Time operators and shift representation of dynamical systems

We introduce and characterise time operators for unilateral shifts and exact endomorphisms. The associated shift representation of evolution is related to the spectral representation by a generalized Fourier transform. We illustrate the results for a simple exact system, namely the Renyi map.

Positive dilations of piecewise monotonic Markov maps

We provide an explicit formula which gives natural extensions of piecewise monotonic Markov maps defined on an interval of the real line. These maps are exact endomorphisms and define chaotic discrete dynamical systems.

Analyticity of smooth eigenfunctions and spectral analysis of the Gauss map

We provide a sufficient condition of analyticity of infinitely differentiable eigenfunctions of operators of the form Uf(x) = integral a(x, y) f(b( x, y)) mu(dy) acting on functions f: [u, v] --> C ( evolution operators of one-dimensional dynamical systems and Markov processes have this form). We estimate from below the region of analyticity of the eigenfunctions and apply these results for studying the spectral properties of the Frobenius-Perron operator of the continuous fraction Gauss map. We prove that any infinitely differentiable eigenfunction f of this Frobenius-Perron operator, corresponding to a non-zero eigenvalue admits a (unique) analytic extension to the set C\(-infinity, 1]. Analyzing the spectrum of the Frobenius Perron operator in spaces of smooth functions, we extend significantly the domain of validity of the Mayer and Ropstorff asymptotic formula for the decay of correlations of the Gauss map.

Extended spectral decompositions of the Renyi map

We present new general methods to obtain spectral decompositions of dynamical systems in rigged Hilbert spaces and investigate the existence of resonances and the completeness of the associated eigenfunctions. The results are illustrated explicitly for the simplest chaotic endomorphism, namely the Renyi map.

Learning soft computing control strategies in a modular neural network architecture

Modelling and control of nonlinear dynamical systems is a challenging problem since the dynamics of such systems change over their parameter space. Conventional methodologies for designing nonlinear control laws, such as gain scheduling, are effective because the designer partitions the overall complex control into a number of simpler sub-tasks. This paper describes a new genetic algorithm based method for the design of a modular neural network (MNN) control architecture that learns such partitions of an overall complex control task. Here a chromosome represents both the structure and parameters of an individual neural network in the MNN controller and a hierarchical fuzzy approach is used to select the chromosomes required to accomplish a given control task. This new strategy is applied to the end-point tracking of a single-link flexible manipulator modelled from experimental data. Results show that the MNN controller is simple to design and produces superior performance compared to a single neural network (SNN) controller which is theoretically capable of achieving the desired trajectory. (C) 2003 Elsevier Ltd. All rights reserved.

Direct, positive feedbacks produce instability in models of interrelationships among soil structure, plants and arbuscular mycorrhizal fungi

Current conceptual models of reciprocal interactions linking soil structure, plants and arbuscular mycorrhizal fungi emphasise positive feedbacks among the components of the system. However, dynamical systems with high dimensionality and several positive feedbacks (i.e. mutualism) are prone to instability. Further, organisms such as arbuscular mycorrhizal fungi (AMF) are obligate biotrophs of plants and are considered major biological agents in soil aggregate stabilization. With these considerations in mind, we developed dynamical models of soil ecosystems that reflect the main features of current conceptual models and empirical data, especially positive feedbacks and linear interactions among plants, AMF and the component of soil structure dependent on aggregates. We found that systems become increasingly unstable the more positive effects with Type I functional response (i.e., the growth rate of a mutualist is modified by the density of its partner through linear proportionality) are added to the model, to the point that increasing the realism of models by adding linear effects produces the most unstable systems. The present theoretical analysis thus offers a framework for modelling and suggests new directions for experimental studies on the interrelationship between soil structure, plants and AMF. Non-linearity in functional responses, spatial and temporal heterogeneity, and indirect effects can be invoked on a theoretical basis and experimentally tested in laboratory and field experiments in order to account for and buffer the local instability of the simplest of current scenarios. This first model presented here may generate interest in more explicitly representing the role of biota in soil physical structure, a phenomenon that is typically viewed in a more process- and management-focused context. (C) 2011 Elsevier Ltd. All rights reserved.

Overcoming the Scalability Challenges of Epidemic Simulations on Blue Waters

Modeling dynamical systems represents an important application class covering a wide range of disciplines including but not limited to biology, chemistry, finance, national security, and health care. Such applications typically involve large-scale, irregular graph processing, which makes them difficult to scale due to the evolutionary nature of their workload, irregular communication and load imbalance. EpiSimdemics is such an application simulating epidemic diffusion in extremely large and realistic social contact networks. It implements a graph-based system that captures dynamics among co-evolving entities. This paper presents an implementation of EpiSimdemics in Charm++ that enables future research by social, biological and computational scientists at unprecedented data and system scales. We present new methods for application-specific processing of graph data and demonstrate the effectiveness of these methods on a Cray XE6, specifically NCSA's Blue Waters system.

Prediction of limit cycle oscillations under uncertainty using a Harmonic Balance method

The Harmonic Balance method is an attractive solution for computing periodic responses and can be an alternative to time domain methods, at a reduced computational cost. The current paper investigates using a Harmonic Balance method for simulating limit cycle oscillations under uncertainty. The Harmonic Balance method is used in conjunction with a non-intrusive polynomial-chaos approach to propagate variability and is validated against Monte Carlo analysis. Results show the potential of the approach for a range of nonlinear dynamical systems, including a full wing configuration exhibiting supercritical and subcritical bifurcations, at a fraction of the cost of performing time domain simulations.

Just a matter of time: Fungi and roots significantly and rapidly aggregate soil over four decades along the Tagliamento River, NE Italy

Fluvial islands are emergent landforms which form at the interface between the permanently inundated areas of the river channel and the more stable areas of the floodplain as a result of interactions between physical river processes, wood and riparian vegetation. These highly dynamical systems are ideal to study soil structure development in the short to medium term, a process in which soil biota and plants play a substantial role. We investigated soil structure development on islands along a 40 year chronosequence within a 3 km island-braided reach of the Tagliamento River, Northeastern Italy. We used several parameters to capture different aspects of the soil structure, and measured biotic (e.g., fungal and plant root parameters) and abiotic (e.g. organic carbon) factors expected to determine the structure. We estimated models relating soil structure to its determinants, and, in order to confer statistical robustness to our results, we explicitly took into account spatial autocorrelation, which is present due to the space for time substitution inherent in the study of chronosequences and may have confounded results of previous studies. We found that, despite the eroding forces from the hydrological and geomorphological dynamics to which the system is subject, all soil structure variables significantly, and in some case greatly increased with site age. We interpret this as a macroscopic proxy for the major direct and indirect binding effects exerted by root variables and extraradical hyphae of arbuscular mycorrhizal fungi (AMF). Key soil structure parameters such as percentage of water stable aggregates (WSA) can double from the time the island landform is initiated (mean WSA = 30%) to the full 40 years (mean WSA = 64%) covered by our chronosequence. The study demonstrates the fundamental role of soil biota and plant roots in aggregating soils even in a system in which intense short to medium term physical disturbances are common.

A compendium of modelling techniques

This Integration Insight provides a brief overview of the most popular modelling techniques used to analyse complex real-world problems, as well as some less popular but highly relevant techniques. The modelling methods are divided into three categories, with each encompassing a number of methods, as follows: 1) Qualitative Aggregate Models (Soft Systems Methodology, Concept Maps and Mind Mapping, Scenario Planning, Causal (Loop) Diagrams), 2) Quantitative Aggregate Models (Function fitting and Regression, Bayesian Nets, System of differential equations / Dynamical systems, System Dynamics, Evolutionary Algorithms) and 3) Individual Oriented Models (Cellular Automata, Microsimulation, Agent Based Models, Discrete Event Simulation, Social Network
Analysis). Each technique is broadly described with example uses, key attributes and reference material.

Disturbance attenuation in linear systems via dynamical compensators: A parametric eigenstructure assignment approach

A simple approach is proposed for disturbance attenuation in multivariable linear systems via dynamical output compensators based on complete parametric eigenstructure assignment. The basic idea is to minimise the H-2 norm of the disturbance-output transfer function using the design freedom provided by eigenstructure assignment. For robustness, the closed-loop system is restricted to be nondefective. Besides the design parameters, the closed-loop eigenvalues are also optimised within desired regions on the left-half complex plane to ensure both closed-loop stability and dynamical performance. With the proposed approach, additional closed-loop specifications can be easily achieved. As a demonstration, robust pole assignment, in the sense that the closed-loop eigenvalues are as insensitive as possible to open-loop system parameter perturbations, is treated. Application of the proposed approach to robust control of a magnetic bearing with a pair of opposing electromagnets and a rigid rotor is discussed.

Enhanced dynamical entanglement transfer with multiple qubits

We present two strategies to enhance the dynamical entanglement transfer from continuous-variable (CV) to finite-dimensional systems by employing multiple qubits. First, we consider the entanglement transfer to a composite finite-dimensional system of many qubits simultaneously interacting with a bipartite CV field. We show that, considering realistic conditions in the generation of CV entanglement, a small number of qubits resonantly coupled to the CV system are sufficient for an almost complete dynamical transfer of the entanglement. Our analysis also sheds further light on the transition between the microscopic and macroscopic behaviors of composite finite-dimensional systems coupled to bosonic fields (like atomic clouds interacting with light). Furthermore, we present a protocol based on sequential interactions of the CV system with some ancillary qubit systems and on subsequent measurements, allowing us to probabilistically convert CV entanglement into "almost-perfect" Bell pairs of two qubits. Our proposals are suited for realizations in various experimental settings, ranging from cavity-QED to cavity-integrated superconducting devices.