Simple Dynamical System Research Articles

A simple model of a Slinky walking down stairs

The motion of a Slinky walking down a set of stairs is modeled by a simple dynamical system with two degrees of freedom undergoing inelastic collisions. Numerical integration of the model’s equations of motion shows that the model’s behavior is similar to observations of the motion of an actual Slinky. In particular, it is found that the model’s motion exhibits a periodic gait, although subject to more restrictive launch conditions than an actual Slinky.

Kinetic Friction by a Small Number of Intervening Inelastic Particles between Rough Surfaces

We investigate a mechanism of the appearance of kinetic friction in granular materials. We consider a small number of intervening inelastic particles between two rough surfaces as one of the simplest dynamical models to study granular friction. The resistance force applied to the upper surface is numerically calculated. We find that the resistance force F ( t ) is scaled as F ′ ( v t ) for a small pulling velocity v . The time average F 0 =< F ( t )> in the limit v →0 is not zero owing to the mutual collisions between the intervening particles. The nonzero F 0 implies the appearance of kinetic friction in this simple dynamical system.

A dynamical systems perspective on the relationship between symbolic and non-symbolic computation

It has been claimed that connectionist (artificial neural network) models of language processing, which do not appear to employ "rules", are doing something different in kind from classical symbol processing models, which treat "rules" as atoms (e.g., McClelland and Patterson in Trends Cogn Sci 6(11):465-472, 2002). This claim is hard to assess in the absence of careful, formal comparisons between the two approaches. This paper formally investigates the symbol-processing properties of simple dynamical systems called affine dynamical automata, which are close relatives of several recurrent connectionist models of language processing (e.g., Elman in Cogn Sci 14:179-211, 1990). In line with related work (Moore in Theor Comput Sci 201:99-136, 1998; Siegelmann in Neural networks and analog computation: beyond the Turing limit. Birkhäuser, Boston, 1999), the analysis shows that affine dynamical automata exhibit a range of symbol processing behaviors, some of which can be mirrored by various Turing machine devices, and others of which cannot be. On the assumption that the Turing machine framework is a good way to formalize the "computation" part of our understanding of classical symbol processing, this finding supports the view that there is a fundamental "incompatibility" between connectionist and classical models (see Fodor and Pylyshyn 1988; Smolensky in Behav Brain Sci 11(1):1-74, 1988; beim Graben in Mind Matter 2(2):29--51,2004b). Given the empirical successes of connectionist models, the more general, super-Turing framework is a preferable vantage point from which to consider cognitive phenomena. This vantage may give us insight into ill-formed as well as well-formed language behavior and shed light on important structural properties of learning processes.

이중 블록 계통의 비선형 지진응답 특성

This paper discusses about modeling method to simulate a nonlinear behavior like sliding or rocking of two stacked body system under earthquake condition. A double body system design can be an option to reduce seismic response of a component in comparison to a single body system for free standing structures. Therefore, according to the priority of components, the structure is to be designed by proper ratio of partition in their height for improvement of seismic capability and structural integrity. Nonlinear modeling and analysis using simple rigid body and dynamic system has been performed to check the trend in such cases. As a result, one of the two bodies can be chosen to reduce the seismic response from energy absorption of the other one by appropriate application of friction ratios not only in slip-slip condition but in slip-rock condition. 기호설명 b : 블록(block)의 폭 c : 감쇠계수(damping coefficient) d : 무게중심의 이격 e : 반발계수(coefficient of restitution) f : 마찰력 g : 중력가속도 [m/s

Small-scale instabilities in dynamical systems with sliding

We demonstrate with a minimal example that in Filippov systems (dynamical systems governed by discontinuous but piecewise smooth vector fields) stable periodic motion with sliding is not robust with respect to stable singular perturbations. We consider a simple dynamical system that we assume to be a quasi-static approximation of a higher-dimensional system containing a fast stable subsystem. We tune a system parameter such that a stable periodic orbit of the simple system touches the discontinuity surface: this is the so-called grazing-sliding bifurcation. The periodic orbit remains stable, and its local return map becomes piecewise linear. However, when we take into account the fast dynamics the local return map of the periodic orbit changes qualitatively, giving rise to, for example, period-adding cascades or small-scale chaos.

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A Simple Model of Cortical Dynamics Explains Variability and State Dependence of Sensory Responses in Urethane-Anesthetized Auditory Cortex

The responses of neocortical cells to sensory stimuli are variable and state dependent. It has been hypothesized that intrinsic cortical dynamics play an important role in trial-to-trial variability; the precise nature of this dependence, however, is poorly understood. We show here that in auditory cortex of urethane-anesthetized rats, population responses to click stimuli can be quantitatively predicted on a trial-by-trial basis by a simple dynamical system model estimated from spontaneous activity immediately preceding stimulus presentation. Changes in cortical state correspond consistently to changes in model dynamics, reflecting a nonlinear, self-exciting system in synchronized states and an approximately linear system in desynchronized states. We propose that the complex and state-dependent pattern of trial-to-trial variability can be explained by a simple principle: sensory responses are shaped by the same intrinsic dynamics that govern ongoing spontaneous activity.

New Microscopic Mechanism for Secondary Relaxation in Glasses

The dynamics of simple molecular systems showing glassy properties has been explored by dielectric spectroscopy and nuclear quadrupole resonance (NQR) on the halogenomethanes CBr2Cl2 and CBrCl3 in their low-temperature monoclinic phases. The dielectric spectra display features which correspond to alpha- and beta-relaxation processes, commonly observed in canonical glass formers. NQR experiments, also performed in the ergodic monoclinic phase of CCl4, enable the determination of the microscopic mechanism underlying the beta dynamics in these simple model glasses: Molecules that are nonequivalent with respect to their molecular environment perform reorientational jumps at different time scales. Thus our findings reveal another mechanism that can give rise to typical beta-relaxation behavior, raising some doubt about the existence of a universal explanation of this phenomenon.

On the use of simple dynamical systems for climate predictions

Over the last few decades, climate scientists have devoted much effort to the development of large numerical models of the atmosphere and the ocean. While there is no question that such models provide important and useful information on complicated aspects of atmosphere and ocean dynamics, skillful prediction also requires a phenomenological approach, particularly for very slow processes, such as glacial-interglacial cycles. Phenomenological models are often represented as low-order dynamical systems. These are tractable, and a rich source of insights about climate dynamics, but they also ignore large bodies of information on the climate system, and their parameters are generally not operationally defined. Consequently, if they are to be used to predict actual climate system behaviour, then we must take very careful account of the uncertainty introduced by their limitations. In this paper we consider the problem of the timing of the next glacial inception, about which there is on-going debate. Our model is the three-dimensional stochastic system of Saltzman and Maasch (1991), and our inference takes place within a Bayesian framework that allows both for the limitations of the model as a description of the propagation of the climate state vector, and for parametric uncertainty. Our inference takes the form of a data assimilation with unknown static parameters, which we perform with a variant on a Sequential Monte Carlo technique (`particle filter’). Provisional results indicate peak glacial conditions in 60,000 years.

Open Source and Commercial Software Platforms

In this article, we consider the dynamics and competition between two software platforms (Open Source and proprietary software). Potential user-developers can adopt one of the two platforms in order to develop and sell new applications based on the platform. We consider the static issue first and then use a simple dynamic system where the dynamics comes from the development efforts (spillovers) made on each platforms. In this context, we first identify the conditions for the two platforms to coexist in the long run. From this baseline, we then consider different strategies for the editor. A first strategy is for the editor to “show the code” of its software, so as to develop more compatible products. A second strategy is to strategically monitor the compatibility degree between the proprietary and OS platform. In both cases, we analyze whether a mixed industry may be sustainable in the long run.

Response of a simple system made complex by attaching subsystems—an analytic study.

A deceptively simple dynamical system comprised of a grounded mass to which are attached N, a large number, sprung masses has received much attention. The reported studies mostly involve physical experiments or numerical simulations. Given the large number of physical parameters needed to describe any single system, such a research agenda requires one to address an excessive number of experiment scenarios. Thus, the motivation for the analytical study reported. The focus of the study is the net force the attached subsystems cause to act on the base system. This force, while “internal” to the coupled system, is a response measure of primary importance since it accomplishes the work that determines the energy flow to and from the base system. As demonstrated, a further reason for the focus is an ability to construct an asymptotically valid, analytic expression for this response measure. Further demonstrated, using the constructed expression, is a resolution of a complex “process,” that describing the energy flow to the system, by the action of an external forcing, and through the system, as a combination of simple “sub‐processes.” The resolution requires describing the complex process in a time/frequency phase space; the sub‐processes are “local” in the phase space.

Catchments as simple dynamical systems: Catchment characterization, rainfall‐runoff modeling, and doing hydrology backward

Water fluxes in catchments are controlled by physical processes and material properties that are complex, heterogeneous, and poorly characterized by direct measurement. As a result, parsimonious theories of catchment hydrology remain elusive. Here I describe how one class of catchments (those in which discharge is determined by the volume of water in storage) can be characterized as simple first‐order nonlinear dynamical systems, and I show that the form of their governing equations can be inferred directly from measurements of streamflow fluctuations. I illustrate this approach using data from the headwaters of the Severn and Wye rivers at Plynlimon in mid‐Wales. This approach leads to quantitative estimates of catchment dynamic storage, recession time scales, and sensitivity to antecedent moisture, suggesting that it is useful for catchment characterization. It also yields a first‐order nonlinear differential equation that can be used to directly simulate the streamflow hydrograph from precipitation and evapotranspiration time series. This single‐equation rainfall‐runoff model predicts streamflow at Plynlimon as accurately as other models that are much more highly parameterized. It can also be analytically inverted; thus, it can be used to “do hydrology backward,” that is, to infer time series of whole‐catchment precipitation directly from fluctuations in streamflow. At Plynlimon, precipitation rates inferred from streamflow fluctuations agree with rain gauge measurements as closely as two rain gauges in each catchment agree with each other. These inferred precipitation rates are not calibrated to precipitation measurements in any way, making them a strong test of the underlying theory. The same approach can be used to estimate whole‐catchment evapotranspiration rates during rainless periods. At Plynlimon, evapotranspiration rates inferred from streamflow fluctuations exhibit seasonal and diurnal cycles that agree semiquantitatively with Penman‐Monteith estimates. Thus, streamflow hydrographs may be useful for reconstructing precipitation and evapotranspiration records where direct measurements are unavailable, unreliable, or unrepresentative at the scale of the landscape.

Control of single neuron activity by global network dynamics in auditory cortex

Event Abstract Back to Event Control of single neuron activity by global network dynamics in auditory cortex The electrical activity of neurons reflects an interplay of sensory input and the brain's own internal dynamics. In sensory cortices, neuronal firing correlates well with characteristics of sensory stimuli; on the other hand, each cortical neuron receives the vast majority of its inputs from other cortical neurons, suggesting that spiking is largely controlled by the activity of the local network. This viewpoint is supported by simultaneous recordings of multiple cells, which show synchronized patterns of spontaneous activity beyond those imposed by the structure of the sensory stimulus. In prior work, we found that the global dynamics of population activity in rat auditory cortex is well approximated by a low-dimensional self-exciting system model, and that the variations in population dynamics with cortical state can be captured by changes in the parameters of this model. Here we investigate the relationship of electrophysiologically identified single neurons to global population dynamics in this context. In the model, population activity is characterized by a pair of 'mean field' variables, v and w, that reflect the average population firing rate and integrated recent past activity, respectively. To predict the spike times of individual neurons from population activity, we use an approach analogous to the computation of hippocampal "place fields", by predicting the mean firing probability of each neuron as a function of position in the v-w plane, which we term the 'activity field.' We find that activity fields for different neurons are quite localized in distinct regions of the v-w plane. We conclude that, while the mean activity of a column is well approximated by a simple low-dimensional dynamical system, individual neurons show widely different relationships to this dynamics. For any neuron, activity fields computed from spontaneous and sensory-evoked activity were similar, further suggesting that the structure of sensory responses reflects the same network dynamics that shape cortical spontaneous activity. This work is supported by NIH Grants MH073245 and R01DC009947. CC was also supported by a Courant Instructorship. Conference: Computational and systems neuroscience 2009, Salt Lake City, UT, United States, 26 Feb - 3 Mar, 2009. Presentation Type: Poster Presentation Topic: Poster Presentations Citation: (2009). Control of single neuron activity by global network dynamics in auditory cortex. Front. Syst. Neurosci. Conference Abstract: Computational and systems neuroscience 2009. doi: 10.3389/conf.neuro.06.2009.03.152 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 02 Feb 2009; Published Online: 02 Feb 2009. Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Google Google Scholar PubMed Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

On Behavioral Arrow-Pratt Risk Process with Applications to Risk Pricing, Stochastic Cash Flows, and Risk Control

We introduce a closed form behavioural stochastic Arrow-Pratt risk process, decomposed into discrete asymmetric risk seeking and risk averse components that run on different local times in ϵ-disks centered at risk free states. Additionally, we embed Arrow-Pratt (“AP”) risk measure in a simple dynamic system of discounted cash flows with constant volatility, and time varying drift. Signal extraction of Arrow-Pratt risk measure shows that it is highly nonlinear in constant volatility for cash flows. Robust identifying restrictions on the system solution confirm that even for small time periods constant volatility is not a measure of AP risk. By contrast, time-varying volatility measures aspects of embedded AP risk. Whereupon maximal AP risk measure is obtained from a convolution of input volatility and idiosyncratic shocks to the system. We provide four applications for our theory. First, we find that Engle, Ng and Rothschild (1990) Factor-ARCH model for risk premia is misspecified because the factor price of risk is time varying and unstable. Our theory predicts that a hyper-ARCH correction factor is required to remove the Factor-ARCH specification. Second, when applied to analysts beliefs about interest rates and volatility, we find that AP risk measure is a feedback control over stochastic cash flows. Whereupon increased risk aversion to negative shocks to earnings increases volatility. Third, we use an oft cited example of Benes, Shepp and Witsenhausen (1980) to characterize a controlled AP diffusion for a conservative investor who wants to minimize the AP risk process for an asset. Fourth, we recover stochastic differential utility functional from the AP risk process and show how it is functionally equivalent to Duffie and Epstein’s (1992) parametrization.

Multilocus selection in subdivided populations I. Convergence properties for weak or strong migration

The dynamics and equilibrium structure of a deterministic population-genetic model of migration and selection acting on multiple multiallelic loci is studied. A large population of diploid individuals is distributed over finitely many demes connected by migration. Generations are discrete and nonoverlapping, migration is irreducible and aperiodic, all pairwise recombination rates are positive, and selection may vary across demes. It is proved that, in the absence of selection, all trajectories converge at a geometric rate to a manifold on which global linkage equilibrium holds and allele frequencies are identical across demes. Various limiting cases are derived in which one or more of the three evolutionary forces, selection, migration, and recombination, are weak relative to the others. Two are particularly interesting. If migration and recombination are strong relative to selection, the dynamics can be conceived as a perturbation of the so-called weak-selection limit, a simple dynamical system for suitably averaged allele frequencies. Under nondegeneracy assumptions on this weak-selection limit which are generic, every equilibrium of the full dynamics is a perturbation of an equilibrium of the weak-selection limit and has the same stability properties. The number of equilibria is the same in both systems, equilibria in the full (perturbed) system are in quasi-linkage equilibrium, and differences among allele frequencies across demes are small. If migration is weak relative to recombination and epistasis is also weak, then every equilibrium is a perturbation of an equilibrium of the corresponding system without migration, has the same stability properties, and is in quasi-linkage equilibrium. In both cases, every trajectory converges to an equilibrium, thus no cycling or complicated dynamics can occur.

Baroclinic splitting of jets

Whether the split of the Somali jet, sometimes seen on monthly mean streamline analyses, is a climatological or a dynamical feature has been the subject of long-standing debate. This paper explores the dynamical conditions leading to a split jet within the framework of a simple barotropic dynamic system. The initial conditions for the dynamical system, along with three other parameters – the jet width, the zonal wavelength, and the latitude of the β-plane, form a parameter space for the problem consisting of a range of solutions for the evolution of the jet. This paper identifies a region in the parameter space in which these solutions support a splitting of the jet. The width and wavelength of the Somali jet determined from observations are such that for most initial conditions the solutions reside near the boundary in parameter space between the split and non-split regions. It is therefore concluded that the splitting of the Somali jet can be a dynamical feature given the observed jet width and wavelength. Whether a split does or does not occur is determined by the parameters defining the initial zonal mean and perturbation flow in the jet, with the solution being highly sensitive to these initial conditions.

Application of optimal control in man power planning

Many techniques are met in the literature (see for instance Bartholomew and Forbes (Statistical Techniques for Manpower Planning. wiley, New York 1979); Gunz (Organiz. Stud. 9(4), 529–554, 1988); Becker and Huselid (Human Resour. Manage. 38, 287–301, 1999); Wagner et al. (J. Manage. Med. 14(5/6), 383–405, 2000); Harris and Ogbonna (J. Business Res. 51, 157–166, 2001); Rogg et al. (J. Manage. 27, 431–449, 2001), among others), for planning the manpower resources. However, we haven’t seen in the literature an empirical study regarding the proper application of optimal control, which considered to be the most efficient method for multi-objective programming. With this in mind, we analyse in this paper the way of applying optimal control for manpower planning. For this purpose, and in order to facilitate the presentation, we first adopted a comparatively simple dynamic system (plant), with analytical presentation of stocks and flows. Next we proceed to the formulation of an optimal control problem, aiming to achieve in the most satisfactory way some preassigned targets. These targets mainly refer to a desirable trajectory of the plant stocks over time, in order to fully satisfy the needs for human resources over the planning horizon. Finally we present a method of solution of the formulated control problem which is based on the use of the generalized inverse Lazaridis (Qual. Quan. 120, 297–306, 1986). We believe that it is very important for successful management, that the policy makers have to know the effect of their polices and to determine the optimal path of the state variables (i.e. the ones describing the system) before the realization of the plan, so as to be able to reform their strategies, reallocate the resources and arrange the infrastructure accordingly, if all these are necessary, as it can be depicted from the optimal control solution.

The dynamical origin of physiological instructions used in birdsong production

In this work we report experimental measurements of pressure patterns used in canary song. We find that these patterns are qualitatively similar to the subharmonic solutions of a simple dynamical system. This is built to account for the activities of subpopulations of neurons arranged in a simple architecture compatible with anatomical observations. The consequences of Hebbian plasticity in the coupling between the driving and the driven systems are outlined.

A critical evaluation of closure methods via two simple dynamic systems

Two simple dynamic systems with cubic nonlinearity and additive Gaussian white noise are used to assess the performance and the usefulness of closure methods in nonlinear random vibration. One of the systems has a single potential well while the other has two potential wells. It is shown that the performance of closure methods is determined by the structure of the moment equations rather than the way in which these equations are closed. For the system with one potential well, any closure method is satisfactory. For the system with two potential wells, closure methods can be inaccurate irrespective of the closure level. It is also shown that moment equations can be augmented with moment inequalities to solve approximately the infinite hierarchy of moment equations.