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Estimating climate response to observed and projected increases in atmospheric greenhouse gases usually requires averaging among multiple independent simulations of computationally expensive global climate models to filter out the spontaneous internal climate variability. Studies have shown that advanced pattern recognition methods allow one to obtain accurate estimates of the forced climate signal from just a handful of such climate realizations. The accuracy of these methods for a fixed ensemble size, however, decreases with an increasing magnitude of the low-frequency, decadal and longer internal climate variability. Here we generalize a previously developed Bayesian methodology of Linear Dynamical Mode (LDM) decomposition for spatially extended time series to enable joint identification and analysis of forced signal and internal variability in ensembles of climate simulations, a methodology dubbed here an ensemble LDM, or ELDM. The new ELDM method is shown to outperform its pattern-recognition competitors by more accurately isolating the forced signal in small ensembles of both toy- and state-of-the-art climate-model simulations. It is able to do so by explicitly recognizing a non-random structure of the internal variability, identified by the ELDM algorithm alongside the optimal forced-signal estimate, which allows one to study possible dynamical connections between the two types of variability. The optimal ELDM filtering provides a unique opportunity for objective intercomparison of decadal and longer climate variability across different global climate models --- a task that proved difficult due to uncertainties associated with the noisy character and limited length of historical climate simulations combined with parameter uncertainties of alternative signal-detection methods.

In this work the interval stability concept is used for study the stability of the Earth’s climate system during last 2.6 million years. For the study we used an empirical model of the Earth’s climate in the Pleistocene [1] which was constructed on the basis of LR04 stack data. By means of this model, we analysed dependence of the Pleistocene climate interval stability on time, and compare the results with the behavior of local linear stability. We find that the global climate resilience to any perturbations decreases throughout the Pleistocene and the nonlinearity plays significant role in this phenomenon. The work was carried out within the framework of the Program for the Development of the Regional Scientifc and Educational Mathematical Center "Mathematics of Future Technologies", project #075-02-2020-1483/1.1. Mukhin, D., Gavrilov, A., Loskutov, E. et al. Bayesian Data Analysis for Revealing Causes of the Middle Pleistocene Transition. Sci Rep 9, 7328 (2019). https://doi.org/10.1038/s41598-019-43867-3

In this work, we demonstrate the performance of a new data-driven method [1] for modeling cross-interacting processes with different time scales represented by time series with different sampling steps. The model is a generalization of a nonlinear stochastic model of evolution operator based on neural networks, designed for the case of time series with constant sampling step, and has complex structure. First, it describes each process by its own stochastic evolution operator with its own time step. Second, it considers possible nonlinear connections within each pair of processes in both directions. These connections are parameterized asymmetrically, depending on which process is faster and which process is slower. They make this model essentially different from the set of independent stochastic models constructed individually for each time scale. All evolution operators and connections are trained and optimized using Bayesian framework, forming a multi-scale stochastic model. We demonstrate the capabilities of the model using two climate data examples. The first example is a climate spatio-temporal data generated by a global climate model running in the middle 19th century external conditions. A second example is spatially distributed reanalysis data describing current climate evolution. In both examples we investigate first how correctly the multiscale model reconstructs the individual evolution of both subsystems, and second whether it reveals the connection between processes that is not captured by models built individually for each process.The work was supported by the Russian Science Foundation (Grant No. 20-62-46056).[1] Gavrilov et al. (2021). Data-driven stochastic model for cross-interacting processes with different time scales, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-4199.

Natural and social systems exhibit complex behavior reflecting their rich dynamics, whose governing laws are not fully known. This study develops a unified data-driven approach to estimate predictability of such systems when several independent realizations of the system’s evolution are available. If the underlying dynamics are quasi-linear, the signal associated with the variable external factors, or forcings, can be estimated as the ensemble mean; this estimation can be optimized by filtering out the part of the variability with a low ensemble-mean-signal-to-residual-noise ratio. The dynamics of the residual internal variability is then encapsulated in an optimal, in a Bayesian sense, linear stochastic model able to predict the observed behavior. This model’s self-forecast covariance matrices define a basis of patterns (directions) associated with the maximum forecast skill. Projecting the observed evolution onto these patterns produces the corresponding component time series. These ideas are illustrated by applying the proposed analysis technique to (1) ensemble data of regional sea-surface temperature evolution in the tropical Pacific generated by a state-of-the-art climate model and (2) consumer-spending records across multiple regions of the Russian Federation. These examples map out a range of possible solutions—from a solution characterized by a low-dimensional forced signal and a rich spectrum of predictable internal modes (1)—to the one in which the forced signal is extremely complex, but the number of predictable internal modes is limited (2). In each case, the proposed decompositions offer clues into the underlying dynamical processes, underscoring the usefulness of the proposed framework.

We propose a new approach to reconstructing complex, spatially distributed systems on the basis of the time series generated by such systems. It allows one to combine two basic steps of such a reconstruction, namely, the choice of a set of phase variables of the system using the observed time series and the development of the evolution operator acting in the chosen phase space with the help of an artificial neural network with special topology. This network, first, maps the initial high-dimensional data onto the lower-dimension space and, second, specifies the evolution operator in this space. The efficiency of this approach is demonstrated by an example of reconstructing the Lorenz system representing a high-dimensional model of atmospheric dynamics.

Currently, causes of the middle Pleistocene transition (MPT) – the onset of large-amplitude glacial variability with 100 kyr time scale instead of regular 41 kyr cycles before – are a challenging puzzle in Paleoclimatology. Here we show how a Bayesian data analysis based on machine learning approaches can help to reveal the main mechanisms underlying the Pleistocene variability, which most likely explain proxy records and can be used for testing existing theories. We construct a Bayesian data-driven model from benthic δ 18O records (LR04 stack) accounting for the main factors which may potentially impact climate of the Pleistocene: internal climate dynamics, gradual trends, variations of insolation, and millennial variability. In contrast to some theories, we uncover that under long-term trends in climate, the strong glacial cycles have appeared due to internal nonlinear oscillations induced by millennial noise. We find that while the orbital Milankovitch forcing does not matter for the MPT onset, the obliquity oscillation phase-locks the climate cycles through the meridional gradient of insolation.

We have found a set of correlation effects, which are due to the inherent dynamics of the spectral density of polarization of an active medium with strong inhomogeneous broadening of the working transition line and occur if the rate of incoherent relaxation of optical dipole oscillations of the active centers is lower than the rate of optical-field attenuation in the laser cavity. Our analysis is based on the numerical studies of the stationary superradiant laser generation during continuous pumping with self-locking of some of the quasistationary modes. For the purposes of studying the detected effects, the methods of comparative analysis of the dynamic spectra of polarization and the field have been developed. In these methods, the time-frequency and space-time empirical modes of the spectra are used, which are determined by orthogonal eigenfunctions of special correlation matrices. The interconnection of the superradiance phenomena and mode self-locking in the considered class of lasers with low-Q cavities is discussed.

An analysis of the climate system is usually complicated by its very high dimensionality and its nonlinearity which impedes spatial and time scale separation. An even more difficult problem is to obtain separate estimates of the climate system’s response to external forcing (e.g. anthropogenic emissions of greenhouse gases and aerosols) and the contribution of the climate system’s internal variability into recent climate trends. Identification of spatiotemporal climatic patterns representing forced signals and internal variability in global climate models (GCMs) would make it possible to characterize these patterns in the observed data and to analyze dynamical relationships between these two types of climate variability.In contrast with real climate observations, many GCMs are able to provide ensembles of many climate realizations under the same external forcing, with relatively independent initial conditions (e.g. LENS [1], MPI-GE [2], CMIP ensembles of 20th century climate). In this report, a recently developed method of empirical spatio-temporal data decomposition into linear dynamical modes (LDMs) [3] based on Bayesian approach, is modified to address the problem of self-consistent separation of the climate system internal variability modes and the forced response signals in such ensembles. The LDM method provides the time series of principal components and corresponding spatial patterns; in application to an ensemble of realizations, it determines both time series of the internal variability modes of current realization and the time series of forced response (defined as signal shared by all realizations). The advantage of LDMs is the ability to take into account the time scales of the system evolution better than some other linear techniques, e.g. traditional empirical orthogonal function decomposition. Furthermore, the modified ensemble LDM (E-LDM) method is designed to determine the optimal number of principal components and to distinguish their time scales for both internal variability modes and forced response signals.The technique and results of applying LDM method to different GCM ensemble realizations will be presented and discussed. This research was supported by the Russian Science Foundation (Grant No. 18-12-00231).[1] Kay, J. E., Deser, C., Phillips, A., Mai, A., Hannay, C., Strand, G., Arblaster, J., Bates, S., Danabasoglu, G., Edwards, J., Holland, M. Kushner, P., Lamarque, J.-F., Lawrence, D., Lindsay, K., Middleton, A., Munoz, E., Neale, R., Oleson, K., Polvani, L., and M. Vertenstein (2015), The Community Earth System Model (CESM) Large Ensemble Project: A Community Resource for Studying Climate Change in the Presence of Internal Climate Variability, Bulletin of the American Meteorological Society, doi: 10.1175/BAMS-D-13-00255.1, 96, 1333-1349 [2] Maher, N., Milinski, S., Suarez-Gutierrez, L., Botzet, M., Dobrynin, M., Kornblueh, L., Kröger, J., Takano, Y., Ghosh, R., Hedemann, C., Li, C., Li, H., Manzini, E., Notz, N., Putrasahan, D., Boysen, L., Claussen, M., Ilyina, T., Olonscheck, D., Raddatz, T., Stevens, B. and Marotzke, J. (2019). The Max Planck Institute Grand Ensemble: Enabling the Exploration of Climate System Variability. Journal of Advances in Modeling Earth Systems, 11, 1-21. https://doi.org/10.1029/2019MS001639[3] Gavrilov, A., Kravtsov, S., Mukhin, D. (2020). Analysis of 20th century surface air temperature using linear dynamical modes. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(12), 123110. https://doi.org/10.1063/5.0028246

The global climate system is an aggregate of a huge number of interacting components, each having an intrinsic time scale. Such a complex dynamical system demonstrates nontrivial behavior and can exhibit a variety of possible modes of evolution. Gradual change of the parameters of the global climate system can lead to transitions (e.g., the Mid-Pleistocene Transition or to abrupt climate changes) from the observed to a new mode.In this work, we investigate the stability of the global climate system against strong sudden perturbations in the last 2.5 million years. This case is fundamentally different from the small perturbations case: in particularly, the system response cannot be described by a linearized evolution operator. To estimate the climate system’s nonlinear stability during the last 2.5 million years, we use a nonlinear data-driven model of climate dynamics in Pleistocene [1] and basin stability criterion [2]. Our results indicate that the stabilityof the Pleistocene climate to large perturbations decreases with time: past climates being much more stable compared to the present one.This work was supported by RFBR grant 19-02-00502.1. D. Mukhin, A. Gavrilov, E. Loskutov, J. Kurths, A. Feigin. “Bayesian Data Analysis for Revealing Causes of the Middle Pleistocene Transition”. ScientificReports, 9 7328 (2019).2. V. Klinshov, S. Kirillov, J. Kurths, V. Nekorkin. “Interval stability for complex systems”. New Journal of Physics, v. 20, p. 043040.

According to recent study [1], the current state-of-the-art climate models lack the substantial part of internal multidecadal climate signal which is observed in the 20th century surface air temperature reanalysis data as a global stadium wave (GSW). In the presented work we further investigate this phenomenon using the recently developed method [2] of empirical spatio-temporal data decomposition into linear dynamical modes (LDMs). The important property of LDMs is their ability to take into account the time scales of the system evolution (they are extracted from observed dataset by the Bayesian optimization technique) better than some other linear techniques, e.g. traditional empirical orthogonal function decomposition. Like any linear decomposition, it provides the time series of principal components and corresponding spatial patterns.We modify the initially developed LDM decomposition to make it possible to take into account a prescribed external forcing (like CO2 emissions, sun activity etc.) and then find part of variability which may be considered as an internal climate dynamics decomposed into set of modes with different time scales, and hence may be helpful in GSW interpretation. The results of applying the method to the 20th century surface air temperature with different ways of forcing inclusion will be presented and discussed.1. Kravtsov, S., Grimm, C., & Gu, S. (2018). Global-scale multidecadal variability missing in state-of-the-art climate models. Npj Climate and Atmospheric Science, 1(1), 34. https://doi.org/10.1038/s41612-018-0044-62. Gavrilov, A., Seleznev, A., Mukhin, D., Loskutov, E., Feigin, A., & Kurths, J. (2018). Linear dynamical modes as new variables for data-driven ENSO forecast. Climate Dynamics. https://doi.org/10.1007/s00382-018-4255-7

We investigate the Middle Pleistocene Transition (MPT) - a rapid change in the periodicity of the Pleistocene glacial cycles from 41 kyr to about 100 kyr, which occurred about a million years ago - using the data-driven model [1]. Here we estimate stability of the model using a novel concept of interval stability [2-4], referring to the behavior of the perturbed model during a finite time interval. In a few words we define the class of 'safe' perturbations after which the system (our data-driven model) returns back to the initial dynamical regime and 'unsafe' perturbation of minimal amplitude needed to disrupt the system.We demonstrate that the MPT is likely associated with decreasing of the climate system's interval stability to rapid disturbances (millennial and shorter). This confirms the statement made in the paper [1] that the main factor in the onset of the long-period glacial cycles is strongly nonlinear oscillations induced by the short-scale variability.D. Mukhin, A. Gavrilov, E. Loskutov, J. Kurths, A. Feigin. Bayesian Data Analysis for Revealing Causes of the Middle Pleistocene Transition. Scientific Reports, 9 7328 (2019). P. Menck, J. Heitzig, N. Marwan, J. Kurths. How basin stability complements the linear-stability paradigm. Nature Phys, 9 89–92 (2013). V. Klinshov, V. Nekorkin, J. Kurths. Stability threshold approach for complex dynamical systems. New Journal Physics, 18 013004 (2016). V. Klinshov, S. Kirillov, J. Kurths, V. Nekorkin. Interval stability for complex systems. New Journal Physics, 18 013004 (2018).

We suggest a new method for construction of data-driven dynamical models from observed multidimensional time series. The method is based on a recurrent neural network (RNN) with specific structure, which allows for the joint reconstruction of both a low-dimensional embedding for dynamical components in the data and an operator describing the low-dimensional evolution of the system. The key link of the method is a Bayesian optimization of both model structure and the hypothesis about the data generating law, which is needed for constructing the cost function for model learning. The form of the model we propose allows us to construct a stochastic dynamical system of moderate dimension that copies dynamical properties of the original high-dimensional system. An advantage of the proposed method is the data-adaptive properties of the RNN model: it is based on the adjustable nonlinear elements and has easily scalable structure. The combination of the RNN with the Bayesian optimization procedure efficiently provides the model with statistically significant nonlinearity and dimension.The method developed for the model optimization aims to detect the long-term connections between system’s states – the memory of the system: the cost-function used for model learning is constructed taking into account this factor. In particular, in the case of absence of interaction between the dynamical component and noise, the method provides unbiased reconstruction of the hidden deterministic system. In the opposite case when the noise has strong impact on the dynamics, the method yield a model in the form of a nonlinear stochastic map determining the Markovian process with memory. Bayesian approach used for selecting both the optimal model’s structure and the appropriate cost function allows to obtain the statistically significant inferences about the dynamical signal in data as well as its interaction with the noise components.Data driven model derived from the relatively short time series of the QG3 model – the high dimensional nonlinear system producing chaotic behavior – is shown be able to serve as a good simulator for the QG3 LFV components. The statistically significant recurrent states of the QG3 model, i.e. the well-known teleconnections in NH, are all reproduced by the model obtained. Moreover, statistics of the residence times of the model near these states is very close to the corresponding statistics of the original QG3 model. These results demonstrate that the method can be useful in modeling the variability of the real atmosphere.The work was supported by the Russian Science Foundation (Grant No. 19-42-04121).

A new data-driven model for analysis and prediction of spatially distributed time series is proposed. The model is based on a linear dynamical mode (LDM) decomposition of the observed data which is derived from a recently developed nonlinear dimensionality reduction approach. The key point of this approach is its ability to take into account simple dynamical properties of the observed system by means of revealing the system’s dominant time scales. The LDMs are used as new variables for empirical construction of a nonlinear stochastic evolution operator. The method is applied to the sea surface temperature anomaly field in the tropical belt where the El Nino Southern Oscillation (ENSO) is the main mode of variability. The advantage of LDMs versus traditionally used empirical orthogonal function decomposition is demonstrated for this data. Specifically, it is shown that the new model has a competitive ENSO forecast skill in comparison with the other existing ENSO models.

A nonlinear decomposition method is applied to the analysis of global sea surface temperature (SST) time series in different epochs related to the Pacific Decadal Oscillation (PDO) since the end of 19th century to present time. This method allows one to extract an optimal (small) number of global nonlinear teleconnection patterns associated with distinct dominant time scales from the original high-dimensional spatially extended data set. In particular, it enables us to reveal ENSO teleconnection patterns corresponding to different PDO cycles during the last 145 years, to uncover four climate shifts connected with PDO phase changes and to reconstruct the corresponding global PDO patterns. We find that SST teleconnections between the ENSO region, extra-tropical Pacific regions and the Indian ocean became fundamentally nonlinear since the second half of 20th century.

We suggest a new method for building data-driven dynamical models from observed multidimensional time series. The method is based on a recurrent neural network with specific structure, which allows for the joint reconstruction of both a low-dimensional embedding for dynamical components in the data and an evolution operator. The key link of the method is a Bayesian optimization of both model structure and the hypothesis about the data generating law, which is needed for constructing the cost function for model learning. First, the performance of the method is successfully tested in the situation when a signal from a low-dimensional dynamical system is hidden in noisy multidimensional observations. Second, the method is used for building the data-driven model of the low frequency variability (LFV) in the quasigeostrophic model of the Earth's midlatitude atmosphere-a high-dimensional chaotic system. It is demonstrated that the key regimes of the atmospheric LFV are reproduced correctly in data simulations by means of the obtained model.

The author investigated the gravitational and the magnetically fields of the Elkon ore unit. The gold-uranium deposits of the Elkon ore district this is unique of his scale and matching with biggest ore units of world. The study of the regularities of forms, location and direction of extension of tectonic ore-controlling structures in magnetic and gravitational fields, the study of their transformation using Surfer 8, Koscad 3D and ArcGIS. As a result, the author obtained get two-dimensional autocorrelation functions, local and regional components of geophysical fields, and their gradients. The most informative transformations of fields are regional component of the magnetically field and local component of the gravitational field by results statistical analysis to study the tectonic structure like Elkon ore unit.

The present paper is the second part of a two-part study on empirical modeling and prediction of climate variability. This paper deals with spatially distributed data, as opposed to the univariate data of Part I. The choice of a basis for effective data compression becomes of the essence. In many applications, it is the set of spatial empirical orthogonal functions that provides the uncorrelated time series of principal components (PCs) used in the learning set. In this paper, the basis of the learning set is obtained instead by applying multichannel singular-spectrum analysis to climatic time series and using the leading spatiotemporal PCs to construct a reduced stochastic model. The effectiveness of this approach is illustrated by predicting the behavior of the Jin–Neelin–Ghil (JNG) hybrid seasonally forced coupled ocean–atmosphere model of El Niño–Southern Oscillation. The JNG model produces spatially distributed and weakly nonstationary time series to which the model reduction and prediction methodology is applied. Critical transitions in the hybrid periodically forced coupled model are successfully predicted on time scales that are substantially longer than the duration of the learning sample.

The article discusses petrography and features of the chemical composition of the Mesozoic alkaline intrusions of the Khokhoi ore field located within the Upper Amga gold-mining district in the Aldan Shield. This gold-mining district is located in the Aldan-Stanovoy Province rich in gold. Tectonic position and magmatic events of this Province are close to those positions and events of Shandong Gold Province of Sino-Korean Craton. The investigation of the genesis of the magmatic rocks and comparison of the Provinces can shed light on the genesis and regularity of location of the gold deposits in them. Intrusions of the Upper Amga gold-mining district passed through the metamorphic rocks of the Precambrian and intruded into the Cambrian carbonate rocks and the Jurassic sandstones. Petrographic observations found that large intrusive massifs of the ore field consist mainly of potassic feldspar and plagioclase. Mineral composition, structure and texture of the rocks correspond to syenite and monzonite. Blocks of melanocratic rocks – presumably tephrites are rare on the surface. Several generations of feldspar and dark-colored minerals are observed within all these rocks, indicating melt fractionation. High Sr and K contents were identified in the rocks. Contents decrease in the series of tephrite-syenite-monzonite. This feature, judging by other petrochemical data, can be associated not only with magmatic differentiation, but also with the melting and interaction of the lithosphere. In sienites, ovoids of quartz and the first-generation potassic feldspar melting margin were found. It could be possibly resulted from the capture of surrounding rock material and changes in the thermodynamic conditions of crystallization. It is found out that potassium and silicon contents in rocks depend inversely on calcium content. Interaction of melts and the host carbonate rocks with removal of silicon and potassium and formation of gold-bearing sericite-quartz and feldspar-quartz metasomatites is assumed. The loss of potassium probably has led to potassium metasomatism at the contacts of intrusions. This metasomatism produced the gold ores.

The article considers features of geochemical composition of the Mesozoic igneous rocks of the Khokhoi ore field, spatially located in the Upper-Amga gold-mining district of the north-western part of the Aldan-Stanovoy shield, at the junction of two large structural units - the Olekma granite-greenstone region and the Aldan granulite-gneiss area, in the zone of meridional deep Amga fault. Magmatic formations are represented by the Northern Boskho, Western Boskho and Upper Khokhoi massifs of the Lebedinsky monzonite-syenite-granite complex, intruding rocks of the crystalline basement and terrigenous-carbonate complex of the Siberian platform cover, and by rare tephrite dikes. Due to the accessibility of the study area, little attention was focused on the mentioned igneous rocks until recently. The study showed that magmatic formations of the ore field are geochemically specialized in lithophile and siderophile elements, and they belong to derivatives of trachy-andesite and andesi-basalt magma. All studied rocks have sufficiently close spectra of rare-earth element distribution, in particular, the rocks are poor in heavy rare-earth elements relative to light rare-earth elements. Presence of low negative Eu-anomaly for the Northern Boskho and Upper Boskho massifs may indicate the fact that they were formed from residual differentiates during fraction crystallization of rock-forming minerals with the involvement of plagioclases. Low Eu/Eu* values also indicate the fact that these rocks were formed during the late stage of the area evolution. Negative niobium anomaly, identified for all igneous rocks of the ore field, indicates magma contamination with continental crust. Considering interpretation of rare-earth elements and their ratio, it was supposed that igneous rocks of the ore field were formed in intracontinental conditions. It was generally concluded that there was an intensive mantle-crust interaction throughout the magmatism evolution of the Khokhoi ore field, beginning from intrusion of tephrite dikes bodies to intrusion of syenite rocks. The presence of long-lived mantle source, as well as contamination with crust material, was responsible for heat and fluid inflow that supplied circulation of thermal spring in the system. Heat and fluids caused additional mobilization of ore material from surrounding rocks, with further formation of gold-sulfide mineralization close to tephrite dikes and the Northern Boskho massif.

We suggest a new nonlinear expansion of space-distributed observational time series. The expansion allows constructing principal nonlinear manifolds holding essential part of observed variability. It yields low-dimensional hidden time series interpreted as internal modes driving observed multivariate dynamics as well as their mapping to a geographic grid. Bayesian optimality is used for selecting relevant structure of nonlinear transformation, including both the number of principal modes and degree of nonlinearity. Furthermore, the optimal characteristic time scale of the reconstructed modes is also found. The technique is applied to monthly sea surface temperature (SST) time series having a duration of 33 years and covering the globe. Three dominant nonlinear modes were extracted from the time series: the first efficiently separates the annual cycle, the second is responsible for ENSO variability and combinations of the second and the third modes explain substantial parts of Pacific and Atlantic dynamics. A relation of the obtained modes to decadal natural climate variability including current hiatus in global warming is exhibited and discussed.

A new empirical approach is proposed for predicting critical transitions in the climate system based on a time series alone. This approach relies on nonlinear stochastic modeling of the system’s time-dependent evolution operator by the analysis of observed behavior. Empirical models that take the form of a discrete random dynamical system are constructed using artificial neural networks; these models include state-dependent stochastic components. To demonstrate the usefulness of such models in predicting critical climate transitions, they are applied here to time series generated by a number of delay-differential equation (DDE) models of sea surface temperature anomalies. These DDE models take into account the main conceptual elements responsible for the El Niño–Southern Oscillation phenomenon. The DDE models used here have been modified to include slow trends in the control parameters in such a way that critical transitions occur beyond the learning interval in the time series. Numerical results suggest that the empirical models proposed herein are able to forecast sequences of critical transitions that manifest themselves in future abrupt changes of the climate system’s statistics.

An approach to the long-term prognosis of qualitative behavior of a dynamic system (DS) is proposed, which is based on the nonlinear-dynamical analysis of a weakly nonstationary chaotic time series (TS). A method for constructing prognostic models using the observed evolution of a single dynamic variable is described, which employs the proposed approach for prediction of bifurcations of low-dimensional DSs. The method is applied to analyze the TS generated by the Roessler system and the system of equations modeling photochemical processes in the mesosphere. The analysis is performed for a TS calculated in the case of a slow variation in the control parameter of the system. The duration of the “observed” TS is limited such that the system demonstrates only one, chaotic, type of behavior without any bifurcations during the observed TS. The proposed algorithm allows us to predict correctly the bifurcation sequences for both systems at times much longer than the duration of the observed TS, to point out the expected instants of specific bifurcation transitions and accuracy of determining these instants, as well as to calculate the probabilities to observe the predicted regimes of the system's behavior at the time of interest.

We present a detailed description of a new approach for the extraction of principal nonlinear dynamical modes (NDMs) from high-dimensional data. The method of NDMs allows the joint reconstruction of hidden scalar time series underlying the observational variability together with a transformation mapping these time series to the physical space. Special Bayesian prior restrictions on the solution properties provide an efficient recognition of spatial patterns evolving in time and characterized by clearly separated time scales. In particular, we focus on adaptive properties of the NDMs and demonstrate for model examples of different complexities that, depending on the data properties, the obtained NDMs may have either substantially nonlinear or linear structures. It is shown that even linear NDMs give us more information about the internal system dynamics than the traditional empirical orthogonal function decomposition. The performance of the method is demonstrated on two examples. First, this approach is successfully tested on a low-dimensional problem to decode a chaotic signal from nonlinearly entangled time series with noise. Then, it is applied to the analysis of 250-year preindustrial control run of the INMCM4.0 global climate model. There, a set of principal modes of different nonlinearities is found capturing the internal model variability on the time scales from annual to multidecadal.

An approach of the empirical modes with a variable spatial-temporal structure is proposed and developed for the analysis of non-stationary nonlinear dynamics of the multimode superradiant lasers with a low-Q cavity and a strong inhomogeneous broadening of lasing transition in an active medium. It is shown that the approach makes it possible to analyze a number of complicated dynamical phenomena in an ensemble of the strongly interacting centers which constitute the active medium and are exposed to CW pumping.

In this work we formulate a consistent Bayesian approach to modeling stochastic (random) dynamical systems by time series and implement it by means of artificial neural networks. The feasibility of this approach for both creating models adequately reproducing the observed stationary regime of system evolution, and predicting changes in qualitative behavior of a weakly nonautonomous stochastic system, is demonstrated on model examples. In particular, a successful prognosis of stochastic system behavior as compared to the observed one is illustrated on model examples, including discrete maps disturbed by non-Gaussian and nonuniform noise and a flow system with Langevin force.

An alternative approach to determining embedding dimension when reconstructing dynamic systems from a noisy time series is proposed. The available techniques of determining embedding dimension (the false nearest-neighbor method, calculation of the correlation integral, and others) are known [H. D. I. Abarbanel, Analysis of Observed Chaotic Data (Springer-Verlag, New York, 1997)] to be inefficient, even at a low noise level. The proposed approach is based on constructing a global model in the form of an artificial neural network. The required amount of neurons and the embedding dimension are chosen so that the description length should be minimal. The considered approach is shown to be appreciably less sensitive to the level and origin of noise, which makes it also a useful tool for determining embedding dimension when constructing stochastic models.

Some recent papers were concerned with applicability of the Bayesian (statistical) approach to reconstruction of dynamic systems (DS) from experimental data. A significant merit of the approach is its universality. But, being correct in terms of meeting conditions of the underlying theorem, the Bayesian approach to reconstruction of DS is hard to realize in the most interesting case of noisy chaotic time series (TS). In this work we consider a modification of the Bayesian approach that can be used for reconstruction of DS from noisy TS. We demonstrate efficiency of the modified approach for solution of two types of problems: (1) finding values of parameters of a known DS by noisy TS; (2) classification of modes of behavior of such a DS by short TS with pronounced noise.