Initial Condition Sensitivity Research Articles

A discrete time neural network model with spiking neurons

We derive rigorous results describing the asymptotic dynamics of a discrete time model of spiking neurons introduced in Soula et al. (Neural Comput. 18, 1, 2006). Using symbolic dynamic techniques we show how the dynamics of membrane potential has a one to one correspondence with sequences of spikes patterns ("raster plots"). Moreover, though the dynamics is generically periodic, it has a weak form of initial conditions sensitivity due to the presence of a sharp threshold in the model definition. As a consequence, the model exhibits a dynamical regime indistinguishable from chaos in numerical experiments.

Revisiting time discretisation of spiking network models

A link is built between a biologically plausible generalized integrate and fire (GIF) neuron model with conductance-based dynamics [1] and a discrete time neural network model with spiking neurons [2], for which rigorous results on the spontaneous dynamics has been obtained. More precisely the following has been shown. i) Occurrence of periodic orbits is the generic regime of activity, with a bounded period in the presence of spike-time dependence plasticity, and arbitrary large periods at the edge of chaos (such regime is indistinguishable from chaos in numerical experiments, explaining what is obtained in [2]), ii) the dynamics of membrane potential has a one to one correspondence with sequences of spikes patterns (raster plots). This allows a better insight into the possible neural coding in such a network and provides a deep understanding, at the network level, of the system behavior. Moreover, though the dynamics is generically periodic, it has a weak form of initial conditions sensitivity due to the presence of the sharp spiking threshold [3]. A step further, constructive conditions are derived, allowing to properly implement visual functions on such networks [4]. The time discretisation has been carefully conducted avoiding usual bias induced by e.g. Euler methods and taking into account a rather complex GIF model for which the usual arbitrary discontinuities are discussed in detail. The effects of the discretisation approximation have been analytically and experimentally analyzed, in detail. Figure 1 A view of the numerical experiments software platform raster-plot output, considering either a generic fully connected network or, here, a retinotopic network related to visual functions (top-left: 2D instantaneous spiking activity).

“Metric” complexity for weakly chaotic systems

We consider the number of Bowen sets necessary to cover a large measure subset of the phase space. This introduces some complexity indicator characterizing different kinds of (weakly) chaotic dynamics. Since in many systems its value is given by a sort of local entropy, this indicator is quite simple to calculate. We give some examples of calculations in nontrivial systems (e.g., interval exchanges and piecewise isometries) and a formula similar to that of Ruelle-Pesin, relating the complexity indicator to some initial condition sensitivity indicators playing the role of positive Lyapunov exponents.

The Sicily Channel Regional Model forecasting system: initial boundary conditions sensitivity and case study evaluation

Abstract. The Sicily Channel Regional Model forecasting system was tested using an optimization package for the initial and lateral boundary conditions. Spurious high frequency oscillations during the spin-up time were successfully reduced both in duration and magnitude by optimizing the time tendency of the free surface elevation using the Variational Initialization and Forcing Platform method developed in the framework of the Mediterranean Forecasting System Towards the Environmental Prediction project. The effect of optimization was most profound for the free surface elevation, where all oscillations with periods shorter than 4 h were suppressed. The overall forecast skill was assessed on a 5 day case study starting on 6 April 2005, characterized by a fast passage of a deepening atmospheric low–pressure field with strong winds and marked wind direction change. We compared the predicted variables with in–situ and remotely sensed data. The forecasts of temperature, including the sea surface temperature, and salinity were quite successful, while the forecasted currents, especially within the surface layer, were not in good agreement with the measurements.

Nonlinearity index of the cayley form

The nonlinearity index is a measure of the nonlinearity of dynamical systems based on computing the initial-condition sensitivity of the state-transition matrix. The Cayley form is a representation for dynamical systems that relates their motion to N-dimensional rotations. The generalized coordinates of the system are used to define an N-dimensional orientation, and a set of quasi velocities is defined equal to the corresponding angular velocity. The nonlinearity index of the Cayley-form representation is computed for an elastic spherical pendulum and a planar satellite example. These results are compared to values for alternative dynamical representations. Additionally, the nonlinearity is evaluated by analyzing how well the linearized equations of each representation capture certain properties of the motion. These results show that the Cayley form can have lower nonlinearity than traditional representations, in particular those representations that suffer from kinematic singularities.

Evaporation en film mince dans un caloduc rainuré

Click to increase image sizeClick to decrease image sizeEvaporative thin film inside a grooved heat pipe: Axially grooved heat pipes are widely used in space applications like thermal control of communications satellites. Inside the heat pipe, evaporation mostly takes place in microscopic region where the meniscus (liquid-vapour interface) is attached to the solid wall. This zone is one of the most important boundary condition to calculate the evaporating heat transfer conductance. An evaporating thin liquid film is commonly used to model this preponderant heat transfer region (typically from 30 to 60% of the heat flux though less than 1% of the surface). This model leads to a system of differential equations that is very tricky to solve due to extreme sensitivity of initial conditions. Nevertheless, relation between wetting contact angle and meniscus curvature inside grooves is re-examined in respect of the obtained results.Congrès Microfluidique 2004

Exact and Approximate Maximum Likelihood Localization Algorithms

Sensors at separate locations measuring either the time difference of arrival (TDOA) or time of arrival (TOA) of the signal from an emitter can determine its position as the intersection of hyperbolae for TDOA and of circles for TOA. Because of measurement noise, the nonlinear localization equations become inconsistent; and the hyperbolae or circles no longer intersect at a single point. It is now necessary to find an emitter position estimate that minimizes its deviations from the true position. Methods that first linearize the equations and then perform gradient searches for the minimum suffer from initial condition sensitivity and convergence difficulty. Starting from the maximum likelihood (ML) function, this paper derives a closed-form approximate solution to the ML equations. When there are three sensors on a straight line, it also gives an exact ML estimate. Simulation experiments have demonstrated that these algorithms are near optimal, attaining the theoretical lower bound for different geometries, and are superior to two other closed form linear estimators.

Optimal information measures for weakly chaotic dynamical systems

The study of dynamical systems suggests that an important indicator for their classification is the quantity of information that is needed to describe their orbits (here the orbits are transated into symbolic sequences by some geometrical construction). This leads to a definition of a complexity of a single orbit. This notion is flexible enough to give a refinement of the classical definition of entropy of a system. This concept is particularly interesting for systems with zero entropy (for motivations and some results see [S. Galatolo, Complexity, initial condition sensitivity, dimension and weak chaos in dynamical systems, Nonlinearity 16 (2003) 1219–1238]).

STOCHASTIC SIMULATION OF CHEMICAL CHUA SYSTEM

A master equation for a scheme of chemical Chua system has been simulated stochastically. Two pairs of reaction simulations, one pair corresponding to deterministic chaos and the other pair to limit cycle, are carried out. The initial conditions differ with only one molecule for the former, while one hundred molecules for the latter. The rapid separation of time traces is shown only in the former case. This marked difference of dynamic behavior between the two cases might reveal the intrinsic random nature of deterministic chemical chaos, i.e. initial-condition sensitivity based on random collision events.

Complexity, initial condition sensitivity, dimension and weak chaos in dynamical systems

We prove quantitative relations between complexity, initial condition sensitivity and dimension which are suitable for 0-entropy dynamical systems. The orbit complexity indicator is a measure of the asymptotic behaviour of the information that is necessary to describe the orbit of a given point. The indicator generalizes Brudno's orbit complexity (which is strongly related to the entropy of the system). The initial condition sensitivity indicators consider the asymptotic behaviour of the speed of divergence of nearby starting orbits. The indicators have non-trivial values also in weakly chaotic (0-entropy) dynamical systems, characterizing various cases of weakly chaotic dynamics. Using constructive mathematics, local relations are proved between generalized orbit complexity, initial condition sensitivity and the dimension of the underlying space or measure. The relations are quantitative and have non-trivial meaning also in the 0-entropy case. The generalized relations are then applied to the study of an important example of weakly chaotic dynamics: the Manneville map.

Effects of Moist Convection on Mesoscale Predictability

In a previous study by the authors, it was shown that the problematic numerical prediction of the 24‐25 January 2000 snowstorm along the east coast of the United States was in some measure due to rapid error growth at scales below 500 km. In particular they found that moist processes were responsible for this strong initial-condition sensitivity of the 1‐2-day prediction of mesoscale forecast aspects. In the present study they take a more systematic look at the processes by which small initial differences (‘‘errors’’) grow in those numerical forecasts. For initial errors restricted to scales below 100 km, results show that errors first grow as small-scale differences associated with moist convection, then spread upscale as their growth begins to slow. In the context of mesoscale numerical predictions with 30-km resolution, the initial growth is associated with nonlinearities in the convective parameterization (or in the explicit microphysical parameterizations, if no convective parameterization is used) and proceeds at a rate that increases as the initial error amplitude decreases. In higherresolution (3.3 km) simulations, errors first grow as differences in the timing and position of individual convective cells. Amplification at that stage occurs on a timescale on the order of 1 h, comparable to that of moist convection. The errors in the convective-scale motions subsequently influence the development of meso- and larger-scale forecast aspects such as the position of the surface low and the distribution of precipitation, thus providing evidence that growth of initial errors from convective scales places an intrinsic limit on the predictability of larger scales.

The renormalization group and optimization of non-extensive entropy: criticality in non-linear one-dimensional maps

We examine the pitchfork and tangent bifurcations in unimodal maps to illustrate a connection between renormalization group (RG) fixed points and entropy extremal properties. We observe that the exact RG solution for the tangent bifurcation is also applicable to the period-doubling cascade and assess its physical meaning. Since the expression for the fixed-point map can be put into the form of the non-extensive expressions for the temporal evolution of phase-space volume and sensitivity of initial conditions, we conclude that the map critical points possess the properties of this formalism. The universality of the RG solution makes this interpretation inclusive to all one-dimensional maps of non-linearity z>1.

Design and Control of a Chained Form Manipulator

Exploiting unique features of nonholonomic systems, innovative and advantageous mechanisms can be designed. We have proposed a nonholonomic manipulator which is a controllable n joint manipulator with only two actuators. In order to create its nonholonomic constraint, a special type of velocity transmission, called the non-holonomic gear, was developed. There are many possible alternatives for designing underactuated manipulators using the nonholonomic gear. In this paper, the chained form manipulator is designed not only to satisfy chained form convertibility, but also to achieve control simplicity. Design requirements are clarified, then we establish the mechanical design of the chained form manipulator. The chained form manipulator is designed with an innovative main power train as well as nonholonomic gears. For experimental verification, a prototype is fabricated. So far, various control strategies for nonholonomic systems have been proposed. In this paper, an efficient motion planning scheme for the chained form is presented to approximate any holonomic path with the feasible nonholonomic path. Furthermore, a new concept of motion planning is proposed through the analysis of the initial-condition sensitivity. Combining these two approaches, the motion planning scheme is constructed towards practical applications. Presented experimental results show the usefulness of the design and the applied control scheme.

Initial Condition Sensitivity and Error Growth in Forecasts of the 25 January 2000 East Coast Snowstorm

Abstract Short- and medium-range (24–96-h) forecasts of the January 2000 U.S. east coast cyclone and associated snowstorm are examined using the U.S. Navy global forecast model and adjoint system. Attention is given to errors on the synoptic scale, including forecast position and central pressure of the cyclone at the verification time of 1200 UTC 25 January 2000. There is a substantial loss of predictive skill in the 72- and 96-h forecasts, while the 24- and 48-h forecasts capture the synoptic-scale features of the cyclone development with moderate errors. Sensitivity information from the adjoint model suggests that the initial conditions for the 72-h forecast starting at 1200 UTC 22 January 2000 contained relatively small, but critical, errors in upper-air wind and temperature over a large upstream area, including part of the eastern Pacific and “well observed” areas of western and central North America. The rapid growth of these initial errors in a highly unstable flow regime (large singular-vector gro...

Mesurer la sensibilité aux conditions initiales dans les processus individuels de premiere insertion professionnelle: Proposition de procédure et application

Measuring the Sensitivity of Initial Conditions in Individual Procedures for Finding First Professional Employment - Proposal for a Process and Application. Individual procedures for obtaining first professional employment are influenced simultaneously by social determinism, chance and individual stratégies. The question is how to unravel these different factors. Starting with the central opposition between chance and determinism -- where strategies create possibilities -- this study proposes an analysis within the framework of bounded chaos. First, a method is proposed for the measurement of sensitive initial conditions (SCI) based on Lyapunov powers associated with obtaining first professional employment. Second, this study shows that the SCI measure permits one to examine the respective roles of chance, determinism and strategy. It then suggests that the SCI measure is a better tool for counseling on scholastic and professional orientation than surveys of entire classes conceming first jobs. The latter remain better adapted to counseling and evaluating educational institutions and curriculums.

Information of sequences and applications

In this short note, we outline some results about complexity of orbits of a dynamical system, entropy and initial condition sensitivity in weakly chaotic dynamical systems. We present a technique to estimate orbit complexity by the use of data compression algorithms. We also outline how this technique has been applied by our research group to dynamical systems and to DNA sequences.

Initial-Condition Sensitivity and Nonlinear Relaxation in Photo-Induced Ionic-to-Neutral Phase Transition in Tetrathiafulvalen- p -Chloranil Crystals

Recent experimental results of the photoinduced ionic-to-neutral (NI) phase transition in tetrathiafulvalen- p -chloranil (TTF-CA) crystals are reviewed with emphasis on the state-sensitive features and nonlinear properties. Frenkel-type and charge transfer (CT)-type excited states can induce the NI transition but with different characteristics; the transition can be induced only above threshold-excitation intensities in the case of CT excitation, whereas it is induced without any threshold for Frenkel-type excitation. The threshold that implies nonlinear processes of CT excited states is strongly state-dependent and temperature-dependent. Femtosecond time-resolved studies have resolved three distinctive sequential steps in the dynamics; formation of the precursor of N-phase domains, the local proliferation of photoinduced changes, and the process of forming N-phase orders. Origin of the nonlinear processes and the mechanism of the NI transition have been discussed based on these results.

INITIAL CONDITION SENSITIVITY IN LATTICE RELAXATION AND DOMAIN FORMATION AFTER PHOTOEXCITATION

We theoretically study the lattice relaxation dynamics of a photogenerated exciton resulting in the macroscopic domain. Especially, we concentrate on the "initial condition sensitivity" of the exciton proliferation. This characteristic is as follows. The generation efficiency of the domain depends sensitively on the way of the initial photoexcitation, even if the total photon energy is same. We consider the 1-dimensional many exciton-Einstein phonon system interacting with a reservoir. Within the Markov approximation, the time evolution of the density matrix is investigated full-quantummechanically. Here, we derive the iterative equation for the proliferation and numerically solve it, so that we can treat multi-exciton states in the large system. Depending on the way of the photoexcitation, the initial distance between photogenerated excitions is assumed to take various values. We have shown that the exciton proliferation successfully occurs only when the initial distance is not too short netiher too long. This sensitivity to the initial distance is due to the nonlinear interaction between excitons.

Theory for photoinduced structural phase transitions

A general concept for photoinduced structural phase transitions is developed in terms of the hidden multistability of the ground state and the proliferations of optically excited states. Taking the ionic→neutral (I - N) phase transition in the organic charge transfer crystal, TTF—CA, as a typical example for this type of transition, we, at first, theoretically show an adiabatic path of this transition, which starts from a single charge transfer exciton in the ionic phase, but finally reaches a neutral domain with a macroscopic size. In connection with this I—N transition, the concept of the initial condition sensitivity is also developed so as to clarify experimentally observed nonlinear characteristics of this material. In the next, using a more simplified model for the many-exciton system, we theoretically study the early time quantum dynamics of the exciton proliferation, which finally results in the domain formation of a large number of ex-citons. For this purpose, we derive a stepwise iterative equation to describe the exciton proliferation, and clarify the origin of the initial condition sensitivity.

Initial Condition Sensitivity in Photoinduced Structural Phase Transitions – An Iterative Equation for Exciton Proliferation during Lattice Relaxation –

We theoretically study lattice relaxation dynamics of photogenerated excitons, which proliferate during this relaxation, and finally result in a macroscopic excited domain in insulating solids. This macroscopic domain formation phenomenon is usually called “photoinduced structural phase transition (PSPT)”. According to recent experiments, occurrence or nonoccurrence of this PSPT is found to depend very sensitively on the way of the photoexcitation, even if the total absorbed photonic energy is same. It is nothing else but “initial condition sensitivity” peculiar to nonlinear systems. We concentrate our attention on this sensitivity, and investigate criticality of this proliferation, using an iterative equation for its time evolution. As one of the models describing such a sensitivity, we consider a 1-dimensional strongly coupled many exciton-Einstein phonon system interacting with a reservoir. In our system, excitons can proliferate through their third order anharmonicity. This anharmonicity originates from the long range Coulomb interaction among electrons and holes constituting excitons. Within the Markov approximation for the reservoir, the time evolution of the density matrix of this exciton-phonon system is investigated full-quantummechanically. In order to overcome numerical complexities, we solve our iterative equation for the proliferation by the following principle. We always focus only on the most front of the expanding photoinduced domain, and the contributions from the other excitons, not in this front, are approximated by a mean field. As the proliferation proceeds, this front moves. Accordingly, we, step by step, iterate this procedure till the excess phonon energy, initially given by the photoexcitation, is exhausted. Thus, we can investigate the whole process of the time evolution of the domain formation, until this domain contains a large number of excitons. As for the aforementioned initial condition sensitivity, we cast the difference of the initial conditions into the difference of the initial distance between adjacent photogenerated excitons. Net proliferation is concluded to occur only when this initial distance is moderate, being not too small nor too large. This sensitivity to the initial distance results from the aforementioned third order anharmonicity.