Stochastic Map Research Articles

Early-warning signals for bifurcations in random dynamical systems with bounded noise

We consider discrete-time one-dimensional random dynamical systems with bounded noise, which generate an associated set-valued dynamical system. We provide necessary and sufficient conditions for a discontinuous bifurcation of a minimal invariant set of the set-valued dynamical system in terms of the derivatives of the so-called extremal maps. We propose an algorithm for reconstructing the derivatives of the extremal maps from a time series that is generated by iterations of the original random dynamical system. We demonstrate that the derivative reconstructed for different parameters can be used as an early-warning signal to detect an upcoming bifurcation, and apply the algorithm to the bifurcation analysis of the stochastic return map of the Koper model, which is a three-dimensional multiple time scale ordinary differential equation used as prototypical model for the formation of mixed-mode oscillation patterns. We apply our algorithm to data generated by this map to detect an upcoming transition.

Tomographic Portrait of Quantum Channels

We formulate the notion of quantum channels in the framework of quantum tomography and address there the issue of whether such maps can be regarded as classical stochastic maps. In particular, kernels of maps acting on probability representation of quantum states are derived for qubit and bosonic systems. In the latter case it results that a single mode Gaussian quantum channel corresponds to non-Gaussian classical channels.

Symmetries Versus Conservation Laws in Dynamical Quantum Systems: A Unifying Approach Through Propagation of Fixed Points

We unify recent Noether-type theorems on the equivalence of symmetries with conservation laws for dynamical systems of Markov processes, of quantum operations, and of quantum stochastic maps, by means of some abstract results on propagation of fixed points for completely positive maps on $$C^*$$ -algebras. We extend most of the existing results with characterisations in terms of dual infinitesimal generators of the corresponding strongly continuous one-parameter semigroups. By means of an ergodic theorem for dynamical systems of completely positive maps on von Neumann algebras, we show the consistency of the condition on the standard deviation for dynamical systems of quantum operations, and hence of quantum stochastic maps as well, in case the underlying Hilbert space is infinite dimensional.

Struwe-like solutions for the Stochastic Harmonic Map flow

We give new results on the well-posedness of the two-dimensional Stochastic Harmonic Map flow, whose study is motivated by the Landau–Lifshitz–Gilbert model for thermal fluctuations in micromagnetics. We first construct strong solutions that belong locally to the spaces $$C([s,t);H^1)\cap L^2([s,t);H^2)$$ , $$0\le s<t\le T$$ . It that sense, these maps are a counterpart of the so-called “Struwe solutions” of the deterministic model. We then provide a natural criterion of uniqueness that extends A. Freire’s Theorem to the stochastic case. Both results are obtained under the condition that the noise term has a trace-class covariance in space.

Iterative Filtering and Smoothing in Nonlinear and Non-Gaussian Systems Using Conditional Moments

This letter presents the development of novel iterated filters and smoothers that only require specification of the conditional moments of the dynamic and measurement models. This leads to generalizations of the iterated extended Kalman filter, the iterated extended Kalman smoother, the iterated posterior linearization filter, and the iterated posterior linearization smoother. The connections to the previous algorithms are clarified and a convergence analysis is provided. Furthermore, the merits of the proposed algorithms are demonstrated in simulations of the stochastic Ricker map where they are shown to have a similar or superior performance to competing algorithms.

Elementary Thermal Operations

To what extent do thermodynamic resource theories capture physically relevant constraints? Inspired by quantum computation, we define a set of elementary thermodynamic gates that only act on 2 energy levels of a system at a time. We show that this theory is well reproduced by a Jaynes-Cummings interaction in rotating wave approximation and draw a connection to standard descriptions of thermalisation. We then prove that elementary thermal operations present tighter constraints on the allowed transformations than thermal operations. Mathematically, this illustrates the failure at finite temperature of fundamental theorems by Birkhoff and Muirhead-Hardy-Littlewood-Polya concerning stochastic maps. Physically, this implies that stronger constraints than those imposed by single-shot quantities can be given if we tailor a thermodynamic resource theory to the relevant experimental scenario. We provide new tools to do so, including necessary and sufficient conditions for a given change of the population to be possible. As an example, we describe the resource theory of the Jaynes-Cummings model. Finally, we initiate an investigation into how our resource theories can be applied to Heat Bath Algorithmic Cooling protocols.

Roosting ecology of Stenodermatinae bats (Phyllostomidae): evolution of foliage roosting and correlated phenotypes

AbstractRoosting ecology has probably shaped several aspects of bat evolution. Although Phyllostomidae species are known to use more types of roosts than any other chiropteran lineage, foliage roosting is almost entirely restricted to the frugivorous members of the subfamily Stenodermatinae. There are relatively few studies on the roosting ecology of stenodermatines other than leaf tent users, and there have been few attempts to reconstruct the evolution of the peculiar foliage‐roosting habits of these fruit bats.Our aim was to reconstruct the evolution of foliage roosting in the Phyllostomidae phylogeny, and to test correlation hypotheses between roost types, pelage markings, and body size.We performed ancestral character reconstructions using stochastic mappings on a molecular phylogeny of Phyllostomidae and reviewed literature records on roosts used by Stenodermatinae. Correlations between roosting habits and pelage patterns were calculated using phylogenetic logistic regressions.Over 1200 records of roost use for 48 Stenodermatinae bat species were found in the literature. Most of the observations consisted of foliage roosting records; the second most common type of roost were caves and crevices, which was followed by holes in standing trees.Our results support a single origin of foliage roosting in Phyllostomidae, which we interpreted as a synapomorphy uniting Rhinophyllinae with the Stenodermatinae. We estimated a minimum of two origins of tent roosting within Phyllostomidae, and our reconstruction suggests that it was basal in the Rhinophyllinae + Stenodermatinae clade.Pelage markings of stenodermatines explained the type of roost, indicating that having white stripes or white spots contrasting with background pelage is related to foliage roosting. Species that use leaf tents are smaller than ones that do not, but group size does not appear to be correlated with tent use.Further studies of foliage‐roosting bat species that do not use leaf tents may shed light on the mode of evolution of this complex behavioural character.

Stochastic maps, continuous approximation, and stable distribution.

A continuous approximation framework for general nonlinear stochastic as well as deterministic discrete maps is developed. For the stochastic map with uncorelated Gaussian noise, by successively applying the Itô lemma, we obtain a Langevin type of equation. Specifically, we show how nonlinear maps give rise to a Langevin description that involves multiplicative noise. The multiplicative nature of the noise induces an additional effective force, not present in the absence of noise. We further exploit the continuum description and provide an explicit formula for the stable distribution of the stochastic map and conditions for its existence. Our results are in good agreement with numerical simulations of several maps.

Accurate and precise prediction of insulin sensitivity variance in critically ill patients

Abstract Background Glycaemic control (GC) in critical care can reduce mortality and improve clinical outcomes. Model based GC allows personalised and effective predictive control. Improving accuracy and precision of patient specific models, parameter identification, and resulting blood glucose predictions would enable more effective model-based GC algorithms. Methods Glycaemic data from 30 critically ill patient episodes was used to fit a model of glucose dynamics. In this model, insulin sensitivity (SI) is identified through time using b-spline basis functions. The number of basis functions, M, relative to the number of data points in each data set, N, determines the level of parameterisation of the model. By fitting the model to all 30 data sets, the identified SI profiles could be used to build stochastic maps of SI changes over time. These maps show how SI is expected to change based on the current SI and the overall behaviour of this cohort. Thus, for a given time step into the future, SI prediction distributions can be found. Stochastic maps were made for varying basis function degree (d) (d ∊ {0, 1, 2}), and ratio of M to N (M:N ∊ {0.3, 0.65, 0.85, 1}). Using small subsets of each data set, many trials were carried out to compare SI prediction distributions to the distribution of the true SI values at the next measurement, for each combination of M and d. The main aim was to observe how the model parameterisation affected the predictive ability of the model. Outcomes from an Akaike Information Criterion (AIC) analysis were compared to the prediction analysis. AIC is a method of model comparison that assesses which of the given models provides the best trade-off between goodness-of-fit and model complexity, based on the expected measurement noise. For each basis function order, an AIC analysis was used to select the best M:N ratio of the four options tested. Results Increasing the parameterisation of the model resulted in lower model-data residuals and thus wider stochastic maps, as SI changed at a faster rate to enable the model to more closely fit to the data. Therefore, increasing M:N resulted in wider prediction distributions. In all cases, when M:N = 1, the prediction distributions were too wide, and when M:N = 0.3 the prediction distributions were too narrow. Increasing the basis function order resulted in tighter prediction distributions, and allowed more accurate predictions to be made with a higher M:N ratio. The ratios that gave the most accurate predictions were M:N = 0.65 when d = 0, M:N = 0.85 when d = 1, and M:N = 1 when d = 2. In contrast, the AIC analysis found that an M:N ratio of approximately 0.45 was optimal in all cases. Conclusions This study presented a new strategy for observing how the level of parameterisation and basis function order affects accuracy of future predictions of SI variability. Applications based on the findings of this research with a larger cohort could lead to improved predictive capability in GC algorithms. Accurate predictions would ultimately allow model-based GC algorithms to be implemented more effectively, increasing clinician confidence and improving patient outcomes.

Mean-field dynamics of a population of stochastic map neurons.

We analyze the emergent regimes and the stimulus-response relationship of a population of noisy map neurons by means of a mean-field model, derived within the framework of cumulant approach complemented by the Gaussian closure hypothesis. It is demonstrated that the mean-field model can qualitatively account for stability and bifurcations of the exact system, capturing all the generic forms of collective behavior, including macroscopic excitability, subthreshold oscillations, periodic or chaotic spiking, and chaotic bursting dynamics. Apart from qualitative analogies, we find a substantial quantitative agreement between the exact and the approximate system, as reflected in matching of the parameter domains admitting the different dynamical regimes, as well as the characteristic properties of the associated time series. The effective model is further shown to reproduce with sufficient accuracy the phase response curves of the exact system and the assembly's response to external stimulation of finite amplitude and duration.

A stochastic hybrid systems based framework for modeling dependent failure processes.

In this paper, we develop a framework to model and analyze systems that are subject to dependent, competing degradation processes and random shocks. The degradation processes are described by stochastic differential equations, whereas transitions between the system discrete states are triggered by random shocks. The modeling is, then, based on Stochastic Hybrid Systems (SHS), whose state space is comprised of a continuous state determined by stochastic differential equations and a discrete state driven by stochastic transitions and reset maps. A set of differential equations are derived to characterize the conditional moments of the state variables. System reliability and its lower bounds are estimated from these conditional moments, using the First Order Second Moment (FOSM) method and Markov inequality, respectively. The developed framework is applied to model three dependent failure processes from literature and a comparison is made to Monte Carlo simulations. The results demonstrate that the developed framework is able to yield an accurate estimation of reliability with less computational costs compared to traditional Monte Carlo-based methods.

Generation of stochastic mobility maps for large-scale route planning of ground vehicles: A case study

This paper describes a simple and efficient methodology to generate a mobility map accounting for two sources of uncertainty, namely measurement errors (RMSE of a Digital Elevation Model) and interpolation error (kriging method). The proposed methodology means a general-purpose solution since it works with standard and publicly-available Digital Elevation Models (DEMs). The different regions in the map are classified according to the geometry of the surface (i.e. slope) and the soil type. A real USGS DEM demonstrates the suitability of the proposed methodology: (1) interpolation of a 26×40-km2 DEM to a finer resolution (30-m to 20-m); (2) analysis of the number of random realizations to account for the variability of the data; (3) efficient computation time (4-million-point DEM requires less than 30min to complete the whole process); (4) route planning using the stochastic mobility map (constraints in slope and soil properties). UNCLASSIFIED: Distribution Statement A. Approved for public release; distribution is unlimited. #27681

Cooperative minimum expected length planning for robot formations in stochastic maps

This paper addresses a tightly integrated multi-robot planning, localization and navigation system in stochastic scenarios. We present a novel motion planning technique for robot formations in such kinds of environments, which computes the most likely global path in terms of a defined minimum expected length (EL). EL evaluates the expected cost of a path considering the probability of finding a non traversable zone and the cost of using an alternative traversable path. A local real time re-planning technique based on the probabilistic model is also developed for the formation when the scenario changes. The formation adapts its configuration to the shape of the free room. The partial views of all the robots are integrated to update the multi-robot localization using a modified EKF based on the measurement differencing technique which improves estimation consistency. As a result, a lower uncertainty map of the local navigation area is obtained for re-planning purposes. Experimental results, both in simulation and in real office-like settings, illustrate the performance of the described approach where a hybrid, centralized–distributed, architecture with wireless communication capabilities is used.

Correlations, operations and the second law of thermodynamics

Completely positive trace preserving (CPTP) maps are essential for the formulation of the second law of thermodynamics. The dynamics of quantum systems initially correlated with their environments are in general not described by such maps. We explore, how this issue can be fixed by describing the classical analogue of this problem. We consider initially correlated probability distributions, whose subsequent system dynamics is ill-described by stochastic maps, and prescribe the correct way to describe the dynamics. We use this prescription to discuss the classical version of the second law, valid for initially correlated probability distributions.

Stochastic evaluation of the impact of sewer inlets’ hydraulic capacity on urban pluvial flooding

Sewer inlet structures are vital components of urban drainage systems and their operational conditions can largely affect the overall performance of the system. However, their hydraulic behaviour and the way in which it is affected by clogging is often overlooked in urban drainage models, thus leading to misrepresentation of system performance and, in particular, of flooding occurrence. In the present paper, a novel methodology is proposed to stochastically model stormwater urban drainage systems, taking the impact of sewer inlet operational conditions (e.g. clogging due to debris accumulation) on urban pluvial flooding into account. The proposed methodology comprises three main steps: (i) identification of sewer inlets most prone to clogging based upon a spatial analysis of their proximity to trees and evaluation of sewer inlet locations; (ii) Monte Carlo simulation of the capacity of inlets prone to clogging and subsequent simulation of flooding for each sewer inlet capacity scenario, and (iii) delineation of stochastic flood hazard maps. The proposed methodology was demonstrated using as case study design storms as well as two real storm events observed in the city of Coimbra (Portugal), which reportedly led to flooding in different areas of the catchment. The results show that sewer inlet capacity can indeed have a large impact on the occurrence of urban pluvial flooding and that it is essential to account for variations in sewer inlet capacity in urban drainage models. Overall, the stochastic methodology proposed in this study constitutes a useful tool for dealing with uncertainties in sewer inlet operational conditions and, as compared to more traditional deterministic approaches, it allows a more comprehensive assessment of urban pluvial flood hazard, which in turn enables better-informed flood risk assessment and management decisions.

The Free Energy Requirements of Biological Organisms; Implications for Evolution

Recent advances in nonequilibrium statistical physics have provided unprecedented insight into the thermodynamics of dynamic processes. The author recently used these advances to extend Landauer’s semi-formal reasoning concerning the thermodynamics of bit erasure, to derive the minimal free energy required to implement an arbitrary computation. Here, I extend this analysis, deriving the minimal free energy required by an organism to run a given (stochastic) map π from its sensor inputs to its actuator outputs. I use this result to calculate the input-output map π of an organism that optimally trades off the free energy needed to run π with the phenotypic fitness that results from implementing π. I end with a general discussion of the limits imposed on the rate of the terrestrial biosphere’s information processing by the flux of sunlight on the Earth.

Noisy homoclinic pulse dynamics.

The effect of stochastic perturbations on nearly homoclinic pulse trains is considered for three model systems: a Duffing oscillator, the Lorenz-like Shimizu-Morioka model, and a co-dimension-three normal form. Using the Duffing model as an example, it is demonstrated that the main effect of noise does not originate from the neighbourhood of the fixed point, as is commonly assumed, but due to the perturbation of the trajectory outside that region. Singular perturbation theory is used to quantify this noise effect and is applied to construct maps of pulse spacing for the Shimizu-Morioka and normal form models. The dynamics of these stochastic maps is then explored to examine how noise influences the sequence of bifurcations that take place adjacent to homoclinic connections in Lorenz-like and Shilnikov-type flows.

Positive and doubly stochastic maps, and majorization in Euclidean Jordan algebras

A positive map between Euclidean Jordan algebras is a (symmetric cone) order preserving linear map. We show that the norm of such a map is attained at the unit element, thus obtaining an analog of the operator/matrix theoretic Russo–Dye theorem. A doubly stochastic map between Euclidean Jordan algebras is a positive, unital, and trace preserving map. We relate such maps to Jordan algebra automorphisms and majorization.

Stochastic mappings and random distribution fields III. Module propagators and uniformly bounded linear stationarity

In this paper, the third in our sequence of papers on multivariate stochastic mappings, we focus on the type of, not necessarily stationary, stochastic mappings that are representable with the aid of a certain operator semigroup, called module propagator. As in the previous papers we obtain neater results for multivariate second order random distribution fields, from which the main result regards the extension to this setting of the theory of uniformly bounded linearly stationary random fields.

Analog quantum computing (AQC) and the need for time-symmetric physics

This paper discusses what will be necessary to achieve the full potential capabilities of analog quantum computing (AQC), which is defined here as the enrichment of continuous-variable computing to include stochastic, nonunitary circuit elements such as dissipative spin gates and address the wider range of tasks emerging from new trends in engineering, such as approximation of stochastic maps, ghost imaging and new forms of neural networks and intelligent control. This paper focuses especially on what is needed in terms of new experiments to validate remarkable new results in the modeling of triple entanglement, and in creating a pathway which links fundamental theoretical work with hard core experimental work, on a pathway to AQC similar to the pathway to digital quantum computing already blazed by Zeilinger's group. It discusses the most recent experiments and reviews two families of alternative models based on the traditional eigenvector projection model of polarizers and on a new family of local realistic models based on Markov Random Fields across space---time adhering to the rules of time-symmetric physics. For both families, it reviews lumped parameter versions, continuous time extension and possibilities for extension to continuous space and time.