Abstract. We present the essential dynmnic model of the •nesospheric photochemical system (PCS) •'md suggest a step-by-step procedure for elaborating such a model of an arbitrary atmospheric PCS. The model demonstrates the same possibilities of nonlinear dynamic behavior and qualitatively the same dynamic characteristics as the corresponding original model, but is much simpler than the latter. We show the adequacy of the essential model compared with the original one in bifurcation diagrams, equilibrimn states, and such new characteristics as correlation dimension and minirotan embedding dimension of a chaotic attractor. The model can be used both for identi•ing and studying the mechanisms of the nonlinear dynmnic behavior of the mesaspheric PCS, as well as for solving a number of problems aimed at revealing nonlinear photochemical phenomena in the actual mesasphere.

Abstract. The photochemical system of the mesopause region is a nonlinear driven oscillator enforced by the diurnal-periodic solar radiation. Under idealized conditions this oscillator can display nonlinear effects such as period doubling cascades or chaos. We investigate what happens if this system is subjected to atmospheric diffusion. A highresolution one-dimensional chemical system of the mesopause region has been established in order to answer this question. Strong diffusion destroys nonlinear effects, but for the lower values of the diffusion coefficient which are still of the order of magnitude of real values, different nonlinear effects occur. The most important effect consisted of the creation of a 2-day subharmonic oscillation. Such a 2-day oscillation of the concentration of chemical active species entails a corresponding oscillation of the chemical heating rates which feeds back to the dynamics of this region. The zonal wind influences the period of the oscillation so that the periods differ from the exact 48-hour value by a few hours depending on the wind velocity and its direction. We call this phenomenon the photochemical Doppler effect. IntroductionThe mesopause region is one of the most interesting and most important regions of the upper atmosphere. Above this region, within the thermosphere, the variations caused by the alteration of the extraterrestrial solar shortwave spectrum (EUV and X ray) and particle precipitation dominate. Below this layer the meteorological influences predominantly deter-

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, (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.

An approach to prognosis of qualitative behavior of an unknown dynamical system (DS) from weakly nonstationary chaotic time series (TS) containing significant measurement noise is proposed. The approach is based on construction of a global time-dependent parametrized model of discrete evolution operator (EO) capable of reproducing nonstationary dynamics of a reconstructed DS. A universal model in the form of artificial neural network (ANN) with certain prior limitations is used for the approximation of the EO in the reconstructed phase space. Probabilistic prognosis of the system behavior is performed using Monte Carlo Markov chain (MCMC) analysis of the posterior Bayesian distribution of the model parameters. The classification of qualitatively different regimes is supposed to be dictated by the application, i.e., it is assumed that some classifier function is predefined that maps a point of a model parameter space to a finite set of different behavior types. The ability of the approach to provide prognosis for times comparable to the observation time interval is demonstrated. Some restrictions as well as possible advances of the proposed approach are discussed.

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.

The impossibility to use the MCMC (Markov chain Monte Carlo) methods for long noisy chaotic time series (TS) (due to high computational complexity) is a serious limitation for reconstruction of dynamical systems (DSs). In particular, it does not allow one to use the universal Bayesian approach for reconstruction of a DS in the most interesting case of the unknown evolution operator of the system. We propose a technique that makes it possible to use the MCMC methods for Bayesian reconstruction of a DS from noisy chaotic TS of arbitrary long duration.

Analysis of numerical solutions for a system of two van der Pol-Duffing oscillators with nonlinear coupling showed that there exist chaotic switchings (occurring at irregular time intervals) between two oscillatory regimes differing by nearly time-constant phase shifts between the coupled subsystems. The analysis includes the investigation of bifurcations of the periodic motions corresponding to synchronization of two subsystems, finding stability regions of synchronization regimes and scenarios of the transitions to chaos.

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