The article considers dynamic processes involving non-linear power-law behavior in such apparently diverse spheres, as demographic dynamics and dynamics of prices of highly liquid commodities such as oil and gold. All the respective variables exhibit features of explosive growth containing precursors indicating approaching phase transitions/catastrophes/crises. The *rst part of the article analyzes mathematical models of demographic dynamics that describe various scenarios of demographic development in the post-phase-transition period, including a model that takes the limitedness of the Earth carrying capacity into account. This model points to a critical point in the early 2050s, when the world population, after reaching its maximum value may decrease afterward stabilizing then at a certain stationary level. The article presents an analysis of the inﬂuence of the demographic transition (directly connected with the hyperexponential growth of the world population) on the global socioeconomic and geopolitical development. The second part deals with the phenomenon of explosive growth of prices of such highly liquid commodities as oil and gold. It is demonstrated that at present the respective processes could be regarded as precursors of waves of the global *nancial-economic crisis that will demand the change of the current global economic and political system. It is also shown that the moments of the start of the *rst and second waves of the current global crisis could have been forecasted with a model of accelerating log-periodic *uctuations superimposed over a power-law trend with a *nite singularity developed by Didier Sornette and collaborators. With respect to the oil prices, it is shown that it was possible to forecast the 2008 crisis with a precision up to a month already in 2007. The gold price dynamics was used to calculate the possible time of the start of the second wave of the global crisis (July–August 2011); note that this forecast has turned out to be quite correct..
Theoretically possible rogue edge wave are studied over cylindrical bottom in the framework of nonlinear shallow water equations in a weakly nonlinear limit. The nonlinear mechanisms (nonlinear dispersion enhancement, modulation instability and multimodal interactions) of possible anomalous edge wave appearance are analyzed.