Nowadays the Lyapunov exponents and Lyapunov dimension have become so widespread and common that they are often used without references to the rigorous definitions or pioneering works. It may lead to a confusion since there are at least two well-known definitions, which are used in computations: the upper bounds of the exponential growth rate of the norms of linearized system solutions (Lyapunov characteristic exponents, LCEs) and the upper bounds of the exponential growth rate of the singular values of the fundamental matrix of linearized system (Lyapunov exponents, LEs). In this work the relation between Lyapunov exponents and Lyapunov characteristic exponents is discussed. The invariance of Lyapunov exponents for regular and irregular linearizations under the change of coordinates is demonstrated.

Abstract-Microwave model for simulation of radiation from the multilayer system "sea surface-sea icesnow cover-atmosphere" is introduced. In the general case, ice and snow cover is modelled by multilayer medium, where every layer is characterized by its specific physical parameters. Electrodynamical properties of each layer are determined from the original authors' model of the effective permittivity of heterogeneous medium. This model takes into account effects of radiation scattering on irregularities of environment. Measurable physical characteristics of sea ice and snow are used as the model input data. This advantage allows using this model for interpretation of remote sensing images of the ice cover in the Polar Regions. Major attention is drawn to comparison of model calculations with satellite data and visual observations from ships. The collection of SSM/I and SSMIS images from GLOBAL-RT data base, and processed visual observations from ships in Arctic cruises were used. Observations data served as the input parameters for electrodynamical model. Comparison of model results with SSM/I images demonstrated good coincidence at various frequencies.

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An analysis of prevalence and associated common risk factors of ED and lower urinary tract symptoms (LUTS) was performed in Russian Federation by cross-sectional multicenter survey. International Index of Erectile Function (IIEF) score and International Prostate Symptom Score (IPSS) were used for data collection in 1225 men between 20 and 77 years interviewed in six regions of Russian Federation. In addition, each participant's social, demographic, lifestyle, sexual and medical history was taken with special emphasis on risk factors for ED. Upon the basis of IIEF erectile domain score interpretation, ED was found in 530 (48.9%) men, consisting of mild and mild to moderate, moderate and severe ED in 375 (34.6%), 78 (7.2%) and 77 (7.1%) respondents, respectively. According to IPSS assessment, LUTSs were present in 649 (59.9%) responders; inclusive 370 (34.2%), 216 (19.9%) and 63 (5.8%) men with mild, moderate and severe LUTS, respectively. Men with both ED and LUTS shared common co-morbidities and lifestyle risk factors with age-adjusted odds ratio between 1.2 and 5.2. In logistic regression model (R(2)=0.361), the strongest associated with ED factor found was IPSS symptom score, followed by hypertension, IPSS-related quality of life, age, diabetes mellitus, obesity and unmotivated fatigue.

In applied investigations, the invariance of the Lyapunov dimension under a diffeomorphism is often used. However, in the case of irregular linearization, this fact was not strictly considered in the classical works. In the present work, the invariance of the Lyapunov dimension under diffeomorphism is demonstrated in the general case. This fact is used to obtain the analytic exact upper bound of the Lyapunov dimension of an attractor of the Shimizu-Morioka system.

Embryonic motility is an important component of development and may be a precursor of posthatching motor behavior. In chickens, it accompanies almost the entire embryogenesis and depends on environmental conditions, whose effects on embryonic motility have been poorly studied. The effect of acute hypoxia (10% O 2 for 10 min) on the temporal parameters of chick embryo motility on incubation days 10 (D10) and 14 (D14) was estimated; the results were compared with data on the heart rate (HR). A force transducer connected with an embryo limb was used to record embryonic movements simultaneously with HR video recording. In the control, the duration of the activity phase (APh) and HR increased, while the inactivity phase (IPh) decreased in the period from D10 to D14. In response to hypoxia, the APh did not change significantly on either day, but the IPh significantly increased on D14 and tended to increase on D10. A distinct pattern of the motor activity response to hypoxia was observed on D14: the IPh increased after 0.5-2.5 min of hypoxia, peaked at a value 6 times as large as the control one, and then partly recovered. Under hypoxia, the mean HR significantly decreased to 87% of the control value and then partly recovered, increasing to 93% on both days studied. The similarity of the hypoxic patterns of IPh and HR on D14 suggests that the hypoxic recovery of HR contributes to the recovery of the embryo motility under hypoxia in late embryos.

One of the main tasks in the study of financial and economic processes is forecasting and analysis of the dynamics of these processes. Within this task lie important research questions including how to determine the qualitative properties of the dynamics (stable, unstable, deterministic chaotic, and stochastic process) and how best to estimate quantitative indicators: dimension, entropy, and correlation characteristics.These questions can be studied both empirically and theoretically. In the empirical approach, one considers the real data represented by time series, identifies patterns of their dynamics, and then forecasts short-and long-term behavior of the process. The second approach is based on postulating the laws of dynamics for the process, deriving mathematical dynamic models based on these laws, and conducting subsequent analytical investigation of the dynamics generated by the models.To implement these approaches, both numerical and analytical methods can be used. It should be noted that while numerical methods make it possible to study complex models, the possibility of obtaining reliable results using them is significantly limited due to calculations being performed only over finite-time intervals, numerical integration errors, and the unbounded space of possible initial data sets. In turn, analytical methods allow researchers to overcome these problems and to obtain exact qualitative and quantitative characteristics of the process dynamics. However, their effective applications are often limited to low-dimensional models (in the modern scientific literature on this subject, two-dimensional dynamic systems are the most often studied).In this paper, we develop analytical methods for the study of deterministic dynamic systems based on the Lyapunov stability theory and on chaos theory. These methods make it possible not only to obtain analytical stability criteria and to estimate limiting behavior (localization of self-excited and hidden attractors, study of multistability), but also to overcome the difficulties related to implementing reliable numerical analysis of quantitative indicators (such as Lyapunov exponents and Lyapunov dimension). We demonstrate the effectiveness of the proposed methods using the "mid-size firm" model suggested recently by V.I. Shapovalov as an example.

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