Ideal turbulence and bifurcations in infinite-dimensional dynamical systems

Abstract

Many effects of the real turbulence can be observed in infinite-dimensional dynamical systems given by certain classes of boundary value problems for linear partial differential equations. The possibility of using one-dimensional maps under investigation of such infinite-dimensional systems allows to understand the mathematical mechanisms of development of complex structures in the solutions of these boundary value problems. We describe the bifurcations in infinite-dimensional systems resulting from the bifurcations in the corresponding one-dimensional maps, namely, the period-doubling bifurcations and the tangent bifurcations, the period-adding bifurcations and the bifurcations subordinate to Farey's rule, and also universal phenomena connected with these bifurcations.