Stochastic self-oscillations and turbulence

Until quite recently, it was thought that turbulence, i.e., stochastic self-oscillations of a continuous medium, was related exclusively to the excitation of an exceedingly large number of degrees of freedom. This review is devoted to the discussion of new ideas on the nature of the unpredictable turbulent motions in dissipative media connected with the discovery of strange attractors, i.e., attractive regions in phase space within which all paths are unstable and behave in a very complex fashion (motions on an attractor of this kind are characterized by a continuous spectrum). Turbulence represented by a strange attractor is described by a finite number of degrees of freedom, i.e., modes whose physical nature may be different. The example of a simple electronic noise generator is used to illustrate how the instability (divergence) of such paths leads to stochastic behavior. The analysis is based on the introduction of a nonreciprocally single-valued Poincare mapping onto itself, which is then used to describe the strange attractors encountered in different physical problems. An example of this mapping is used to demonstrate the discrete, symbolic, description of dynamic systems. The properties of such systems which indicate their stochastic nature, for example, positive topologic entropy and hyperbolicity are discussed. Specific physical mechanisms leading to the appearance of stochastic behavior and characterized by a continuous time spectrum are discussed. Strange attractors that appear in the case of parametric instability of waves in plasmas, laser locking by an external field, and so on, are demonstrated. "Attractor models" of hydrodynamic turbulence are reviewed, and in particular, finite-dimensional hydrodynamic models of convection in a layer and of Couette flow between rotating cylinders are constructed and found to exhibit stochastic behavior.