3D systems via stochastic sensitivity synthesis

In this paper, we consider a nonlinear controlled system forced by stochastic disturbances. The problem addressed is to design a feedback regulator that can stabilize an equilibrium of the closed-loop system and, around this equilibrium, to synthesize a required dispersion of random states of the corresponding system. We use a stochastic sensitivity function technique to approximate the stationary probabilistic distribution of these random states. We also develop a new method for stabilization based on the stochastic sensitivity synthesis. A constructive description of the attainability set of the stochastic sensitivity matrices for a 3D system is given. The effectiveness of the new approach is demonstrated by the 3D stochastic Chen system. It is shown that the new regulator provides a low level of sensitivity and can suppress both regular and chaotic oscillations.

Hopf bifurcation and uncontrolled stochastic traffic-induced chaos in an RED-AQM congestion control system

We study the Hopf bifurcation and the chaos phenomena in a random early detection-based active queue management (RED-AQM) congestion control system with a communication delay. We prove that there is a critical value of the communication delay for the stability of the RED-AQM control system. Furthermore, we show that the system will lose its stability and Hopf bifurcations will occur when the delay exceeds the critical value. When the delay is close to its critical value, we demonstrate that typical chaos patterns may be induced by the uncontrolled stochastic traffic in the RED-AQM control system even if the system is still stable, which reveals a new route to the chaos besides the bifurcation in the network congestion control system. Numerical simulations are given to illustrate the theoretical results.

Stabilization of stochastic cycles and chaos suppression for nonlinear discrete-time systems

In this paper we consider a nonlinear discrete-time control system with regular and chaotic dynamics forced by stochastic disturbances. The problem addressed is the design of the feedback regulator which stabilizes a limit cycle of the closed-loop deterministic system and synthesizes a required dispersion of random states for the corresponding stochastic system. To solve this problem, we propose a new method based on the stochastic sensitivity function technique. This function approximates a dispersion of random states distributed around deterministic cycle. Explicit formulas for the intercoupling between stochastic sensitivity function and considered system parameters are worked out. The problem of the design of the required stochastic sensitivity function for cycles by feedback regulators is solved. Coefficients of the feedback regulator are constructed and corresponding attainability sets are described. The effectiveness of the proposed approach is demonstrated on the stochastic Verhulst model. It is shown that constructed regulators provide a low level of sensitivity and suppress chaotic oscillations.

Digital simulation of road–vehicle systems

This paper investigates dynamics of road–vehicle systems via stochastic numerics, applying discrete integration schemes of first and second order. The ride on rough roads generates vertical car vibrations whose root mean squares become resonant for critical speeds. The investigations are extended to nonlinear wheel suspensions with cubic–progressive springs. For weak but still positive damping, the car vibrations become unstable in overcritical speed ranges detected by means of perturbation equations whose top Lyapunov exponent can become positive in the case of parameter resonances. This indicates that the stationary car vibrations bifurcate into stochastic chaos.