Numerical Studies of Statistical Management Decisions in Conditions of Stochastic Chaos

The research presented in this article is dedicated to analyzing the acceptability of traditional techniques of statistical management decision-making in conditions of stochastic chaos. A corresponding example would be asset management at electronic capital markets. This formulation of the problem is typical for a large number of applications in which the managed object interacts with an unstable immersion environment. In particular, this issue arises in problems of managing gas-dynamic and hydrodynamic turbulent flows. We highlight the features of observation series of the managed object’s state immersed in an unstable interaction environment. The fundamental difference between observation series of chaotic processes and probabilistic descriptions of traditional models is demonstrated. We also present an additive observation model with a chaotic system component and non-stationary noise which provides the most adequate characterization of the original observation series. Furthermore, we suggest a method for numerically analyzing the efficiency of conventional statistical solutions in the conditions of stochastic chaos. Based on numerical experiments, we establish that techniques of optimal statistical synthesis do not allow for making effective management decisions in the conditions of stochastic chaos. Finally, we propose several versions of compositional algorithms focused on the adaptation of statistical techniques to the non-deterministic conditions caused by the specifics of chaotic processes.

Chaotic Features of Decomposed Time Series from Tidal River Water Level

This study assessed the characteristics of water-level time series of a tidal river by decomposing it into tide, wave, rainfall-runoff, and noise components. Especially, the analysis for chaotic behavior of each component was done by estimating the correlation dimension with phase-space reconstruction of time series and by using a close returns plot (CRP). Among the time series, the tide component showed chaotic characteristics to have a correlation dimension of 1.3. It was found out that the water level has stochastic characteristics showing the increasing trend of the correlation exponent in the embedding dimension. Other components also showed the stochastic characteristics. Then, the CRP was used to examine the characteristics of each component. The tide component showed the chaotic characteristics in its CRP. The CRP of water level showed an aperiodic characteristic which slightly strayed away from its periodicity, and this might be related to the tide component. This study showed that a low water level is mainly affected by a chaotic tide component through entropy information. Even though the water level did not show chaotic characteristics in the correlation dimension, it showed stochastic chaos characteristics in the CRP. Other components showed stochastic characteristics in the CRP. It was confirmed that the water level showed chaotic characteristics when it was not affected by rainfall and stochastic characteristics deviating from the bounded trajectory when water level rises due to rainfall. Therefore, we have shown that the water level related to the chaotic tide component can also have chaotic properties because water level is influenced by chaotic tide and rainfall shock, thus it showed stochastic chaos characteristics.

Stochastic Chaos and Markov Blankets

In this treatment of random dynamical systems, we consider the existence—and identification—of conditional independencies at nonequilibrium steady-state. These independencies underwrite a particular partition of states, in which internal states are statistically secluded from external states by blanket states. The existence of such partitions has interesting implications for the information geometry of internal states. In brief, this geometry can be read as a physics of sentience, where internal states look as if they are inferring external states. However, the existence of such partitions—and the functional form of the underlying densities—have yet to be established. Here, using the Lorenz system as the basis of stochastic chaos, we leverage the Helmholtz decomposition—and polynomial expansions—to parameterise the steady-state density in terms of surprisal or self-information. We then show how Markov blankets can be identified—using the accompanying Hessian—to characterise the coupling between internal and external states in terms of a generalised synchrony or synchronisation of chaos. We conclude by suggesting that this kind of synchronisation may provide a mathematical basis for an elemental form of (autonomous or active) sentience in biology.

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.

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.

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.

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.