Local dimension and finite time prediction in coupled map lattices

Forecasting, for obvious reasons, often become the most important goal to be achieved. For spatially extended systems (e.g. atmospheric system) where the local nonlinearities lead to the most unpredictable chaotic evolution, it is highly desirable to have a simple diagnostic tool to identify regions of predictable behaviour. In this paper, we discuss the use of the bred vector (BV) dimension, a recently introduced statistics, to identify the regimes where a finite time forecast is feasible. Using the tools from dynamical systems theory and Bayesian modelling, we show the finite time predictability in two-dimensional coupled map lattices in the regions of low BV dimension. © Indian Academy of Sciences.

Damage detection in nonlinear structures using discrete-time volterra series

Structural damage identification is basically a nonlinear phenomenon; however, nonlinear procedures are not used currently in practical applications due to the complexity and difficulty for implementation of such techniques. Therefore, the development of techniques that consider the nonlinear behavior of structures for damage detection is a research of major importance since nonlinear dynamical effects can be erroneously treated as damage in the structure by classical metrics. This paper proposes the discrete-time Volterra series for modeling the nonlinear convolution between the input and output signals in a benchmark nonlinear system. The prediction error of the model in an unknown structural condition is compared with the values of the reference structure in healthy condition for evaluating the method of damage detection. Since the Volterra series separate the response of the system in linear and nonlinear contributions, these indexes are used to show the importance of considering the nonlinear behavior of the structure. The paper concludes pointing out the main advantages and drawbacks of this damage detection methodology. © (2013) Trans Tech Publications.

Nonlinear features identified by Volterra series for damage detection in a buckled beam

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Seizure prediction and control in epilepsy

The first part of my thesis presents an overview of the different approaches used in the past two decades in the attempt to forecast epileptic seizure on the basis of intracranial and scalp EEG. Past research could reveal some value of linear and nonlinear algorithms to detect EEG features changing over different phases of the epileptic cycle. However, their exact value for seizure prediction, in terms of sensitivity and specificity, is still discussed and has to be evaluated. In particular, the monitored EEG features may fluctuate with the vigilance state and lead to false alarms. Recently, such a dependency on vigilance states has been reported for some seizure prediction methods, suggesting a reduced reliability. An additional factor limiting application and validation of most seizure-prediction techniques is their computational load. For the first time, the reliability of permutation entropy [PE] was verified in seizure prediction on scalp EEG data, contemporarily controlling for its dependency on different vigilance states. PE was recently introduced as an extremely fast and robust complexity measure for chaotic time series and thus suitable for online application even in portable systems. The capability of PE to distinguish between preictal and interictal state has been demonstrated using Receiver Operating Characteristics (ROC) analysis. Correlation analysis was used to assess dependency of PE on vigilance states. Scalp EEG-Data from two right temporal epileptic lobe (RTLE) patients and from one patient with right frontal lobe epilepsy were analysed. The last patient was included only in the correlation analysis, since no datasets including seizures have been available for him. The ROC analysis showed a good separability of interictal and preictal phases for both RTLE patients, suggesting that PE could be sensitive to EEG modifications, not visible on visual inspection, that might occur well in advance respect to the EEG and clinical onset of seizures. However, the simultaneous assessment of the changes in vigilance showed that: a) all seizures occurred in association with the transition of vigilance states; b) PE was sensitive in detecting different vigilance states, independently of seizure occurrences. Due to the limitations of the datasets, these results cannot rule out the capability of PE to detect preictal states. However, the good separability between pre- and interictal phases might depend exclusively on the coincidence of epileptic seizure onset with a transition from a state of low vigilance to a state of increased vigilance. The finding of a dependency of PE on vigilance state is an original finding, not reported in literature, and suggesting the possibility to classify vigilance states by means of PE in an authomatic and objectic way. The second part of my thesis provides the description of a novel behavioral task based on motor imagery skills, firstly introduced (Bruzzo et al. 2007), in order to study mental simulation of biological and non-biological movement in paranoid schizophrenics (PS). Immediately after the presentation of a real movement, participants had to imagine or re-enact the very same movement. By key release and key press respectively, participants had to indicate when they started and ended the mental simulation or the re-enactment, making it feasible to measure the duration of the simulated or re-enacted movements. The proportional error between duration of the re-enacted/simulated movement and the template movement were compared between different conditions, as well as between PS and healthy subjects. Results revealed a double dissociation between the mechanisms of mental simulation involved in biological and non-biologial movement simulation. While for PS were found large errors for simulation of biological movements, while being more acurate than healthy subjects during simulation of non-biological movements. Healthy subjects showed the opposite relationship, making errors during simulation of non-biological movements, but being most accurate during simulation of non-biological movements. However, the good timing precision during re-enactment of the movements in all conditions and in both groups of participants suggests that perception, memory and attention, as well as motor control processes were not affected. Based upon a long history of literature reporting the existence of psychotic episodes in epileptic patients, a longitudinal study, using a slightly modified behavioral paradigm, was carried out with two RTLE patients, one patient with idiopathic generalized epilepsy and one patient with extratemporal lobe epilepsy. Results provide strong evidence for a possibility to predict upcoming seizures in RTLE patients behaviorally. In the last part of the thesis it has been validated a behavioural strategy based on neurobiofeedback training, to voluntarily control seizures and to reduce there frequency. Three epileptic patients were included in this study. The biofeedback was based on monitoring of slow cortical potentials (SCPs) extracted online from scalp EEG. Patients were trained to produce positive shifts of SCPs. After a training phase patients were monitored for 6 months in order to validate the ability of the learned strategy to reduce seizure frequency. Two of the three refractory epileptic patients recruited for this study showed improvements in self-management and reduction of ictal episodes, even six months after the last training session.

Nonlinear / electromagnetic co-simulation and co-design of microwave links in highly polluted and / or strongly inhomogeneous environments

The objective of the Ph.D. thesis is to put the basis of an all-embracing link analysis procedure that may form a general reference scheme for the future state-of-the-art of RF/microwave link design: it is basically meant as a circuit-level simulation of an entire radio link, with – generally multiple – transmitting and receiving antennas examined by EM analysis. In this way the influence of mutual couplings on the frequency-dependent near-field and far-field performance of each element is fully accounted for. The set of transmitters is treated as a unique nonlinear system loaded by the multiport antenna, and is analyzed by nonlinear circuit techniques. In order to establish the connection between transmitters and receivers, the far-fields incident onto the receivers are evaluated by EM analysis and are combined by extending an available Ray Tracing technique to the link study. EM theory is used to describe the receiving array as a linear active multiport network. Link performances in terms of bit error rate (BER) are eventually verified a posteriori by a fast system-level algorithm. In order to validate the proposed approach, four heterogeneous application contexts are provided. A complete MIMO link design in a realistic propagation scenario is meant to constitute the reference case study. The second one regards the design, optimization and testing of various typologies of rectennas for power generation by common RF sources. Finally, the project and implementation of two typologies of radio identification tags, at X-band and V-band respectively. In all the cases the importance of an exhaustive nonlinear/electromagnetic co-simulation and co-design is demonstrated to be essential for any accurate system performance prediction.

Electrothermal characterization and nonlinear modelling of GaN transistors for telecommunication circuit design

This thesis starts showing the main characteristics and application fields of the AlGaN/GaN HEMT technology, focusing on reliability aspects essentially due to the presence of low frequency dispersive phenomena which limit in several ways the microwave performance of this kind of devices. Based on an equivalent voltage approach, a new low frequency device model is presented where the dynamic nonlinearity of the trapping effect is taken into account for the first time allowing considerable improvements in the prediction of very important quantities for the design of power amplifier such as power added efficiency, dissipated power and internal device temperature. An innovative and low-cost measurement setup for the characterization of the device under low-frequency large-amplitude sinusoidal excitation is also presented. This setup allows the identification of the new low frequency model through suitable procedures explained in detail. In this thesis a new non-invasive empirical method for compact electrothermal modeling and thermal resistance extraction is also described. The new contribution of the proposed approach concerns the non linear dependence of the channel temperature on the dissipated power. This is very important for GaN devices since they are capable of operating at relatively high temperatures with high power densities and the dependence of the thermal resistance on the temperature is quite relevant. Finally a novel method for the device thermal simulation is investigated: based on the analytical solution of the tree-dimensional heat equation, a Visual Basic program has been developed to estimate, in real time, the temperature distribution on the hottest surface of planar multilayer structures. The developed solver is particularly useful for peak temperature estimation at the design stage when critical decisions about circuit design and packaging have to be made. It facilitates the layout optimization and reliability improvement, allowing the correct choice of the device geometry and configuration to achieve the best possible thermal performance.

A Phase Space Box-counting based Method for Arrhythmia Prediction from Electrocardiogram Time Series

Arrhythmia is one kind of cardiovascular diseases that give rise to the number of deaths and potentially yields immedicable danger. Arrhythmia is a life threatening condition originating from disorganized propagation of electrical signals in heart resulting in desynchronization among different chambers of the heart. Fundamentally, the synchronization process means that the phase relationship of electrical activities between the chambers remains coherent, maintaining a constant phase difference over time. If desynchronization occurs due to arrhythmia, the coherent phase relationship breaks down resulting in chaotic rhythm affecting the regular pumping mechanism of heart. This phenomenon was explored by using the phase space reconstruction technique which is a standard analysis technique of time series data generated from nonlinear dynamical system. In this project a novel index is presented for predicting the onset of ventricular arrhythmias. Analysis of continuously captured long-term ECG data recordings was conducted up to the onset of arrhythmia by the phase space reconstruction method, obtaining 2-dimensional images, analysed by the box counting method. The method was tested using the ECG data set of three different kinds including normal (NR), Ventricular Tachycardia (VT), Ventricular Fibrillation (VF), extracted from the Physionet ECG database. Statistical measures like mean (μ), standard deviation (σ) and coefficient of variation (σ/μ) for the box-counting in phase space diagrams are derived for a sliding window of 10 beats of ECG signal. From the results of these statistical analyses, a threshold was derived as an upper bound of Coefficient of Variation (CV) for box-counting of ECG phase portraits which is capable of reliably predicting the impeding arrhythmia long before its actual occurrence. As future work of research, it was planned to validate this prediction tool over a wider population of patients affected by different kind of arrhythmia, like atrial fibrillation, bundle and brunch block, and set different thresholds for them, in order to confirm its clinical applicability.

Experimental validation of a nonlinear μ FE model based on cohesive-frictional plasticity for trabecular bone

Trabecular bone is a porous mineralized tissue playing a major load bearing role in the human body. Prediction of age-related and disease-related fractures and the behavior of bone implant systems needs a thorough understanding of its structure-mechanical property relationships, which can be obtained using microcomputed tomography-based finite element modeling. In this study, a nonlinear model for trabecular bone as a cohesive-frictional material was implemented in a large-scale computational framework and validated by comparison of μFE simulations with experimental tests in uniaxial tension and compression. A good correspondence of stiffness and yield points between simulations and experiments was found for a wide range of bone volume fraction and degree of anisotropy in both tension and compression using a non-calibrated, average set of material parameters. These results demonstrate the ability of the model to capture the effects leading to failure of bone for three anatomical sites and several donors, which may be used to determine the apparent behavior of trabecular bone and its evolution with age, disease, and treatment in the future.

NONLINEAR TIME SERIES/SYSTEM ANALYSIS (STATE-SPACE, DYNAMICS, INTERVENTION, RESILIENCE, KALMAN)

A discussion of nonlinear dynamics, demonstrated by the familiar automobile, is followed by the development of a systematic method of analysis of a possibly nonlinear time series using difference equations in the general state-space format. This format allows recursive state-dependent parameter estimation after each observation thereby revealing the dynamics inherent in the system in combination with random external perturbations.^ The one-step ahead prediction errors at each time period, transformed to have constant variance, and the estimated parametric sequences provide the information to (1) formally test whether time series observations y(,t) are some linear function of random errors (ELEM)(,s), for some t and s, or whether the series would more appropriately be described by a nonlinear model such as bilinear, exponential, threshold, etc., (2) formally test whether a statistically significant change has occurred in structure/level either historically or as it occurs, (3) forecast nonlinear system with a new and innovative (but very old numerical) technique utilizing rational functions to extrapolate individual parameters as smooth functions of time which are then combined to obtain the forecast of y and (4) suggest a measure of resilience, i.e. how much perturbation a structure/level can tolerate, whether internal or external to the system, and remain statistically unchanged. Although similar to one-step control, this provides a less rigid way to think about changes affecting social systems.^ Applications consisting of the analysis of some familiar and some simulated series demonstrate the procedure. Empirical results suggest that this state-space or modified augmented Kalman filter may provide interesting ways to identify particular kinds of nonlinearities as they occur in structural change via the state trajectory.^ A computational flow-chart detailing computations and software input and output is provided in the body of the text. IBM Advanced BASIC program listings to accomplish most of the analysis are provided in the appendix. ^

Pushover analysis for the seismic response prediction of cable-stayed bridges under multi-directional excitation

Cable-stayed bridges represent nowadays key points in transport networks and their seismic behavior needs to be fully understood, even beyond the elastic range of materials. Both nonlinear dynamic (NL-RHA) and static (pushover) procedures are currently available to face this challenge, each with intrinsic advantages and disadvantages, and their applicability in the study of the nonlinear seismic behavior of cable-stayed bridges is discussed here. The seismic response of a large number of finite element models with different span lengths, tower shapes and class of foundation soil is obtained with different procedures and compared. Several features of the original Modal Pushover Analysis (MPA) are modified in light of cable-stayed bridge characteristics, furthermore, an extension of MPA and a new coupled pushover analysis (CNSP) are suggested to estimate the complex inelastic response of such outstanding structures subjected to multi-axial strong ground motions.

Assessment of expected damage on buildings subjected to Lorca earthquake through an energy-based seismic index method and nonlinear dynamic response analyses

The city of Lorca (Spain) was hit on May 11th, 2011, by two consecutive earth-quakes of magnitudes 4.6 and 5.2 Mw, causing casualties and important damage in buildings. Many of the damaged structures were reinforced concrete frames with wide beams. This study quantifies the expected level of damage on this structural type in the case of the Lorca earth-quake by means of a seismic index Iv that compares the energy input by the earthquake with the energy absorption/dissipation capacity of the structure. The prototype frames investigated represent structures designed in two time periods (1994–2002 and 2003–2008), in which the applicable codes were different. The influence of the masonry infill walls and the proneness of the frames to concentrate damage in a given story were further investigated through nonlinear dynamic response analyses. It is found that (1) the seismic index method predicts levels of damage that range from moderate/severe to complete collapse; this prediction is consistent with the observed damage; (2) the presence of masonry infill walls makes the structure very prone to damage concentration and reduces the overall seismic capacity of the building; and (3) a proper hierarchy of strength between beams and columns that guarantees the formation of a strong column-weak beam mechanism (as prescribed by seismic codes), as well as the adoption of counter-measures to avoid the negative interaction between non-structural infill walls and the main frame, would have reduced the level of damage from Iv=1 (collapse) to about Iv=0.5 (moderate/severe damage)

Coupled Nonlinear Ginzburg-Landau and Mechanics Model for Martensitic Transformations in Polycrystals

Las transformaciones martensíticas (MT) se definen como un cambio en la estructura del cristal para formar una fase coherente o estructuras de dominio multivariante, a partir de la fase inicial con la misma composición, debido a pequeños intercambios o movimientos atómicos cooperativos. En el siglo pasado se han descubierto MT en diferentes materiales partiendo desde los aceros hasta las aleaciones con memoria de forma, materiales cerámicos y materiales inteligentes. Todos muestran propiedades destacables como alta resistencia mecánica, memoria de forma, efectos de superelasticidad o funcionalidades ferroicas como la piezoelectricidad, electro y magneto-estricción etc. Varios modelos/teorías se han desarrollado en sinergia con el desarrollo de la física del estado sólido para entender por qué las MT generan microstructuras muy variadas y ricas que muestran propiedades muy interesantes. Entre las teorías mejor aceptadas se encuentra la Teoría Fenomenológica de la Cristalografía Martensítica (PTMC, por sus siglas en inglés) que predice el plano de hábito y las relaciones de orientación entre la austenita y la martensita. La reinterpretación de la teoría PTMC en un entorno de mecánica del continuo (CM-PTMC) explica la formación de los dominios de estructuras multivariantes, mientras que la teoría de Landau con dinámica de inercia desentraña los mecanismos físicos de los precursores y otros comportamientos dinámicos. La dinámica de red cristalina desvela la reducción de la dureza acústica de las ondas de tensión de red que da lugar a transformaciones débiles de primer orden en el desplazamiento. A pesar de las diferencias entre las teorías estáticas y dinámicas dado su origen en diversas ramas de la física (por ejemplo mecánica continua o dinámica de la red cristalina), estas teorías deben estar inherentemente conectadas entre sí y mostrar ciertos elementos en común en una perspectiva unificada de la física. No obstante las conexiones físicas y diferencias entre las teorías/modelos no se han tratado hasta la fecha, aun siendo de importancia crítica para la mejora de modelos de MT y para el desarrollo integrado de modelos de transformaciones acopladas de desplazamiento-difusión. Por lo tanto, esta tesis comenzó con dos objetivos claros. El primero fue encontrar las conexiones físicas y las diferencias entre los modelos de MT mediante un análisis teórico detallado y simulaciones numéricas. El segundo objetivo fue expandir el modelo de Landau para ser capaz de estudiar MT en policristales, en el caso de transformaciones acopladas de desplazamiento-difusión, y en presencia de dislocaciones. Comenzando con un resumen de los antecedente, en este trabajo se presentan las bases físicas de los modelos actuales de MT. Su capacidad para predecir MT se clarifica mediante el ansis teórico y las simulaciones de la evolución microstructural de MT de cúbicoatetragonal y cúbicoatrigonal en 3D. Este análisis revela que el modelo de Landau con representación irreducible de la deformación transformada es equivalente a la teoría CM-PTMC y al modelo de microelasticidad para predecir los rasgos estáticos durante la MT, pero proporciona una mejor interpretación de los comportamientos dinámicos. Sin embargo, las aplicaciones del modelo de Landau en materiales estructurales están limitadas por su complejidad. Por tanto, el primer resultado de esta tesis es el desarrollo del modelo de Landau nolineal con representación irreducible de deformaciones y de la dinámica de inercia para policristales. La simulación demuestra que el modelo propuesto es consistente fcamente con el CM-PTMC en la descripción estática, y también permite una predicción del diagrama de fases con la clásica forma ’en C’ de los modos de nucleación martensítica activados por la combinación de temperaturas de enfriamiento y las condiciones de tensión aplicada correlacionadas con la transformación de energía de Landau. Posteriomente, el modelo de Landau de MT es integrado con un modelo de transformación de difusión cuantitativa para elucidar la relajación atómica y la difusión de corto alcance de los elementos durante la MT en acero. El modelo de transformaciones de desplazamiento y difusión incluye los efectos de la relajación en borde de grano para la nucleación heterogenea y la evolución espacio-temporal de potenciales de difusión y movilidades químicas mediante el acoplamiento de herramientas de cálculo y bases de datos termo-cinéticos de tipo CALPHAD. El modelo se aplica para estudiar la evolución microstructural de aceros al carbono policristalinos procesados por enfriamiento y partición (Q&P) en 2D. La microstructura y la composición obtenida mediante la simulación se comparan con los datos experimentales disponibles. Los resultados muestran el importante papel jugado por las diferencias en movilidad de difusión entre la fase austenita y martensita en la distibución de carbono en las aceros. Finalmente, un modelo multi-campo es propuesto mediante la incorporación del modelo de dislocación en grano-grueso al modelo desarrollado de Landau para incluir las diferencias morfológicas entre aceros y aleaciones con memoria de forma con la misma ruptura de simetría. La nucleación de dislocaciones, la formación de la martensita ’butterfly’, y la redistribución del carbono después del revenido son bien representadas en las simulaciones 2D del estudio de la evolución de la microstructura en aceros representativos. Con dicha simulación demostramos que incluyendo las dislocaciones obtenemos para dichos aceros, una buena comparación frente a los datos experimentales de la morfología de los bordes de macla, la existencia de austenita retenida dentro de la martensita, etc. Por tanto, basado en un modelo integral y en el desarrollo de códigos durante esta tesis, se ha creado una herramienta de modelización multiescala y multi-campo. Dicha herramienta acopla la termodinámica y la mecánica del continuo en la macroescala con la cinética de difusión y los modelos de campo de fase/Landau en la mesoescala, y también incluye los principios de la cristalografía y de la dinámica de red cristalina en la microescala. ABSTRACT Martensitic transformation (MT), in a narrow sense, is defined as the change of the crystal structure to form a coherent phase, or multi-variant domain structures out from a parent phase with the same composition, by small shuffles or co-operative movements of atoms. Over the past century, MTs have been discovered in different materials from steels to shape memory alloys, ceramics, and smart materials. They lead to remarkable properties such as high strength, shape memory/superelasticity effects or ferroic functionalities including piezoelectricity, electro- and magneto-striction, etc. Various theories/models have been developed, in synergy with development of solid state physics, to understand why MT can generate these rich microstructures and give rise to intriguing properties. Among the well-established theories, the Phenomenological Theory of Martensitic Crystallography (PTMC) is able to predict the habit plane and the orientation relationship between austenite and martensite. The re-interpretation of the PTMC theory within a continuum mechanics framework (CM-PTMC) explains the formation of the multivariant domain structures, while the Landau theory with inertial dynamics unravels the physical origins of precursors and other dynamic behaviors. The crystal lattice dynamics unveils the acoustic softening of the lattice strain waves leading to the weak first-order displacive transformation, etc. Though differing in statics or dynamics due to their origins in different branches of physics (e.g. continuum mechanics or crystal lattice dynamics), these theories should be inherently connected with each other and show certain elements in common within a unified perspective of physics. However, the physical connections and distinctions among the theories/models have not been addressed yet, although they are critical to further improving the models of MTs and to develop integrated models for more complex displacivediffusive coupled transformations. Therefore, this thesis started with two objectives. The first one was to reveal the physical connections and distinctions among the models of MT by means of detailed theoretical analyses and numerical simulations. The second objective was to expand the Landau model to be able to study MTs in polycrystals, in the case of displacive-diffusive coupled transformations, and in the presence of the dislocations. Starting with a comprehensive review, the physical kernels of the current models of MTs are presented. Their ability to predict MTs is clarified by means of theoretical analyses and simulations of the microstructure evolution of cubic-to-tetragonal and cubic-to-trigonal MTs in 3D. This analysis reveals that the Landau model with irreducible representation of the transformed strain is equivalent to the CM-PTMC theory and microelasticity model to predict the static features during MTs but provides better interpretation of the dynamic behaviors. However, the applications of the Landau model in structural materials are limited due its the complexity. Thus, the first result of this thesis is the development of a nonlinear Landau model with irreducible representation of strains and the inertial dynamics for polycrystals. The simulation demonstrates that the updated model is physically consistent with the CM-PTMC in statics, and also permits a prediction of a classical ’C shaped’ phase diagram of martensitic nucleation modes activated by the combination of quenching temperature and applied stress conditions interplaying with Landau transformation energy. Next, the Landau model of MT is further integrated with a quantitative diffusional transformation model to elucidate atomic relaxation and short range diffusion of elements during the MT in steel. The model for displacive-diffusive transformations includes the effects of grain boundary relaxation for heterogeneous nucleation and the spatio-temporal evolution of diffusion potentials and chemical mobility by means of coupling with a CALPHAD-type thermo-kinetic calculation engine and database. The model is applied to study for the microstructure evolution of polycrystalline carbon steels processed by the Quenching and Partitioning (Q&P) process in 2D. The simulated mixed microstructure and composition distribution are compared with available experimental data. The results show that the important role played by the differences in diffusion mobility between austenite and martensite to the partitioning in carbon steels. Finally, a multi-field model is proposed by incorporating the coarse-grained dislocation model to the developed Landau model to account for the morphological difference between steels and shape memory alloys with same symmetry breaking. The dislocation nucleation, the formation of the ’butterfly’ martensite, and the redistribution of carbon after tempering are well represented in the 2D simulations for the microstructure evolution of the representative steels. With the simulation, we demonstrate that the dislocations account for the experimental observation of rough twin boundaries, retained austenite within martensite, etc. in steels. Thus, based on the integrated model and the in-house codes developed in thesis, a preliminary multi-field, multiscale modeling tool is built up. The new tool couples thermodynamics and continuum mechanics at the macroscale with diffusion kinetics and phase field/Landau model at the mesoscale, and also includes the essentials of crystallography and crystal lattice dynamics at microscale.

Models for stochastic climate prediction

There has been a recent burst of activity in the atmosphere/ocean sciences community in utilizing stable linear Langevin stochastic models for the unresolved degree of freedom in stochastic climate prediction. Here several idealized models for stochastic climate modeling are introduced and analyzed through unambiguous mathematical theory. This analysis demonstrates the potential need for more sophisticated models beyond stable linear Langevin equations. The new phenomena include the emergence of both unstable linear Langevin stochastic models for the climate mean and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances. The strategy for stochastic climate modeling that emerges from this analysis is illustrated on an idealized example involving truncated barotropic flow on a beta-plane with topography and a mean flow. In this example, the effect of the original 57 degrees of freedom is well represented by a theoretically predicted stochastic model with only 3 degrees of freedom.

Intermediate-term earthquake prediction.

An earthquake of magnitude M and linear source dimension L(M) is preceded within a few years by certain patterns of seismicity in the magnitude range down to about (M - 3) in an area of linear dimension about 5L-10L. Prediction algorithms based on such patterns may allow one to predict approximately 80% of strong earthquakes with alarms occupying altogether 20-30% of the time-space considered. An area of alarm can be narrowed down to 2L-3L when observations include lower magnitudes, down to about (M - 4). In spite of their limited accuracy, such predictions open a possibility to prevent considerable damage. The following findings may provide for further development of prediction methods: (i) long-range correlations in fault system dynamics and accordingly large size of the areas over which different observed fields could be averaged and analyzed jointly, (ii) specific symptoms of an approaching strong earthquake, (iii) the partial similarity of these symptoms worldwide, (iv) the fact that some of them are not Earth specific: we probably encountered in seismicity the symptoms of instability common for a wide class of nonlinear systems.