New scenario of the transition to chaos in degenerated systems

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During the term of the present grant a systematic study of the soft-mode turbulence (SMT) and related topics has been carried out. The following results have been obtained. In case of electroconvection in homeotropically aligned nematic layer SMT is caused by slow random long-wavelength modulations of a roll pattern. The temporal autocorrelation function for components of the order parameter is calculated and expressed in terms of probability density for random drift velocity of the pattern. It is shown that despite the problem has at least two different characteristic times associated with the slow pattern dynamics, only one of them enters into the autocorrelation function.The simplest nonlinear equation exhibiting SMT is the so-called Nikolaevskii equation. We employ numerical integration of this model to obtain detailed quantitative description of SMT. It is shown that SMT is characterized by a smooth interplay of different spatial scales, with defect generation being unimportant. T … More he Lyapunov exponents are calculated for several system sizes for fixed values of the control parameter ε. The Lyapunov dimension and the Kolmogorov-Sinai entropy are calculated and both shown to exhibit extensive and microextensive scaling. The distribution functional is shown to satisfy Gaussian statistics at small wavenumbers and small frequency. It is shown that if such a system undergoes instability against spatially periodic perturbations with a finite wavenumber, interplay of short-wavelength modes associated with the instability and long-wavelengths modes generated by the symmetry transformation affects the dynamics of the system dramatically. In particular, it may result in direct transition from a spatially uniform state to SMT, analogous to second order phase transition in equilibrium systems. Deep connection between SMT and the structure of the the dispersion equation for the relevant stability problem is revealed. A general phenomenological theory of hydrodynamic waves in regions with smooth loss of convexity of isentropes is developed. The theory is based upon the fact that for most media these regions in the p-V plane are anomalously small. The corresponding generalized Burgers equation is derived and analyzed. The exact solution of the equation for steady shock waves of rare faction is obtained and discussed.The dynamics of actual market prices is a very specific example of dynamical chaos. We apply our knowledge in the theory of dynamical chaos to quantitative analysis of this type of chaos. A number of particular examples (exchange rates USD vs. JPY, XAU vs. USD, oil, etc.) is considered in detail. It allows predicting the price dynamics in future with high accuracy. The main advantage of the approach is that the prediction error does not increase in the course of time. Less